Warrantless Self-evidence: Anti-foundationalist Stance On Logic

WARRANTLESS SELF-EVIDENCE: ANTI – FOUNDATIONALIST STANCE ON LOGIC

A Thesis Presented to The Faculty of the Division of Philosophy Department of Humanities College of Arts and Sciences University of the Philippines Los Baños

In Partial Fulfillment of the Requirements for the Degree of

BACHELOR OF ARTS IN PHILOSOPHY

by

MARIA DIORY FAJARDO RABAJANTE

April 2007

2

The Author

…was born four days before the first anniversary of the First People Power revolution in the Philippines, in a town where Jose Rizal's mother was detained, in a province famous for Buko pie, kesong puti and uraro; is not the youngest nor the eldest among one subtracted to four children of the seventh child among twelve of her parents and of the third child among five of his parents; was educated in a Catholic school since age four until age 16; was admitted to a statesubsidized university that is supposed to be a training ground for the future leaders of the country to join the crowd of bluffers, brainstormers and philosophylovers and…

and…

and…

(a history and an infinity of possibilities).

3

Preface If it was Hume who made Kant awakened from the latter's dogmatic slumbers, it was being bumped on a wall that awakened me from whatever sort of slumber I had while I was walking. I cannot claim that the slumber I had is a dogmatic one, because I had such slumber while on a journey. I had a confidence with the ideas that I was then upholding because I was confident with the path where I was walking. But being bumped made me ask why I was bumped. Was it because I was having a slumber while walking? Or because I was mistaken with my manner of walking? Or was it simply because there is a big wall where I was walking? Is the blame to be rightfully given to the ideas that I had upheld, the approach that I had taken or the loophole in the ideas and/or theories that I had observed? I guess I have to blame the three for making me now walk on a different path and have a new slumber (again) while walking. I should not be blamed for sleeping while walking (I literally am not). I would rather sleep or strongly uphold ideas while walking or going on with my life. The slumber I have is determined by my manner of walking; the ideas that I uphold are determined by my approach towards different things. Bumping on a wall is indicative of the fact that I already have to open my eyes and decide whether the wall is too large that it already means I have reached a dead end. If this means that I have to take a new route, then I would. But when I walk on my new route, I would again take a slumber. Such is the rule. I cannot walk without sleeping. I cannot just go on without upholding beliefs that may serve as burden in knowing whether there is a wall in the road. I have to take the challenge of walking while sleeping. What follows is a story of a new journey that I have - a walking while sleeping in a new route. Along my old journeys, a lot had influenced me with the approaches that I was taking and had helped me recognize the wall/s in the paths that I was taking. It is of my pleasure to send them my gratitude. Though he does not know me, I would like to thank Robert Pirsig for allowing me to ride in his motorcycle. His "Zen and the Art of Motorcycle Maintenance" is a great motorcycle ride that made me challenge the way I was taking my old journey. Though I do not assent to much of his claims, he has influenced my act of recognizing the wall in my old route. I also thank Mr. John Ian Boongaling for challenging my views. He made us write a paper about the question "can logical principles be falsified by experience?" for our PHLO 150 (Epistemology) class. His comments on my paper made me want to write an enquiry on logic. This thesis is a product of such want. He also served as the critic of this thesis. Pirsig's book and my short essay on logic for PHLO 150 class are the two initial influences in my writing of this thesis. Much gratitude is also given to my adviser, Prof. Arlyn Pinpin-Macapinlac for patiently scrutinizing my work and for giving my thesis a chance of being scrutinized, from the content to the technical matters, despite her strict schedule. Her motherly aura inspires us, her advisees, to work harder (and think harder). I also thank Mr. Robert Bass of the Coastal Carolina University for answering my questions. He is the only person in the online community to whom I have sent an e-mail that answered my questions. Prof.

4

Efren Alverio had made comments on the initial draft of my introduction. I thank him for giving comments even if he does not know me personally. Of course, I have to express gratitude to Ernesto Macaloi for spending conversations with me regarding logic. He has also referred my thesis to Prof. Alverio for the latter's comments. Our conversations had influenced my claims a lot. I also had conversations with Mickey Nieto and Jouh Pabalate regarding this thesis. Their comments are very much considered in my creation of arguments. The philosophy community in UPLB serves as my inspiration for this thesis. I thank Ms. Aleli Caraan, Ms. Donna Nuguid and Mr. Nicolo Masakayan for giving my brain a hard time whenever we are given with paper assignments. I also thank Dr. Nicolito Gianan and Ms. Omega Otinero for being nice to me even though I have never been their student. I thank them for their continued service to the university despite the problem that the philosophy program is facing. I thank the UPLB Sophia Circle for its continued act of promoting philosophy in a science-oriented university. I thank these particular people: Sandra Panganiban, Amabel Raflores, Andrew Estialbo, Joanna Ebora, Megan Cruz, Clare Vega, Chinky Cuevas, Katrina Esguerra, Kezia Evangelista, Joanne Aquino, Sairah Saipudin, Rey Araja and Shiela Abucay. They made my last year in the university colorful. And of course, I cannot not thank Florence de Guzman, Karl Christian Abalos and Franco Azucena. They are my closest friends in the university. Every conversation with them is a philosophical discussion that sometimes translates to an exchange of corny jokes. Conversations with Florence whenever we are on a jeepney ride from Los Baños to Sta. Cruz always give me new insights for my thesis. We share the same view on "institutionalization." KC and I also had an exchange of ideas regarding our theses. We actually theorized that Florence's work is a common ground among our theses. Franco never failed to remind me of how great I am. Thesis writing was made easier because the four of us are helping each other. Finally, I thank my family. I thank my mom, Rosemarie, for understanding that I was so busy with this thesis (which excuses me from household chores). My dad, Jojo, for telling me to rest even if I did not want to (because I'd rather continue working for this thesis). Kuya Jomar had read parts of this thesis and made some comments. Jaja for cheering me up after receiving stresses from this thesis. To the Grand Perceiver, Unmoved Mover, Uncaused Cause, Allah, Yahweh, God, Love (because it is said "God is Love") and to whatever else one might call Him or Her, this is an offering. Logic is often times used as a tool in reasoning. We are taught of its certainty. But before we use logic and look at it the way we are taught about it, it is proper to examine it and the issues underlying it. We do not have to continue this old slumber if it would only make us bump into a wall. What you will read is a journey. Come, walk with me. Happy trip!

Maria Diory Fajardo Rabajante

5

"What makes his world so hard to see clearly is not its strangeness but its usualness. Familiarity can blind you too." - Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance

6

The thesis entitled WARRANTLESS SELF-EVIDENCE: ANTIFOUNDATIONALIST STANCE ON LOGIC is presented and submitted by Maria Diory Fajardo Rabajante under the supervision of Prof. Arlyn V. Pinpin-Macapinlac.

ABSTRACT The focal point of investigation of this thesis is the Problem of First Principles of Logic or the issue on why such principles, if there really are such principles, need not be proven. The conclusion that was drawn led to refutations of Foundationalism and the appeal to self-evidence in logic. The refutations were supported by a defense of two claims - that there is no warrant to argue for the self-evidence of first principles of logic because there are no such principles and that there is no warrant to argue for the selfevidence of anything in logic because there is nothing in logic that could be taken as necessary. The first claim established the untenability of Foundationalism in logic by arguing for the triviality of an appeal to axiomatization in logic. Foundationalism, which is an appeal to axiomatization, argues for a linear justification structure in logic. This thesis had, however, shown that the justification structure of logic could be described as a mutual support system where there is no axiom that could serve as the sole starting-point of derivation and sole end-point of justification. The non-existence of axioms means that there are no axioms that are self-evident. The second claim furthered the refutation on self-evidence by claiming for the impossibility of necessary truthfulness or necessity of logic and logical principles. This necessity or truthfulness in all possible worlds is the essential condition in order for a

7

logical principle to serve as a proof of its own and be self-evident. Such construal on the necessity of logic was proven untenable by arguing for the dependence of logic on the human mind. Logical principles are possible only in a world where there is a rational mind that is able to recognize and understand the relations among objects in the world. Logic, though is about the world and/or reality, is not anchored on reality or on what reality necessarily is but on how the community of rational minds interprets and understands reality.

8

INTRODUCTION

"Foundationalism does not hold true in logic, and its appeal to self-evidence is warrantless."

One of the most interesting, and perhaps also among the most difficult, problems in philosophy is the Problem of First Principles or the issue on why first principles need not be proven. One who wishes to answer the question should also be involved on the issue on whether there really are first principles. We can look at a first principle in two ways as an end-point of justification and as a starting-point of derivation. As an end-point, a first principle serves as the last and ultimate justification or proof of principles, if we wish to avoid an infinite regress of justifications. As a starting-point, a first principle serves as an axiom - a principle not derived from any other principle but serves as the ultimate principle from which other principles are derived. An essential condition for a principle to be genuinely counted as a first principle is the property of being self-evident. Ascribing the property of being self-evident to propositions, particularly to the fundamental principles of logic is tantamount to giving them, not only the claim to certainty, but also the claims to infallibility and incontrovertibility. This is, however, indicative of the fact, or of the idea if not a fact, that we have reached the limits of justification. The terms certain, infallible and incontrovertible are often used to describe the self-evident propositions in making an account of their truthfulness in the absence of proof. In their (accidental or unaccidental) avoidance of skepticism and infinite regress, self-evident propositions put a stop in every why's or every question for further justifications, which legitimizes their being foundational propositions of certain philosophical edifices.

9

An appeal to self-evidence does not necessarily imply an appeal to Foundationalism. Where self-evidence merely accounts for the being evident of a proposition in its own right, Foundationalism accounts for the derivation of propositions or propositional knowledge from the self-evident basic or first principles. Suppose, for example, it is argued that the uttered proposition "A triangle has three sides" is self-evident. In arguing for the self-evidence of such proposition, it is not necessarily claimed that other propositions or knowledge claims must be derived from this proposition, though such may be the case. Hence, in enquiring on the "appeal to self-evidence of foundationalist edifices," we are not only taking into consideration the "being true in the absence of proof" of the proposition in question, but also its axiomatic feature as a considered foundational or basic proposition. It is the case, however, that Foundationalism necessarily assumes self-evidence, for it is the self-evident proposition that Foundationalism considers as the foundation of a philosophical edifice. In this enquiry, the philosophical edifice that is in question is the formal system of logic. Foundationalism is a theory of justification in epistemology. It is premised from the notion that there are first principles that serve as the foundation of knowledge. Foundationalism, however, is not only limited to being a theory in epistemology. There are also foundationalist arguments regarding other fields in philosophy, particularly in the field of logic. Just like other foundationalist claims on knowledge, philosophers, which dates back from Aristotle, followed by Gottlob Frege and later on by Roderick Chisholm, claimed that there are first or basic principles in logic that are self-evident from which other principles of logic are derived. This foundationalist stance on logic is similar to

10

Euclid's axiomatization of geometry. Euclidean geometry presupposes postulates or axioms wherein other principles of such geometry are derived. We could see from here the aspect of a first principle as the starting-point of derivation. In terms of justification of principles or propositions, a foundationalist philosopher argues that the justification cannot go on an infinite regress. If we want to be warranted in our belief or in believing a particular proposition or principle (let's say P1), we ask for its justification, proof, or grounding. We would then be given another proposition or principle (P2). And again, in order to be warranted in holding P2, we would ask for its justification, let's say P3 and so on. A foundationalist argues that this could not go on an infinite regress, which he thinks is vicious. He argues that there must be a point in the whole process of justification where we have to stop. Such point would be the first principle, with an aspect of being the end-point of justification. The two features of a self-evident proposition - being an end-point of justification and being a starting-point of derivation, are only two ways by which we could look at a selfevident proposition. They may be necessary features of a self-evident first principle, but not a necessary feature of all that could be taken as self-evident. Foundationalism argues for a formal system of logic as an axiomatic system where the axioms or the first principles are said to be self-evident. In addition to the selfevidence of the first principles, foundationalists argue for the self-evidence of rules of inference - the very rules of deriving other principles of logic from the first principles. Appeal to self-evidence in logic, then, applies not only to the first principles but also to the rules of inference. In such a case where there is a distinction between a first principle and a rule of inference, it could be accounted that the features of being an end-point of

11

justification and starting-point of derivation or simply, the property of being noninferentially justified are present only in a first principle. Though a chapter would be devoted to the discussion of self-evidence and its very definition, the author would like to make an initial disambiguation of it and an initial exposition of the sort of self-evidence that will be examined so as to make clear the scope of this enquiry. Take note that the examination of first principles of other foundationalist edifices (as it is in a foundationalist edifice that there are said to be self-evident first principles) is not within the scope of this enquiry. The focus is on the realm of logic only. First, self-evidence is previously stated as "being true in the absence of proof." What this means is that a self-evident thing is not in need of proof. Such a self-evident thing does not depend on any other thing in order to be true. It carries its own evidence. This sort of evidence is what will be examined in this enquiry. We must enquire on how the evidence of a first principle and a rule of inference differ from the evidence of other principles of logic. If, in natural language, or in the usual application of, the term "evident," we mean "obvious" and/or "easily noticeable," then an evident thing must be obvious and easily noticeable to someone (a subject). Such is why self-evidence is usually regarded as being unavoidable for any subject (any rational mind) to believe in the truthfulness of something (proposition, principle, rule of inference or anything that is worthy of being called self-evident) that is self-evident, once such self-evident thing is understood. It is commonly argued that the being evident-in-itself (the sense of being self-evident) produces the property of being evident-to-us (obviousness). In the latter part of this enquiry, however, it will be shown that self-evidence and obviousness are not the same, but for now, an analysis of the usual application of the term "evident" will help us.

12

An evident thing must always make itself present, obvious and easily noticeable to a subject, whether or not this subject notices this evident thing. Hence, this enquiry would largely deal with our epistemic relation to the so-called first principles of logic and to the rules of inference. Second, since self-evidence is being true in the absence of proof, it could be argued that self-evidence is also applicable to other systems, not only to the system of logic. In this light, there must be an initial distinction between the realm of logic (the conceptual) and the realm of the material (the perceptual). Traditionally, logic is taken as that which is necessary and independent of experience. As E. C. Ewing puts it, "laws of logic must be known a priori or not at all."1 In this enquiry, the sort of self-evidence that is in question is that which is attributed to logic. If we are to put it in Gottfried Leibniz terms, there are two sorts of self-evidence, the evidence of the primary truths of fact, and that of the primary truths of reason. The primary truths of fact are those that are referred to when speaking of the material realm (the realm of those that are knowable a posteriori), while the primary truths of reason are those that are referred to when speaking of logic, and are the focus of this enquiry. The self-evidence of truths of fact has, certainly, different form and implications, and hence must not be confused with the object of our enquiry. Though the main focus of this enquiry is the Problem of First Principles of Logic or the selfevidence of the first principles or primary truths, the issue of the self-evidence of the rules of inference will also be touched. It is by virtue of these that other principles of logic are derived from the basic ones.

1

A. C. Ewing, "In Defense of A Priori Knowledge," in The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman (Canada: Wadsworth, Thomson Learning Inc., 2003), 386.

13

Third, it must be noted that self-evidence is not synonymous to a prioricity, analyticity, or necessity. They are closely connected to each other but it would be misleading to consider them as synonymous to each other. A priori and its antithesis, a posteriori are simply about knowledge or about how we acquire knowledge. A prioricity and a posterioricity, then, are epistemological categories. Analyticity, on the other hand, is merely about tautology. It, and its antithesis, syntheticity, are semantical categories, or on how the predicate of a proposition stands in relation to the subject of such proposition. On the other side, necessity and contingency refer to how a proposition is or might be true. They are metaphysical categories. But self-evidence is never an epistemological, semantical or even a metaphysical category. It is a category of justification or on how something is or might be justified. Lastly, to speak of the first principles of logic as not self-evident is not tantamount to saying that they are not true. It simply means that the evidence for their truthfulness is not a sort of self-evidence. In this enquiry, it would be argued that nothing in logic could be accounted as self-evident, or a proof of its own, because nothing in logic can be true independently of a human mind. Logic is simply about how our minds are predisposed to think or to reason and not about norms of thought. The first principle is also called first proposition, basic principle, basic proposition, primary truth, postulate or axiom. Which word or phrase to use is a matter of choice; and on that account, there must be no confusion whenever they are used interchangeably for they do mean the same thing. The phrase "first principle" was derived from Aristotle; "basic principle" was from Tyler Burge's writings about, and translations of, Frege's works; "primary truth" was from Leibniz; while the terms "postulate" and "axiom" were

14

from mathematics and geometry. All of them are used in foundationalist theories, are referring to the principle that serves as both the starting-point of derivation and end-point of justification and are said to be self-evident. A self-evident thing, which carries its own evidence, must be necessary. This means that it must be true, not only in this actual world, but also in all possible worlds. Necessity, then, is presupposed in an appeal to self-evidence in logic. But it is not a mere presupposition, just as mere premise for a conclusion. It is the essential condition for something to be worthy of the definition of 'self-evident' in logic. Without it, principles of (as well as rules of inference in) logic could not satisfy the very definition of selfevidence. Hence, if the non-existence of necessity is proven, then logic could not claim an appeal to self-evidence. The triviality in necessity would mean that self-evidence in logic is untenable. Necessity of logic is premised from the very idea that the logical principles are "out there." It is their "being out there" or their independence of any human mind that they are said to be true in all possible worlds, which is discoverable through rational reflection or a priori insight. A negation of necessity, therefore, would largely affect the notion that the truthfulness of logic is independent of human mind. If we are to follow modus tollens (a rule of inference), without necessarily claiming it as self-evident, the negation of necessity must lead to a negation of the "being out there" of the truthfulness of logic or of any logical principles and rules. This enquiry works on a claim that there is no warrant to claim for the self-evidence of the first principles of logic. As proving such would mean an exploration of the essential constitution and epistemic status of logic, we would be led to examining also the

15

so-called self-evidence of the rules of inference. Hence, the second claim, which is quite similar to the first one, which the author would like to prove, is that there is no warrant to claim for the self-evidence of anything in logic. To prove the claim would mean a refutation of necessity in logic. Its refutation, however, would mean a refutation of the "being out there" of logical principles or rules. This enquiry, then, would also have to explore the implications of the two claims. The method of proving the warrantlessness of an appeal to self-evidence in logic is outlined as follows: [i] Deny the existence of synthetic a priori propositions, thereby reducing all propositions of logic to analytic; [ii] Deny that a prioricity implies necessity; [iii] Prove that all propositions of logic, which are analytic, cannot be taken as necessary even though a priori; [iv] Prove (iii) by claiming that the so-called primary truths of logic are mere predispositions of the mind, and hence cannot be thought of as true when there is no mind; and [v] Defend the implication of (iv) that the truths of logic are not "out there" or not independent of human mind against any possible criticism. This enquiry could also amount to an examination of the general nature of logic, by examining (as aforesaid) our epistemic relation with the basic principles. The notion of logic as norms of thought is a corollary of the "being out there" of logic. Logic is generally thought of as normative; that it is "how we ought to think," because the truths of logic are already out there - necessary and are discoverable a priori. Defenders of such

16

notion reject the idea that logic is essentially descriptive; they reject that logic is simply "how we think," because to claim such would reduce logic to mere psychological laws of thought. Chapter 1 of this enquiry would be an exposition of the traditional conception of the essential constitution and epistemic and metaphysical status of logic. Chapter 2 is devoted to the disambiguation of the term 'self-evident' and an exposition on what it is in logic that are said to be self-evident. This chapter would discuss not only the appeal to self-evidence but also the appeal to Foundationalism in the system of logic. Chapter 3 is intended to show the triviality in the concept of self-evidence if would be applied in logic. This would then deal with the refutation of necessity in logic. And finally, on Chapter 4, an alternative view on what logic is, as a corollary of the accounts on Chapter 3, will be discussed and defended. For a more active and less technical discourse, let the author, in the succeeding parts of this enquiry, use a first person perspective. Let the term "I" be used instead of "the author."

17

CHAPTER 1 LOGIC AND THE TRADITIONAL CONCEPTION OF ITS EPISTEMIC AND METAPHYSICAL STATUS

The very definition of logic is largely a matter of contention. Nevertheless, we can remove vagueness from it by stating that it is about "reasoning." It is commonly asserted that the sense of this "reasoning" is connected to an account on normativity - that logic is how we ought to reason, as opposed to the view that logic is simply a description of how we reason. Logic is a realm of the a priori. This is how logic would most likely be viewed if we are to preserve the a priori / a posteriori or the truths of reason / truths of fact dichotomies. It is said to be independent of experience in the sense of not owing validity from an empirical verification - a sort of verification necessary for a posteriori truths or truths of fact. In this sense, logical principles have a sort of irrefutability - a sort of necessary truthfulness. The traditional notion on logic is that epistemically, it is a priori, and metaphysically, it is necessary.

§ 1.1 The signification of the "a priori" The Latin terms "a priori" and "a posteriori" primarily refer to how a thing or a proposition is or might be known. Literally, a priori is referred to as "from what is prior" to distinguish from "from what is posterior," the literal sense of a posteriori. The distinction between the two coincides with the non-empirical knowledge / empirical

18

knowledge distinction and Leibniz's truths of reason / truths of fact distinction. The empirical knowledge and truths of fact are the a posteriori while the non-empirical knowledge or the truths of reason are the a priori. The terms "a priori" and "a posteriori" are used to signify ways of knowing a proposition or argument or to indicate how a concept is acquired. An a priori argument is that which is knowable a priori, while an a posteriori argument is that which is knowable a posteriori or a combination of a priori and a posteriori premises. To say that a thing is knowable a priori is to say that it is knowable independently of sense experience, while to say that something is knowable a posteriori is to say that it is knowable dependently of sense experience. Consequently, a priori concept is a concept acquired independently of sense experience and whose justification arises from pure thought or reason. This does not, however, necessarily mean that it is innate, though there are theories such as Plato's and Leibniz's that take a priori knowledge to be innate. Traditional examples of the a priori are "All bachelors are unmarried," "5 + 2 = 7" and "If all men are mortal and Socrates is a man, then Socrates is mortal." In contrast, a posteriori concept is obtained from experience, which includes the perceptual, memorial, and beliefs in scientific claims. Examples of a posteriori propositions are "The cat is on the mat," "It is raining" and "The color of my blouse is red." We can see from the examples that a posteriori propositions are capable of being falsified by simply referring to experience, while a priori propositions are not. The validity of an a priori proposition, if there really is an a priori proposition, cannot be drawn from experience.

19

§ 1.2 Intuitive Induction A priori knowledge is said to be acquired through pure thought or rational reflection. This process is, as Chisholm puts it, "Intuitive Induction."2 It is different from the ordinary "induction" in the sense that the ordinary induction is only a sort of generalization. We take particular events, then generalize from them. The truthfulness of the generalization or the conclusion is only derived from the particular instances. Intuitive induction, on the other hand, takes particular instances as mere occasion of knowing the things that are a priori. They are only incidental to an a priori conclusion and are our means of knowing the necessity of such conclusion. For example, we may know from the perception of particular things what it is for a thing to be red and not to be red, and what it is for a thing to be green and not to be green. Having such knowledge, we are now able, through rational reflection, to infer that necessarily, "being red excludes being green or being not-red."

§ 1.3 Logical Truths as Necessary Truths As earlier said, the necessity / contingency dichotomy is metaphysical. The dichotomy refers to the truth-value of a given proposition. A necessary proposition is one that is true in all possible worlds, while a contingent proposition is one that is true only in the actual world. Leibniz spoke of his truths of fact (e.g. I am conscious now) as contingent.3 It is because they are true only in this world. The truths of reason or the truths of logic (e.g. A = A), on the other hand, are necessary. They cannot be "not true" in 2

Roderick Chisholm, "Truths of Reason," in The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman (Canada: Wadsworth, Thomson Learning Inc., 2003), 411. 3

Gottfried Wilhelm Leibniz, New Essays on Human Understanding, ed. and trans., Peter Remnant and Jonathan Bennett (Cambridge: Cambridge University Press, 1981), 362.

20

any possible world. Defenders of the necessity of logic would, in fact, argue that logical truths are true even in the world where nothing exists. Though the necessary / contingent distinction is metaphysical and the a prioricity / a posterioricity dichotomy is epistemological, they are closely related. It may be thought that if a given proposition is necessary then it must be knowable a priori because sense experience can only tell us about the actual world or what is the case and not about what is necessarily the case or what always ought to be the case. This leads us to the traditional claim that logic is about rules or norms on how we ought to reason since it talks about what is necessarily true - a truthfulness that is not dependent on the actual world or on any experience of it.

§ 1.4 Normativity in Logic If we say that logical truths are necessary, that they are laws of truth and/or of reality, and that how we ought to reason must be parallel to these logical truths, then we cannot decide on or ascribe truthfulness or falsity to logic. Even if it is argued that an a priori proposition is innate (an issue, which I am agnostic about), it still does not mean that the truthfulness of such proposition is decided or ascribed by me or by any rational mind. It just means that the knowledge of it (and of its necessity) is innately known. Without necessarily making an account on whether a necessary a priori proposition of logic is innate (such that we re-learn or re-discover it) or discovered, it can still be inferred, as it is still compatible with these two different claims, that a necessary a priori proposition of logic is independent of the human mind.

21

If the whole of logic is a priori, and we can observe from it that some of its principles or propositions are derived from others then we will discover that there are logical principles that are (considered as) more epistemically privileged than others. These principles that have epistemic privilege status are the basic (first) principles or axioms that have epistemic privilege by virtue of being self-evident. An a priori system, then, is grounded on self-evident a priori principles. In addition to the self-evidence of axioms of logic, the rules of inference or the rules by which we derive logical principles from axioms are also said to be self-evident. These axioms and rules of inference are already there, independent of our minds, and can be immediately known. Hence, even if they are immediately known to us, we cannot be mistaken about them because they speak of what necessarily is the case. Their incontrovertibility is reason enough to make them as the basic presuppositions of how we ought to think or reason.

§ 1.5 Analytic and Synthetic A Priori Another dichotomy that is quite related to the issue of a prioricity and a posterioricity is the analytic / synthetic dichotomy. The dichotomy is semantical, and is about how the predicate of a proposition stands in relation to the subject of such proposition. The key in understanding the dichotomy is the notion of "conceptual containment." In an analytic proposition, the predicate is contained in the subject. Such proposition has a form of an explicit redundancy, whose denial would lead to a self-contradiction. Hence, the principle that governs all analytic judgments is the principle of noncontradiction. Kant accounted that even when the concepts in an analytic proposition are empirical, all analytic propositions are a priori. The concepts in analytic propositions are

22

just analyzed.4 Examples of analytic propositions are "[A1] All bachelors are unmarried males," "[A2] All that are pregnant are women," and "[A3] Everything that is a square is a rectangle." A synthetic proposition, on the other hand, is a proposition whose predicate is not contained in the subject but is adding something to it. Examples of such are "[S1] Los Baños is an hour drive away from Sta. Cruz," "[S2] The Prince of Korea is gorgeous" and "[S3] Everything that is a square is a thing that has a shape." We can see that being an hour drive away from Sta. Cruz is not contained in (or is not the very definition of) the subject, which is Los Baños, but it does tell something about it. The same is true with the second example; being gorgeous only adds something to the Prince of Korea, but it is not contained in it. The two examples are a posteriori propositions because they are knowable through sense experience. The third example (S3), which is one of Chisholm's, is quite difficult to consider as a synthetic proposition. One may argue that its form is quite the same with A3, which is an analytic proposition. Chisholm argued that they (S3 and A3) are of different semantical forms, though he argues that they are both epistemologically a priori - we need not look at experience to know if they are true. S3, therefore, is an example of a synthetic a priori proposition. But how are A3 and S3 or analytic propositions and synthetic a priori propositions semantically different? Let us start with an analysis of A3. It was stated that an analytic proposition has a form of explicit redundancy. This is usually taken to mean that the subject speaks of itself when it speaks of the predicate that it talks about an identity, such that the statement "A bachelor is an unmarried male" 4

Immanuel Kant, "Prolegomena to Any Future Metaphysic, Section VII.1," in "A Priori Knowledge," The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman (Canada: Wadsworth, Thomson Learning Inc., 2003), 372.

23

has the form same as "A is A." But in the case of A3, or even in example A2, the predicate is not actually identical to the subject. Explicit redundancy, then, refers not only to "identity." Chisholm argues that it refers to the "predicate being analyzed out of the subject." This is what Chisholm means with being analyzed out: … a term is analyzed out of a subject-term provided the subject-term is such that either it is itself the predicate-term or it is a conjunction of independent terms one of which is the predicate term. 5 In the case of "All squares are rectangles," the subject-term is said to be analyzed out such that it is a conjunction of logically independent terms, one of which is the predicateterm. To further understand how it happens, consider the proposition below, which is a translation of "Everything that is a square is a rectangle": Translation of A3: Everything that is an equilateral thing and a rectangle is a rectangle. Notice that in the given translation, it was made obvious to us that A3 is analytic. Though rectangle is not identical with square, square is a conjunction of being equilateral and being rectangle. We can also do this translation in understanding the analyticity of A2. We will see that being pregnant is a conjunction of having a child in the womb and being woman. In the case of S3 however, the predicate "a thing that has a shape" is not analyzed out of the subject "everything that is a square." One may argue that "everything that is a square" is a conjunction of something (let us say, Ф) and "everything that has a shape." But in such case, we cannot say that "everything that has a shape" is analyzed out of "everything that is a square" because we do not know yet what Ф stands for, thereby

5

Roderick Chisholm, "Truths of Reason," 418.

24

making us incapable of knowing whether "everything that has a shape" is logically independent of Ф. The phrase that we should substitute to Ф must produce a definition of "everything that is a square" when combined with "everything that has a shape," in order to analyze out "everything that is a square." Suppose we substitute the phrase "everything that is equilateral and a rectangle" to Ф, in order to have a definition of "everything that is a square." The proposition would be: "Everything that is equilateral and a rectangle and everything that has a shape is everything that has a shape." This, however, would produce problems. First, the phrase "everything that has a shape" is not logically independent of, but is actually logically implied by, "everything that is equilateral and a rectangle." Second, "everything that is equilateral and a rectangle" is already a definition of "everything that is a square"; hence, "everything that has a shape" can be considered as something that is only added to, though logically implied by, "everything that is a square." Hence, S3 is a synthetic proposition, but a priori, such that it can be known without referring to experience. Other examples of synthetic a priori are "5 + 2 = 7," "Everything red is colored" and "A thing that is blue all over is not pink." One needs not to refer to experience or sense experience to know them, but they are shown to be not analytic. Having discussed the notion that there are synthetic a priori propositions, it is reasonable for us to expect that advocates of such notion argue that logic is not purely analytic. It may be the case that there are analytic propositions that are presupposed as first or basic principles, but the rules of inference can be seen as synthetic.

25

Chapter 2 FOUNDATIONALISM: APPEAL TO SELF-EVIDENCE IN LOGIC

The account that logic is a norm of thought or of correct reasoning is premised from the very idea that logic speaks as a law of truth or of reality (not of material reality, but of what necessarily is). This approach to logic is a construal of the idea that the system of logic is metaphysically necessary that is knowable a priori. In fact, an appeal to necessity in logic always implies an appeal to a priori insight because mere sense perception can only give us the knowledge of particular things in the actual world. If an idea of innateness of knowledge of the necessity of logic is added to the idea of a priori insight, the knowledge is still described as that which is "discovered" in the sense of being "rediscovered." In this account, whether or not the knowledge of logical principles is innate, if logical principles are said to be necessary, the sense of a priori insight is a matter of discovery (or re-discovery). In such case, logical principles (or logic) are (is) said to be "out there" - independent of human mind. We must recognize this in our analysis of the appeal to Foundationalism in logic. Foundationalism's essential premise, self-evidence, assumes the "being out there" of the said to be self-evident propositions, and hence also assumes the necessity of self-evident propositions and a priori insight about them. The "being out there" of a self-evident proposition or principle would mean that such principle carries its own evidence or justification. It is evident in its own right and owes no justification from other principles. A rational mind is certain of a self-evident principle because such principle is evident in its own right, not because it is evident by virtue of a

26

rational mind certain of it. A self-evident principle needs not a subject to be true, but the subject would find it unavoidable to believe in the truthfulness of a self-evident principle.

§ 2.1 Disambiguation of the term "self-evident" Self-evidence is a technical term, which makes its definition open to some stipulation. However, it carries a function specific to it in that which there are many cases of its application that are easy to recognize. Robert Bass calls self-evidence as "epistemic privilege" and self-evident truths or principles as "epistemically privileged class of truths." With self-evident, he is saying, "no sane or rational person who understands it can deny it."6 Chisholm holds the same, only that he calls the self-evident as "self-presenting" (a term he borrowed from Meinong).7 As we have previously noted, self-evidence can also be applied to truths of fact (or the realm of the a posteriori). In such a manner, let me first distinguish the selfevidence in truths of reason from the self-evidence in truths of fact. The self-presenting (or the self-evident), according to Chisholm, is immediately evident. A thing is evident if such thing is beyond reasonable doubt for a rational mind and if it is epistemically preferable (it is preferable to believe in it than not believe in it). The self-presenting truths or propositions in the truths of fact refer to propositions about someone's (a rational mind's) own state of mind at a time. This includes being in a certain sensory state.8 Bertrand Russell has a congruent idea about this sort of self-evidence by 6

Robert Bass. 9 Dec 2006. E-mail to the author. 10 Dec 2006; Robert Bass. 11 Dec 2006. E-mail to the author. 14 Dec 1006. 7

Roderick Chisholm, "The Problem of Criterion," in The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman (Canada: Wadsworth, Thomson Learning Inc., 2003), 15. 8

27

which we know matters of fact. He actually calls this "perceptual evidence." As an example, he stated: If you slip on a piece of orange peel and hit your head with bump on the pavement, you will have little sympathy with a philosopher who tries to persuade you that it is uncertain whether you are hurt. 9 The self-evident (self-presenting or epistemically privileged truths) in the truths of reason or in logic, on the other hand, is that which has, according to Russell, "conceptual evidence." This conceptual evidence is what Bass refers to when he said that no rational person would deny a (conceptually) self-evident proposition once it is (or the terms in the proposition are) understood. Chisholm stated that a (conceptually) self-evident proposition could not be known unless it is evident, and one could not believe it unless he understands it. We can take Russell's account as an encapsulation of self-evidence: "where it is present, we cannot help believing."10 However, such definition applies only to selfevidence, but not to conceptual self-evidence or self-evidence in the realm of the a priori system of logic, which is the object of our enquiry. For a complete definition of the 'selfevident' in logic, we can take Tyler Burge's analysis of Frege's writings. Tyler Burge outlines the essential components of that which is self-evident. The essential components are as follows: 1. (Rationality) An ideally rational mind would be rational in believing it. 2. (Unprovabilty) This rationality in believing it need not depend for its rationality on inferring it from other truths - or reasoning about its relation to other truths; it derives merely from understanding it. Ibid. 9

Bertrand Russell, Human Knowledge: Its Scope and Limits, 172. 10

Ibid.

28

3. (Unavoidability) Belief in it is unavoidable for an ideally rational mind that fully and deeply understands it. 11 The three capture the essentiality of the 'self-evident'. The first component, which I aptly call rationality,12 signify that it is reasonable for us (or for any rational mind) to believe in a proposition that is self-evident. This is what Bass means when he states that no rational or sane person could deny a self-evident proposition. This is what Chisholm means when he says that self-evident propositions are epistemically preferable. It is the case, therefore, that any system that is grounded on self-evident propositions has a normative character. By appealing to self-evidence, which essentially has this character, logic is able to gain legitimacy in measuring sanity or rationality. Any rational being must believe and accept the truthfulness of a self-evident proposition if he really understands it. It seems, then, that normativity is implied by inability of a rational mind to deny the truthfulness of a proposition. The second component, which I call unprovability, points to the property of being incapable and being not in need of proof. It is what we mean when we say that a selfevident thing is a thing that is "an evidence of its own self." A self-evident principle has a sort of self-justification. It justifies its own self. It carries its own evidence and owes no evidence to other principles. It is its own evidence for its truth and certainty. Such is the case because, as Aristotle said, a self-evident principle is "prior" and "better known." 13 It is by virtue of being unprovable that a basic or first principle can become non11

Tyler Burge, "Frege on knowing the foundation," Gottlob Frege, philosophy of mathematics, internet, available from http://findarticles.com. 12

The term 'rationality', in this sense, is not meant to refer to 'reason' or 'reasoning' but on being 'reasonable'. The use of the term is patterned from how Tyler Burge discusses Frege's accounts on 'selfevidence'. 13

Aristotle, "Posterior Analytics, 71b - 72a," in Thales to Ockham, ed. Walter Kaufmann, (New Jersey, USA: Prentice Hall Inc., 1968), 288 - 289.

29

inferentially justified or can serve as a starting-point of derivation and an end-point of justification. A self-evident principle that justifies its own self does not need any other principle to be justified, but it can justify and infer other principles. Unavoidability points to the property of a self-evident proposition that when it is understood, a rational mind cannot not believe in it. This might seem that self-evidence is immediate obviousness, but as Frege argued, self-evidence is not a sort of obviousness. These three properties all point to the unconditional truthfulness or necessity of logical propositions or principles.

§ 2.2 Self-evidence vs. Obviousness According to Frege, self-evidence is not synonymous with subjective obviousness or immediate psychological certainty. This means that self-evidence is an account of justification of logical truths or principles that is independent of human minds or of psychological considerations. "Self-evidence is an intrinsic property of the basic truths, values, and thoughts expressed by definitions. It is intrinsic in that it is independent of relations to actual individuals."14 Self-evidence is never dependent on a human mind, rather it is evident from itself. To further understand this, I distinguish between being evident-in-itself, the sense of the self-evident, and being evident-to-us, the sense of being obvious to a rational mind. The evident-in-itself would only imply the being evident-to-us if it is fully understood, otherwise, no rational mind could have considered a thing that is evident-in-itself as evident-to-him (or evident-to-us). Defenders of the notion of self-evidence, on the other 14

Frege as cited by Tyler Burge in Tyler Burge, 'Frege on knowing the foundation', Gottlob Frege, philosophy of mathematics.

30

hand, would argue that it is always the case that a thing evident-to-us and is not made evident to us by other things is evident-in-itself. Corollarily, it must not be misconstrued that evidence-to-us of a logical principle points to its self-evidence. As stated in the previous section, Chisholm argued that a selfevident proposition could not be known unless it is evident. This means that it is not by virtue of a human mind that a logical principle is said to be self-evident; rather, it must be capable of serving as evidence of its own self and of its own truthfulness. It must be further noted that when we speak of a logical principle's self-evidence, and hence of its being an evidence of its own self and truthfulness, we are referring to the evidence of both the logical principle itself and its truthfulness. When we speak of selfevidence, we cannot separate the logical principle from its truthfulness. As a corollary of what was discussed, a self-evident logical principle would not become self-evident if it were not true. Hence, it does not make sense to separate the question of a logical principle to the question of its truthfulness when we say that such logical principle is selfevident. The proposition "A thing is itself," for example, when construed as a self-evident logical principle, is the same as saying "It is the case that a thing is itself" or "It is true that a thing is itself." Corollarily, since this enquiry is about self-evidence of logical principles, it is inevitable that we also touch the issue on such principles' truthfulness. Likewise, as Tomas Alvira, Luis Clavell and Tomas Melendo pointed out, a selfevident logical principle is not a "sort of built-in intellectual framework for understanding reality."15 Rather, it is "evident by itself to everyone."16 This means that it is not by virtue 15

Tomas Alvira, Luis Clavell, and Tomas Melendo, Metaphysics, trans. Fr. Luis Sapan, ed. Fr. M. Guzman (Manila: Sinag-tala Publishers, Inc., 1991), 35. 16

Ibid.

31

of our capability to recognize a logical principle that makes it self-evident, but by virtue of its own capability to become evident-in-itself.

§ 2.3 The relevant sense of "Necessity" and "Being Out There" As previously noted, the notion of logic's necessity, as a corollary of logic's "being out there" is an essential condition for self-evidence in logic. In fact, the three essential components of self-evidence in logic (rationality, unprovability and unavoidability) are all pointing to the concept of necessity. Rationality points to being reasonable in believing a self-evident proposition because such proposition is true in all possible worlds. Unprovability points to the being incapable of, and being not in need of proof of a self-evident proposition, as a corollary of truthfulness in all possible worlds, which is independent of any demonstration or proof that may be outlined by any rational mind. Through intuitive induction, it will be realized that a self-evident proposition has a sort of truthfulness independent of other propositions, which means that it is necessary rather than contingent. Lastly, unavoidability points to a rational mind's being incapable of denying a self-evident proposition once it is understood. This means that a self-evident proposition, once understood, cannot be denied because its truthfulness is independent of us or of any psychological considerations or human representations (Logic is "out there"); and hence, it has a necessary truthfulness. It is, thus, by virtue of being necessary and "being out there" that a proposition can be evident-in-itself (or self-evident). Such is because if a proposition is not necessary and not "out there," then its being evident-to-us is only an evidence-to-us and not an evidence-in-itself. A proposition's truthfulness and evidence for truthfulness will, thus, be

32

dependent on us (or on how we are predisposed to think) - an idea that is widely rejected as a corollary of affirming the necessity and "being out there" of logic.

§ 2.4 Foundationalism as Axiomatization in Logic To say that an edifice is an axiomatic system is to say that such edifice is grounded on certain foundations from which other parts of such edifice are based. Axiomatization, therefore, attempts to locate the "axioms" or the foundations of an edifice. The term "axiom" is derived from the Greek term "axioma," which means "that which is considered self-evident."17 The modern usage of the term, however, is not anymore restricted to that which is self-evident. In mathematics, axioms need not be self-evident, but they are principles or propositions from which the theorems of mathematics are derived. At any rate, an axiom is that which yields theorems, whether or not such axiom is self-evident. In such case, appeal to axiomatization may be said to be either foundational (appeal to axiomatization that does not necessarily claim that axioms are self-evident) or foundationalist (appeal to axiomatization that claims that axioms are and must be selfevident). The traditional notion on logic, as previously discussed, is an appeal to a foundationalist axiomatization or simply Foundationalism. It is commonly viewed that the axioms or first principles of logic are self-evident - an idea that is a corollary of appeal to necessity and "being out there" of logic. Foundationalism in logic may be likened to a building that is grounded on strong foundations - foundations that are self-evident. These foundations are what we call first

17

Wikipedia, Axiom, internet, available from http://en.wikipedia.org/wiki/Axiom.

33

principles, basic principles or axioms. Other principles in logic follow a priori from axioms, and hence, it can be argued that not all that are a priori are self-evident. Let me call the principles that are derived from axioms as theorems. It was previously discussed how something in logic might be called as self-evident. A proposition, in order to be basic, must satisfy everything that is essential in a self-evident proposition. Its unprovability points to its being both the starting-point of derivation and end-point of justification. In addition to self-evidence of axioms, rules of inference, the rules that make us infer other principles from the basic ones, are also said to be selfevident. Other philosophers do not distinguish rules of inference from axioms. For the purpose of our enquiry, let me initially assume (and later on, try to refute) that they are different. It is because if they are indeed different, and we have proven that axioms are not self-evident, it may still be argued that the rules of inference are self-evident, and hence, there is still self-evidence in logic. Below is an illustration of the analogy between a building and the system of logic: Another way of explaining axiomatization in logic is by comparing it to a chess game. Let all the possible positions in a chess game be the propositions in logic. The initial position of the chess game would be the axioms, while the rules of the game would be the rules of inference. The rules of the game allow for moving from the initial position. All possible positions that are executed from the initial position and are executed according to the rules are the theorems of logic. We cannot simply move any

34

characters in the chess game (variables in logic) in any way we want; we must follow the rules in order to play the game right. This shows the normative character of logic. As noted in the previous chapter, there are axioms (at least some of the axioms of logic) that are analytic. An example of which is the principle of identity; i.e. A thing is itself (A → A). Another principle that could be taken as an axiom is the principle of noncontradiction, i.e. It is not the case that a thing is the not-itself (~ [A • ~A]). This principle, however, is argued by some philosophers like Louis Pojman as semantically synthetic.18 It is for the reason that the "not-itself" is not contained in the negation of the case of a thing, though it is logically implied by it. Rules of inference, on the other hand, are generally regarded as synthetic a priori. The classical example of a valid argument "All men are mortal and Socrates is a man, therefore Socrates is mortal" is argued as synthetic because at least one of its premises is synthetic. Even if we say that "All men are mortal" is analytic, "Socrates is a man" is synthetic, which actually makes its conclusion synthetic. The rule of inference that this argument follows, hypothetical syllogism, is clearly shown as synthetic.

18

Louis J. Pojman, "A Priori Knowledge," in The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman (Canada: Wadsworth, Thomson Learning Inc., 2003), 368.

35

CHAPTER 3 UNTENABILITY OF SELF-EVIDENCE IN LOGIC

The two claims that I am working on are as follows: (1) there is no warrant to claim for the self-evidence of the first principles of logic, and (2) there is no warrant to claim for the self-evidence of anything in logic. The method of proving the warrantlessness of an appeal to self-evidence in logic is outlined as follows: [i] deny the existence of synthetic a priori propositions, thereby reducing all logical propositions to analytic; [ii] deny that a prioricity implies necessity; [iii] prove that all logical propositions, which are analytic, cannot be taken as necessary even though a priori; [iv] prove (iii) by claiming that the so-called primary truths of logic are mere predispositions of the mind, and hence cannot be thought of as true when there is no mind; and [v] defend the implication of (iv) that the truths of logic are not "out there" or not independent of human mind against any possible criticism.

§ 3.1 Step 1: Deny the existence of synthetic a priori propositions Denying the existence of synthetic a priori propositions, and hence also denying the existence of synthetic a priori logical principles or propositions, would mean that logic, which is a realm of the a priori, is wholly analytic. The idea that all logical principles are analytic would support the idea that there is no axiomatization in logic, thereby automatically making an appeal to Foundationalism wrong. Furthermore, the analysis of analyticity would lead us to inferring that logic is a priori because it is wholly analytic, and not because of its necessity (which is highly questionable).

36

3.1.1 There are no synthetic a priori propositions There may be borderline cases where it seems difficult to distinguish between analyticity and syntheticity, but this does not mean that we have to reject the analytic/synthetic distinction. "Synonymy," "definition" and "analyticity" are not problems if it is clear to us how the terms and/or concepts in a proposition are used. There are cases wherein propositions are clearly analytic and cases wherein propositions are clearly synthetic. The proposition "A is A," for example, is clearly analytic, as it clearly shows an explicit redundancy (where the predicate is also the subject and the subject is also the predicate). "Proxima Centauri is 4.3 light years away from the solar system" is clearly synthetic. Proxima Centauri will still be Proxima Centauri if it is not 4.3 light years away, and even if it is discovered that it is just a light year away. But in the case of "A is A," A (the subject) cannot be thought of as not-A (the predicate). The problem is not with the analytic/synthetic distinction, but with the borderline cases - the synthetic a priori propositions. I call synthetic a priori propositions as the borderline cases because they are those that seem synthetic, and those that seem knowable a priori, which is a property of analytic propositions. As we have previously noted, the key in understanding the distinction is the "conceptual containment." In a synthetic proposition, the predicate is not conceptually contained in the subject. What being not conceptually contained means is that a concept (the subject-term) will still be complete even if we do not attach the predicate-term to it. Proxima Centauri will still be Proxima Centauri if it is discovered that it is not 4.3 light

37

years away from the solar system. Hence, in knowing the truthfulness of a synthetic proposition, we refer to sense experience. If we happen to discover that Proxima Centauri is not 4.3 light years away, the statement "Proxima Centauri is 4.3 light years away from the solar system" will be not true; but Proxima Centauri will still be itself. Such is also the case with the examples given in Chapter 1. Los Baños will still be Los Baños if it is not an hour drive away from Sta. Cruz because the creation of new roads may speed up the travel time from Sta. Cruz to Los Baños (S1). The Prince of Korea will still be the Prince of Korea if he happens to be not gorgeous; a change in his physical appearance does not make him the not-Prince of Korea (S2). The third example (S3), "Everything that is a square is everything that has a shape," however, is not only epistemically, but is also semantically different from the previous examples (an idea that is different from that of the defenders of synthetic a prioricity). Before analyzing my refutation of the synthetic a priori, let us first analyze the analytic propositions. In an analytic proposition, the predicate is conceptually contained in the subject. Having conceptual containment is having explicit redundancy and having the predicate terms being analyzed out of the subject. Chisholm argues that this conceptual containment is not only an explication of identity (that the subject and predicate are identical or equal to each other), an idea that which I assent to. However, his idea that there is no conceptual containment or explicit redundancy in the case where the predicate is logically implied by the subject is problematic. Chisholm only argues such to save the notion of "synthetic a priori propositions." There is no reason why conceptual containment must be restricted only with the identicality (being identical) of the predicate

38

with the subject and the characteristic of the subject as the "conjunction of independent terms, one of which is the predicate term."19 Following Kant's argument that the guiding principle of an analytic proposition is the principle of non-contradiction, we could say that conceptual containment happens when the subject would not be itself if its predicate does not define it or if its predicate is not an essential characteristic of it. Chisholm is right in saying that there would be such a conceptual containment if the subject is the predicate (e.g. A is A) and if the subject is a conjunction of logically independent terms, one of which is the predicate (e.g. B is C, iff B is a conjunction of C and D, etc.) But it is wrong to restrict conceptual containment with these two, for there will still be conceptual containment even if the subject is a conjunction of logically dependent terms, one of which is the predicate, or that the predicate is logically implied by the subject. We can use S3 as our springboard of analysis here: S3: Everything that is a square is everything that has a shape. Translation: Everything that is equilateral and rectangle is everything that has a shape. Everything that is a square is a conjunction of logically independent terms "everything that is equilateral and rectangle." But it is also a conjunction of hidden terms / concepts that are logically dependent of, or logically implied by "everything that is a square" (or "everything that is equilateral and rectangle"). Examples of such hidden concepts are, "everything that has a shape," "everything that has four right angles," "everything that has at least two pairs of parallel lines," "everything that has at least four 19

Roderick Chisholm, "Truths of Reason," 411.

39

perpendicular lines," etc. We will observe that if we negate at least one of these hidden concepts, we are also negating the subject. It is because these concepts are contained in the very definition of the subject. Hence, they are not phrases that are simply added to the subject. Propositions such as this one have conceptual containment / explicit redundancy and are therefore not synthetic a priori, but rather analytic. Mathematical propositions are argued by Kant as synthetic a priori. The example that he had given in his Prolegomena is "7 + 5 = 12." I assent to the idea that this proposition is a priori because we do not need to consult experience to know if such is the case. I do not, however, assent to the idea that it is synthetic. Kant argues that we must make intuitive leaps to know such proposition's syntheticity. We may be tempted in thinking that it is analytic, but as he argues, "7 + 5" contains not the number "12," but merely the union of "7" and "5." If it does contain number “12,” Kant argues, then it must also contain "9 + 3" and "16 - 4." But Kant argues that "7 + 5" does not contain them and the other possible combinations that yield number "12." It may be argued, however, that "7 + 5" does indeed contain "12" and other possible combinations. Henri Poincaré argues for the analyticity of mathematical propositions by arguing in terms of the demonstration Leibniz had made: Let us see how Leibniz tried to show that two and two make four. I assume the number one to be defined and also the operation x + 1, i.e., the adding of unity to a given number x… I next define the numbers 2, 3, 4 by the equalities: (1) 1 + 1 = 2; (2) 2 + 1 = 3; (3) 3 + 1 = 4, and in (4) the same way I define the operation x+2 by the relation; (4) x + 2 = (x + 1) + 1. Given this, we have: 2 + 2 = (2 + 1) + 1 [def.4] (2 + 1) + 1 = 3 + 1 [def.2] 3 + 1 = 4 [def.3]

40

whence 2 + 2 = 4… It cannot be denied that this reasoning is purely analytical.20 From the previous passage, by only using the operations "x + 1" and "x + 2," we have seen how "2 + 2" contains "(2 + 1) + 1," "3 + 1" and hence "4." By substituting "x" in definition 4 with "2," we could see that definition 2, which is "2 + 1 = 3," is contained in definition 4 (where "2 + 1" in definition 2 is "x + 1" in definition 4). Definition 3 is seen as contained in both definitions 2 and 4. "2 + 2 = 4" is, therefore, analytic - one claim of Leibniz that I do not disagree with. If we are patient enough to do the same demonstration with "7 + 5," we would say that "7 + 5 = 12" is analytic. If "7 + 5" contains the union of "7" and "5," it also contains "12" because "12" is the union of "7" and "5." Likewise, "7 + 5 = 12" passes the test of analyticity - the principle of noncontradiction. If "7 + 5" is not "12," there would be a contradiction. Arithmetic, then, is (as Poincaré puts it) a "gigantic tautology." I am not in the position to say that such is the case for the whole of mathematics; but one of its domains - arithmetic, is hereby shown as analytic. Since this does not amount to the proof of analyticity of mathematics, I am leaving the possibility that mathematics (as a whole) may be synthetic, and thus, a posteriori. As noted, the key in understanding analyticity / syntheticity distinction is the "conceptual containment." We are able to see this containment by simply referring to the terms of the proposition. Analytic propositions, then, are simply dependent on how the terms are used (or on the term's meanings). With this, we could deduce that the proposition "It is raining or it is not raining" (that has the form A v ~ A) is analytic, which would seem to be neither analytic nor synthetic for Kant because this is not a proposition 20

Henri Poincaré, "Chapter 1, On The Nature of Mathematical Reasoning," in Science and Hypothesis.

41

that has a subject-predicate form. Analyticity, then, is largely dependent on the use of language. All that are taken as synthetic a priori have form similar to S3. It is, in fact, the case that such propositions are a priori, but this a prioricity is due to their being analytic (in the sense that their predicate is logically implied by the subject). "14" is logically implied by "7 + 7;" "being not red of an object" is logically implied by "being blue all over" and "being colored" is logically implied by "being red." This logical implication points to the analyticity of propositions, as their subjects would not be them if they are also not their predicates or if they are not their logical implications. It is by virtue of being analytic that a proposition is known a priori, and not by virtue of being necessary. This will be further explained later; for the meantime, let us first discuss the implications of this to the formal system of logic.

3.1.2 Logic is not an axiomatic system One counter-argument against the claim that all propositions of logic are analytic is by stating that we are in fact using synthetic a priori propositions, (e.g. Ben is tall) in arguments in logic (e.g. If Ben is not short, Ben is tall. Ben is not short. Therefore, Ben is tall.) However, such argument misleads us into thinking that propositions of logic are about any statement that are worthy of being called as proposition (statements that are capable of truth-value). Logic is not about particular propositions. It is about the form of propositions. Likewise, it is not about the form of propositions such as "Ben is tall" but about the form of the propositions that are also arguments, such as "If Ben is not short, Ben is tall. Ben is not short. Therefore, Ben is tall" and "If all men are mortal, and

42

Socrates is a man, then Socrates is mortal." "Ben is tall" and "Socrates is a man" are propositions, but they are not included in what I call logical propositions, for they are concerned with their truth-value, which is established by verification from experience. Logic is concerned with logical propositions, logical truths or logical propositions (they are all referring to the same thing), that take the form of what people call valid arguments. Logical propositions are the forms these valid arguments take. Clearly then, logic is not about particular propositions and what I call logical propositions do not include such particular propositions. Particular propositions such as "Ben is tall" (which is synthetic) and "Everything that is a square is a rectangle" (which is analytic) may be used in logic not because of their syntheticity or analyticity, but because they may be used in an argument, and may be substituted with variables, such as A, B, X, P, etc.21 It is, in fact, by virtue of being concerned with the form and not with particular propositions that logical propositions are called as principles. A principle is something by which other things are said to be based. In such case, logical propositions are also called as logical principles because logic is traditionally viewed as normative - that our reasoning must be based on these logical propositions. Because logic is traditionally viewed as a system of necessary propositions that could serve as justification of their own selves, these necessary propositions of logic are called principles because their necessary truthfulness legitimizes them as the starting-point of derivation and end-point of justification. The notion "principle" is, hence, due to the traditional attribution of 21

Hence, Quine's arguments on the untenability of analytic/synthetic distinction due to problems in synonymy and interchangeability may not be used to refute my claim. I am talking about the form of analytic statements (that are also arguments) given that their antecedents imply their consequents. Logic is not concerned with whether 'bachelor' and 'unmarried' indeed refer to the same thing and whether the two particular terms are interchangeable. Logic is rather concerned with the form that it purports to take analyticity or tautology.

43

necessary truthfulness to logic and its implication - the self-justification of the necessary truths or propositions. And because logical propositions are traditionally called logical principles, let me use them synonymously. This act of using them synonymously has no significant value except for pointing out that I am referring to what people traditionally claim as logical principles when I speak of logical propositions. This should not be taken to mean that I intend to give same signification for the terms "proposition" and "principle" or to ascribe a necessary truthfulness to logical propositions. Now that this is made clear, my use of the term "logical principle" to refer to "logical propositions" should not be taken as a contradictory account when I claim for the non-necessity of logic and for the absence of self-justification of logical propositions. As a corollary of having no synthetic a priori propositions, we can infer that there are no synthetic a priori propositions in logic. As Ludwig Wittgenstein puts it "the propositions of logic are tautologies."22 With tautology, there is a repetition, such that the predicate is also stated in the subject. In the language of logicians, a tautological proposition must be demonstrated as true in all combinations in its truth table. Below are examples of tautological propositions (all of which are logical propositions):

PRINCIPLE OF IDENTITY A→A A → A T T T F T F

~ T T

PRINCIPLE OF NONCONTRADICTION ~ ( A • ~A) (A • ~ T F F F F T

22

Ludwig Wittgenstein, Tractatus Logico-Philosophicus, internet, available from http://www.kfs.org/~jonathan/witt/tlph.html, 6.1.

A) T F

44

[(A T T F F

MODUS PONENS [(A→B)•A]→B → B) • A] T T T T F F F T T T F F T F F F

→ T T T T

B T F T F

P T F

DOUBLE NEGATION P≡~~P ≡ ~ ~ T T F T F T

P T F

If we were to construct a truth table for all propositions of logic, we would see their tautological form. It is, in fact, by virtue of this tautological form that there are logicians that could infer the idea that logical propositions are unconditionally true. Wittgenstein, himself has argued for this unconditional truthfulness of tautologies. We must not, however, be misled into thinking that this unconditional truthfulness points to eternal truthfulness or absolute objectivity (or simply to metaphysical necessity, which is truthfulness in all possible worlds). A proposition is unconditionally true insofar as it is a tautology. Logical propositions, then, are unconditionally true because they are logical propositions, and hence, are constructed and/or posited as tautologies. To say that A implies B (or A → B) is to say that B is contained in A. If we are given a proposition, then we are also given all its properties or logical implications, just as we are given with "everything that has a shape" when given with "everything that is a square." What's stated in the predicate is already stated in the subject.

45

We cannot argue that there are logical propositions, which are not about particular propositions, that are derived from the most basic logical propositions (or axioms). There is no sense in distinguishing theorems from axioms. Wittgenstein states that "All the propositions of logic are of equal status: it is not the case that some of them are essentially derived propositions."23 With not being essentially derived, we mean that there is no logical proposition that is derived solely from other logical proposition that serve as its sole starting point of derivation. Axioms and theorems are of equal status; hence, it is fallacious to compare logic to a building. Theorems do not have axioms as their foundations; they are of equal footing. Wittgenstein further argues in his TLP (Tractatus Logico-Philosophicus) that every logical proposition shows itself as a tautology. It does not make sense, therefore, to argue that one is essentially derived from the other. One may counter this by pointing to the use of proof in logic; that proofs show that there are propositions that justify others. I argue, on the other hand, that proofs in logic are simply demonstrations so that we could recognize tautologies in logic. Wittgenstein describes a proof as "merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases."24 Likewise, Axioms / Rules of Inference dichotomy makes no sense. They are of equal status, of the same form and are pointing to one thing - tautology. A foundationalist would argue that the rules of inference are indeed not derived from axioms because they are both said to be self-evident (as we have noted in Chapter 2). I argue that rules of inference are not essentially derived propositions but I do not assent to the idea that it is because they are self-evident. When I say that they are not essentially derived, I mean 23

Ibid., 6.127. 24

Ibid., 6.1262.

46

that there is no axiom that could serve as their sole proof or justification. Logical propositions, which are all tautologies, are of equal status; they are translatable to each other. Below is a demonstration of how a considered rule of inference (modus ponens) can be seen as translatable to principle of identity, which is considered as first principle or axiom: A→B A ————— Therefore B

or

[(A → B) • A] → B

[ (A → B) • A ] → B Modus Ponens [ (~A v B) • A ] → B Implication, antecedent's first conjunct ~ [ (~A v B) • A ] v B Implication [ ~ (~A v B) v ~A ] v B De Morgan's, first disjunct [ (~ ~A • ~B) v ~A ] v B De Morgan's, first disjunct's first disjunct (~ ~A • ~B) v [ ~A v B ] Association of disjunction [ ~ (~A v B) ] v [ (~A v B) ] De Morgan's, first disjunct *We are now left with ~ (~A v B) as our first disjunct and (~A v B) as our second disjunct. *For our convenience, let us substitute X to (~A v B). *In substituting X to (~A v B), we will get ~X v X. Translation of [ ~ (~A v B) ] v [ (~A v B) ] ~XvX by substituting X to (~A v B) X→X Implication *We could then see that modus ponens is just the same (and just as tautologous) as the principle of identity (X → X). And hence, there is no point to make a distinction between what we call a rule of inference and an axiom. With the given demonstration, one might argue that this shows that modus ponens is inferable from the principle of identity. I argue that such is the case. This does not, however, mean that the principle of identity is the sole justification or proof of modus ponens (that could make it as the latter's starting-point of derivation). As I have noted, they are translatable to each other, which means that the principle of identity is also

47

inferable from modus ponens. Below, which is just an inversion of the demonstration above, shows that the principle of identity is inferable from modus ponens: X→X ~XvX *Substitute ( ~ A v B ) to X. [ ~ (~A v B) ] v [ (~A v B) ] (~ ~A • ~B) v [ ~A v B ] [ (~ ~A • ~B) v ~A ] v B [ ~ (~A v B) v ~A ] v B ~ [ (~A v B) • A ] v B [ (~A v B) • A ] → B [ (A → B) • A ] → B

Principle of Identity Implication Translation of ~ X v X by substituting ( ~ A v B ) to X De Morgan's, first disjunct Association of disjunction De Morgan's, first disjunct's first disjunct De Morgan's, first disjunct Implication Implication, antecedent's first conjunct

We have now seen that both are inferable from each other. Being inferable of one proposition from the other does not make it essentially inferred or essentially derived from this other proposition. Such is because they are translatable to each other. With being translatable, I am referring to their logical equivalence. There is logical equivalence between propositions when they have the same logical content. This sameness in logical content can be seen in their sameness in truth-values in the truth table. Below is a proof of the two logical proposition's logical equivalence: *X → X is a simplified ( ~ A v B ) → ( ~ A v B ). Principle of Identity can, thus, be expressed as ( ~ A v B ) → ( ~ A v B ). *Modus Ponens is expressed as [ (A → B) • A ] → B. *Prove that: ( ~ A v B ) → ( ~ A v B ) ≡ [ (A → B) • A ] → B. It must be proven that both have the same truth values in all combinations. ~ F F F T

A T T F F

v T T T F

B T F T F

( ~ A v B ) → ( ~ A v B ) ≡ [ (A → B) • A ] → B ~ A v B ≡ A → F T T T T T F T T F T F F F T T F T T F F F F T

→ T T T T

B T F T F

• T F F F

A T T F F

→ T T T T

*This shows that each proposition is a tautology and that they are logically equivalent with each other, such that they have the same logical content.

B T F T F

48

If it is true that rules of inference and axioms are self-evident, then they must not be inferable from each other. We have, however, shown that such is not the case. It could not also be argued that modus ponens is a theorem that is essentially derived from the principle of identity because they are logically equivalent and that the principle of identity is also inferable from modus ponens. This shows that they serve as justification for each other. There is no logical proposition that is more basic that serves as the starting-point of derivation and end-point of justification. Each logical proposition justifies each other, and is derivable or inferable from each other. We can start at any logical proposition (which is a tautology) to get to another logical proposition (which is also a tautology). It, therefore, makes no sense to speak of first principle, basic principle, primary truth or axiom in logic except in analyzing whether such exists. It is clear that there is no axiom from which other logical propositions or principles are derived. Hence, whenever we speak of logical propositions or principles, we are speaking of all propositions in logic that are of equal status. We do not anymore specifically refer to axioms or rules of inference because they are just the same. It may perhaps be argued that "A is A" or the principle of identity is more basic than "[(A → B) • A] → B" or the modus ponens, in the sense that it deals with only one variable (A), while the latter deals with two variables (A and B). It may likewise be argued that "A is A" is axiomatic in system L (logic) such that, though it and the modus ponens are both tautologies, its dealing with only one variable makes it more obvious, thereby seemingly putting modus ponens in a second order (in terms of obviousness). This counter-argument does not, however, refute my claims. First, the apparent

49

obviousness of "A is A" cannot prove its self-evidence as we have previously shown that self-evidence is not about obviousness but about the capability of a principle to serve as a proof and/or evidence of its own self. Second, the apparent obviousness of "A is A" is not a mark of its being axiomatic because the obviousness of a proposition or principle does not make it the starting-point of derivation and end-point of justification. Likewise, obviousness does not make it independent of other principles (independent in the sense that no other principle could serve as its justification). Axiomatization is the process employed by logicians by which they locate the axioms. It is the case, however, that no axiom can be seen in the system of logic. Appeal to axiomatization sees logic as a building (the one illustrated in Chapter 2) or as an inverted pyramid whose foundation serves as the justification of the principles that are on top of it. This stance on logic sees the justification structure of logic as linear. In the case of non-independence of logical principles with each other, a linear justification structure is untenable. Those that are commonly taken as axioms are not independent of each other as each of them implies each other. Negating "A is A" would mean negating "[(A → B) • A] → B." But this does not mean that it is axiomatic because negating "[(A → B) • A] → B" would also mean negating "A is A." A logical principle makes sense only if it is in connection with other propositions or principles in a logical system. The justification structure in logic, then, is not linear. It is much like of a web - a mutual support system, where each logical principle supports each other and the system as well. Likewise, the system itself serves as the justification of principles. And because the principles are justified by the system and within the system and are supported by every principle in the

50

system, there are no principles that are basic or that have the paramount certainty by virtue of being the foundation. All principles of logic are of equal status. Since axiomatization is not possible in logic, Foundationalism in logic, is also shown as untenable. This then proves my first claim that there is no warrant to claim for the selfevidence of first principles in logic, because there are no such principles. They are of equal footing with other logical principles. However, as I have earlier pointed out, appeal to self-evidence does not necessarily imply an appeal to Foundationalism. In denying Foundationalism through refutation of axiomatization, what I have denied is the notion that there are axioms, first principles, basic principles or primary truths that could serve as the starting-point of derivation and end-point of justification (or be non-inferentially justified). This does not, however, destroy the notion of self-evidence - rationality, unprovability and unavoidability (all of which are pointing to logic's necessity). Hence, an analysis of the metaphysical category "necessity" (or necessary), which is truthfulness in all possible worlds, is needed.

§ 3.2 Step 2: Deny that a prioricity implies necessity Necessity implies a prioricity because if there is such a thing that is necessary then such could not be known through sense experience because sense experience can only tell us about the actual world. If a proposition is true in all possible worlds, it must be known only through rational reflection (known a priori). However, a prioricity does not imply necessity. It does not mean that if a proposition is knowable independently of experience, then it is true in all possible worlds.

51

Intuitive induction (or a priori insight) is argued by Chisholm as the method by which we do not only know things without referring to experience, but also a method by which we know the things that are necessary (or necessarily true). Such is because a thing that is necessarily true implies that it is knowable a priori. However, if there is nothing (no proposition / principle) in logic that is necessary, it does not follow that logical truths or propositions are not knowable a priori. I argue that they still are because they are analytic statements. This analyticity of propositions is not tantamount to their being necessary. Saul Kripke has a congruent idea regarding this. He argues that a priori propositions need not be necessary. A prioricity only means that a proposition can be known independently of experience; it does not mean that it must be true in all possible worlds. He gave as an example the proposition "The standard meter bar in Paris is one meter long." Such proposition is knowable a priori because the bar in question (the bar in Paris) defines the length of a meter. But Kripke argues that the proposition will not be true in all possible worlds. The proposition will be false in a world where the bar is subjected to extreme heat or damage, which would make a change in its length.25

§ 3.3 Step 3: Prove that no logical proposition can be taken as necessary Step 1 successfully reduced all statements of logic to analytic. And since analyticity implies a prioricity, we have further supported the idea that logic is the realm of the a priori. Step 2 successfully discussed that being a priori needs not mean being necessary, which we could see as not inconsistent with the analysis in Step 1. In proving the non-

25

Saul A. Kripke, "A Priori Knowledge, Necessity and Contingency," in The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman (Canada: Wadsworth, Thomson Learning Inc., 2003), 425 – 426.

52

necessity of logic, it must be proven that there is at least one possible world where logical propositions are not true.

3.3.1 Logical truths are not necessary truths Defenders of the necessity of logic argue that it is impossible for us to conceive of a world where the propositions of logic are not true. Such is premised from the idea that if logical propositions are not true, then they are false; and from the idea that logical propositions cannot be false because such is an absurdity.26 I, on the other hand, argue that we cannot possibly think of a world where the propositions of logic are false because they show how we think. And as long as they are how we think, they cannot be conceived as false. Let this actual world, where we consider propositions of logic as true, be called World Ω. I do not reject the idea that logical propositions in World Ω are true. What we have to know is whether their truthfulness is a sort of truthfulness in all possible worlds. To know such, let us imagine of a possible world, World α. World α is a world where only one object exists. Let this object be called A. Defenders of the necessity of logic would argue that logical propositions are true in World α. Such is premised from the idea that A is indeed itself (A → A as necessarily true). However, in such a world where only A exists, it makes no sense to assert that "a thing is itself (A → A)," which is an assertion on equality (equality of A to itself) or of a relation in the form of logical implication (→). In this sense, the relation of A to A is "equality" or "identity." Even if we can assert that "A is" or "A exists," we still cannot 26

It is commonly argued that if PNC (principle of non-contradiction) is not true, then it is false. But we cannot argue that it is false, because if we do such, we also affirmed PNC by saying that 'it is true that PNC is false, hence it is false that PNC is true.'

53

assert that "A is A." "A exists" is an assertion on existence; "A is A" is an assertion on equality. Equality is manifested only in cases wherein there is something that is not-equal (just as relation to the self is manifested only in cases wherein there is a relation of the self to the not-self). There can only be equality, even an equality to the self (or a relation to the self) if A is in relation to other object/ or when there is something (a B or a C, etc.) that could serve as the not-A. Otherwise, only "A exists," which is not a logical proposition, is true. But even "A exists" is not a necessary truth; it cannot be true in World Ф where nothing exists. As a corollary of this, the principle of identity is also not true in World Ф. Hence, it is not necessary. But it is not only the principle of identity that is not necessary. All propositions in logic are. In World α, there is no not-A that A could have a relation to, and hence ~ (A • ~A) or the principle of non-contradiction is not true (just as A → A is not). But they are also not false. They cannot be false because there cannot be something that could serve as ~A, and there is no relation with other object/s that could serve as the "•" (and) and even "→" (a relation in the form of logical implication). Other logical operations (i.e. "v" and "↔" or "≡") that refer to relations among the concepts/terms/variables in logic could also not be asserted. The principles of logic simply have no truth-value in this world as well as in World Ф, where nothing exists. Likewise, we can clearly see that we do not affirm even the principle of excluded middle (A v ~A, or A is either true or false) because the logical propositions, including the principle of excluded middle itself are both not true and not false. Hence, the counter-argument that it is absurd to argue that the principles of logic (particularly the principles of non-contradiction and identity) are not true does not

54

refute my claim. When I say that these principles are not true, I am not claiming that they are false; and we may dismiss the impression of absurdity. One possible counter-argument to my possible world argument is the logician's answer to the question 'What happens when an irresistible force meets an immovable object?' His answer would be 'everything,' for it is the case (as many, if not all, logicians would argue) that 'logical impossibility is impossible.' This sense of impossible means that logical impossibility is unthinkable. Such is the case why we cannot conceive of a square circle and/or a five-sided triangle. Logic, hence, sets what is possible. However, this idea does not actually contradict my point. Logic is how we think, but this does not mean that we cannot think of a possible world where the logical propositions are not true. When we say that logical impossibility is unthinkable, we are saying that we cannot think of a radically different logic. That is why I did not construct radically different logical propositions when I thought of possible worlds α and Ф. If one could properly observe, when I think of the possible worlds α and Ф, I am employing how my mind works and hence the logic that I am subscribed to (the logic that I am using). This does not however rule out the possibility of construing a world where the logical propositions in World Ω are not true. Such is because when we say that logical impossibility is unthinkable, we are supporting the idea that logic is dependent on the thing that is thinking or the mind and not on what is metaphysically necessary (or on what the reality, which is outside of us, necessarily is). Hence, in a world where such thinking thing does not exist, it can be construed that logical propositions are not possible (but not false).

55

3.3.2 Analyticity without Necessity From the previous arguments, we could see that [1] logic is wholly analytic, and [2] logic is not necessary. Logic is analytic because it is about tautologies. Its propositions or principles show the tautological form of what we call valid arguments that are about tautologies (or at least purport to be about tautologies). It is not necessary because it is not true in all possible worlds. Its non-truthfulness in all possible worlds means that logical propositions are not evident-in-themselves. Logical propositions are about reality. However, they are not anchored on reality, but on how we think or reason and/or understand reality. Hence, they are primarily about how we understand or view reality, though the initial concern is to make an account on reality. What we can claim about a logical proposition or principle is that it is expressed as a tautology in language, and anchored on how we understand reality and not on what reality necessarily is. In the case of non-truthfulness of logical propositions in Worlds α and Ф, one may ask when the logic has started. I argue that logical principles started when there is a rational mind that is already able to recognize his relations with other objects in the world. The system of logic started when there is a tacit agreement between rational minds that are able to recognize the relations that there are in the world. These rational minds have the same background, and hence, same predisposition of the mind when it comes to recognizing the relations in the world.

56

Chapter 4 The Fourth and Fifth Steps: LOGIC AS ABSTRACTED FROM THE PREDISPOSITION OF THE MIND AND A DEFENSE OF SUCH

Since we have already destroyed the notion of necessity in logic, it seems easy for us to destroy the notion of self-evidence of logic that which is pointing to logic's necessity. But let us suppose that there is a very unyielding defender of self-evidence of logic, who would not give up such notion even if the notion of logic's necessity has collapsed. He would then argue that logical propositions are self-evident despite of the contingency of logic. He would argue that even if logical propositions are not true in World α, he would say that they are true in World Ω. He would not only say that they are true, he would definitely say that they are self-evident. Suppose this is an acceptable argument, even if we have established that "self-evidence" as applied to logic points to logic's necessity, that the necessity of logic is the necessary/essential condition for self-evidence. However, if we accept this defender's argument, we have to redefine "self-evidence" in logic because this defender is not defending the traditional notion on self-evidence in logic. He may claim that a proposition may still be evident-in-itself even if contingent. However, the sense of evident-in-itself is not only about the proposition's being true and the obviousness of the truth of such proposition. It is also about being independent of other propositions. But if it is contingent, its truthfulness must be dependent on some conditions. This is an idea that is incompatible with the original sense of "self-evidence." It is clear then that he is applying a new definition to self-evidence, which makes him

57

incapable of refuting my claims because he is playing a different language, which uses the same terms differently. What I am refuting is the self-evidence, which goes hand in hand with necessity of logic. Clearly then, I have not only proven, but have also defended my second claim - there is no warrant to claim for the self-evidence of anything in logic, because there is no logical proposition that is necessary. Let us now suppose that this defender is not actually a defender of self-evidence, but my adversary who would always stand in opposition to my arguments. This adversary of mine could challenge how I view logic by asking whether logic is about how we understand the world or whether it is about how the world is. Surely then, he would argue that logic is about how the world is. Arguing such, he may still argue that logic is "out there" (independent of how we think or how we understand the world). But this "being out there" of logic is not the one that implies necessity and self-evidence, because we have already refuted them. The sense of this "being out there" that my imaginary adversary is arguing means that the truthfulness of a logical proposition, e.g. A → A, is contingent, but independent of any rational mind. It is contingent because it is not true in Worlds α and Ф, but he claims that its truth is not due to a rational mind that recognizes the relations among objects in the world. "A is A" because A is indeed A, he argues. I, on the other hand, argue that it might be the case that A is indeed A or that the truthfulness of A → A is independent of me. But how can we know of it if it transcends our intellectual framework or our mind's predisposition? Even if A is indeed A, our act of knowing it is still based on how our mind works or predisposed to think or to reason or how our intellectual framework is structured. I consider the predisposition/s of the mind

58

and intellectual framework as synonymous with each other. They are what make us recognize the relations among objects in the world, and hence make us understand the world. Leibniz argues that there are two great principles from which our "reasonings" are founded on. These are the principles of non-contradiction and of sufficient reason. Then he proceeds by saying that there are two kinds of truths, that of reason (which he argues as necessary that is different from my adversary's claim) and that of fact (which he argues as contingent).27 In terms of his assertion on the great principles of reasoning, I do not concur. We may have a congruent idea that our reasonings have some structure by which we understand the world. I do not, however, assent to the idea that there are particular built-in predispositions in the mind (such as the principles of non-contradiction and sufficient reason). I argue that they are / may be parts of our mind's predisposition or intellectual framework, but such is due to our background. His idea on the two great principles as the foundations of our reasoning is actually pointing to these two principles as innate in our minds and not as mere predispositions of our minds as affected by our background. We may question, however, the innateness of these principles. Suppose that the object A in World α is a rational mind (or a rational being with a rational mind). Having no relations that exist (relations to other object/s and to the self), there could be no traces of the two principles. A, as a rational mind, would not be able to think in accordance to the principles of non-contradiction and sufficient reason. Advocates of innateness of principles, such as Leibniz and Plato, would answer by saying that the knowledge of these principles is a matter of "recollection" or "re27

Gottfried Wilhelm Leibniz, "The Monadology," in Bacon to Kant, ed. Walter Kaufmann, (New Jersey, USA: Prentice Hall Inc., 1968), 208.

59

learning"; that we only recollect or re-learn these already known principles once we have already recognized relations among objects in the world. Such construal that we would know that we already know of these principles once we are able to recognize relations, however, does not actually point to the idea that we innately know them. It does not support, though is compatible with, the idea that we are re-learning these principles. Rather, it points to the idea that we are learning them once we recognize relations. The notion on "recollection" or "re-learning" is groundless; but the account that we start to think in accordance to these two principles after recognizing relations is pointing to the account that our minds have started to be framed or predisposed to think or to reason in such way. Such is also the case with Chisholm's "intuitive induction." When we reflect upon things and say that they are what they are and not what they are not, we do not know that they are necessarily what they are and necessarily what they are not. We just think and/or understand that they are, and hence assert that they are. I assent that there is an intuitive induction that exists, but this sort of induction is not an occasion for knowing what a thing necessarily is, but an occasion for us to merely think and postulate what a thing necessarily is. Likewise, I do not assent to the idea that we can claim how the world is (that there exist truths of reason whose truthfulness are independent of us) when we claim that our reasonings or minds are structured in certain ways. It is as if there is and must be a necessary correspondence between how our minds work and how the world is. I argue that our minds' predisposition is affected by our background, which we use in our

60

recognition of the relations among objects in the world. But this does not mean that our minds directly capture how the world is, and/or understand the world as how it really is. The background that I am talking about refers to the condition of our relations with other objects or other rational minds in the world.28 Though it may be argued that our minds already have their own structure that have functions of their own, this background affects how our minds work. The background or our condition in the world shapes how the mind thinks or how the intellectual framework works. The structure of the mind alone, or the mind as not affected by the background, cannot yield belief in logical propositions. It only provides for the possibility of being molded by the background and to be predisposed in thinking in a particular manner. The mind alone does not and cannot capture anything in the world (or the reality that is outside of us). It must have a relation to the world or to the reality in order to make an account of the world, whether or not such account is identical with what necessarily is (or what the case in the world indeed is). This relation of the mind to the world is furnished by the background while the accounts that the mind makes in his understanding of the world or the reality are expressed through language. It is by virtue of having a same background that people have the same frame of mind / predisposition of the mind / intellectual framework. It is by virtue of language that the mind can make an account of the reality and express such account in tautologies. Sameness of background, and hence, sameness of intellectual framework, would mean a tacit agreement that embraces between rational minds. This tacit agreement marks the formation of the system of logic.

28

Richard Epstein used the term 'background' almost synonymously with agreement. I, on the other hand, refer to background as almost synonymous with our 'situatedness' in the world. This 'situatedness' is that which yields the same intellectual framework and tacit agreement.

61

If all that we have and could claim for are our intellectual frameworks by which we understand the world, our claim to the truthfulness of logical propositions are only based on these intellectual frameworks. The logical propositions that we have in logic are mere formalization / systematization or simply abstraction from how we think or are predisposed to think. Below is an outline of how the system of logic and logical propositions came to be. [1] Sameness of background or sameness of intellectual frameworks produces a tacit agreement among rational minds. [2] Tacit agreement leads to formalization / systematization of logical principles through making an abstraction of what the intellectual frameworks of such background holds. The systematized / formalized logical principles or propositions together make a system - the system of logic. The logical principles that are abstracted from intellectual frameworks are expressed in language; hence those that are expressed in language as analytic are those that the mind has interpreted, understood and posited as tautologies. [3] The systematization produces "objectivity" and "normativity" in logic. I do not reject the normativity of logic, but unlike the traditional claim on logic's normativity, I argue that logic's normative character is derived from the claimed objectivity of the abstracted and systematized predispositions of the mind. Unlike the traditional notion on logic, I do not argue that logic is essentially normative, which is premised on the notion that logic is about necessary truths. Logic is essentially descriptive; it is about the description of how we think or are predisposed to think. Richard Epstein has a congruent idea on having objectivity in logic as a corollary of having implicit agreements (which shows our background). He states "objectivity arises

62

because we'd all go mad (or be mad) if we didn't act in conformity with some implicit agreements and rules."29 He gave as an example the game of chess.30 Suppose that my opponent announced 'Checkmate', and suppose that he really is in the position of saying such according to the rules of chess. I cannot, of course, just say 'I do not agree.' Epstein states: If I do say, 'I don't agree', either through perversity or my actually not regarding such a proposition as checkmate… I'll find no one to play chess with me… if I do not agree with the community's language agreements and assumptions, then I cannot get anyone to talk with me. We are all built roughly the same, and we have to count on that in exchanging information about our experience.31 We can make three assertions at this point. First, logic's normative character is due to its being systematized after we had a tacit agreement (which is due to sameness of our background). The systematization of logical principles and the creation of a system of logic produce objectivity in such system of logic and logical principles. Logical principles and logic itself are given existence outside of us because they are systematized after abstracting from how we are predisposed to think or how our intellectual framework works. This is, however, not similar to claiming that logic is already "out there." Logic is abstracted from, and hence about, what we are predisposed to think, thereby making it dependent on us (on rational minds that had a tacit agreement). Because logic is given an outside existence and objectivity, it can claim a normative character in the sense that it

29

Richard Epstein, The Semantic Foundations of Logic, Propositional Logics, 2d ed. (New York: Oxford University Press, 1995), 402. 30

this chess game example is different from the previous chess game example, which is used to make an analogy with the traditional conception on the system of logic. 31

Richard Epstein, The Semantic Foundations of Logic,402.

63

tells us that we ought to think in accordance to where our logic is abstracted from, or to our roughly similar intellectual frameworks. Second, because logic is a product of our background, it may be argued that there is a sort of relativity in logic because it is not always the case that we share the same background. For example, those who have the background that holds that a thing that is true is not false argue for Classical Logic. On the other hand, for those who have a different background, and who have pondered on paradoxes and examples, from which a two-valued logic is not applicable, would argue for Many-Valued Logics. For Classical logicians, the principle of excluded middle is absolutely true; for the proponents of ManyValued Logics, such principle is not always the case. Łukasiewicz, for example, argues that a two-valued logic (a logic that considers only two values - true and false, an example of which is the Classical Logic) "is incompatible with the view that some propositions about the future are not predetermined."32 Take as an example the statement "I will be graduating from college in April of this year (2007)," which we may say as determined neither positively or negatively (because it is about the future). If we say that the statement is true, then the possibility that I will not graduate is ruled out (but I am not an all-knowing being to know that it is impossible that I will not graduate in April). If we say that it is false, then we rule out the possibility that I will graduate in April even if I am able to meet all the requirements for graduation (but again, I am not an all-knowing being). Except that I am not a graduating senior college student, the statement would not be either true or false. Such statement (and other statements of the same form) and the consideration of such affect the background of those who argue for Many-Valued logics.

32

Ibid., 318.

64

For those who are not bothered by statements of such form and/or those who deal with these statements differently, consider other logic (e.g. Classical logic) as more appropriate. As Epstein puts it, "what one pays attention to in reasoning determines which logic is appropriate."33 Lastly, I argue that it is not possible for an individual to create his own logic. As noted, there must be a tacit agreement where the rational minds have the same background and hence same predisposition of the mind. He may assert that he has successfully created a new system of logic with new background assumptions, and assert that he has refuted the background assumptions that logics (systems of logic) have. However, such would only amount to mere assertion. One cannot just disagree with the community's language and background assumptions as we have noted in the chess game example. Even though there is relativity in logic, there is also a unity to it. Its relativity is manifested on being anchored on our backgrounds, which yields the possibility of various logics (systems of logic). Its unity is manifested in the common ground or roughly the same background among different systems of logic (and different backgrounds these systems are anchored on). The different systems of logic more or less have the same background. They take as a fundamental assumption the consistency or lack of contradictions within their system. But such fundamental assumption of these systems of logic is not due to lack of contradiction in the world (which is outside of us), but mainly due to how our minds conceive the relations among objects in this world. We state that there are no contradictions because we cannot conceive of contradictions.

33

Ibid., Preface.

65

The systematized logical propositions that are abstracted from how we are predisposed to think are tautologies. What the mind does is to posit tautologies, which actually makes us think that a thing is itself and cannot be the not itself. Such is what makes logic a priori. Analyticity of logic is dependent on how we are framed to think and on what we posit as tautologies. When we say that [(A → B) • A] → B is analytic, we say that we are framed to think that whenever [(A → B) • A], then B. What we determine as tautologies and express in language as tautologies are what our intellectual framework has produced (is framed to think). In the case of synthetic propositions, therefore, the mind does not posit tautology. Synthetic propositions are established a posteriori, where no intuitive induction or a priori insight occurs.

66

CONCLUSION

In proving the two claims (that there is no warrant to claim for the self-evidence of first principles of logic and that there is no warrant to claim for the self-evidence of anything in logic), we started with discussing how we may look at a first principle of logic. Arguing that there is a first principle or axiom in logic is arguing for axiomatization in logic, that is, that there are principles, which we call first principles or axioms, that serve as the starting-point of derivation and end-point of justification in logic. The notion of being both the starting-point of derivation and end-point of justification or simply being non-inferentially justified, however, is an illusion. Because logic is wholly analytic and is simply about tautology, all principles of logic are of equal status. There are no significant differences among axioms or first principles, theorems or those that are derived from axioms and rules of inference or those that make us derive from axioms. These three are equal to each other, translatable to each other, and are pointing to the same thing - tautology. In such case, we have proven our first claim that there is no warrant to claim for the self-evidence of first principles of logic, because there are no such principles. Those that are called first principles are in equal footing with other principles of logic. The idea that there is no first principle that could justify other principles does not, however, mean that the principles of logic are not justified. It is just that all principles in the system of logic justify each other. Logic is a mutual support system and not an axiomatic system that presupposes axioms that serve as both the starting-point of derivation and end-point of justification and cannot be justified within and by the system.

67

In denying the idea that logic is an axiomatic system, we have also denied Foundationalism in logic because Foundationalism is an appeal to axiomatization in logic. It argues that there are foundations (axioms) that are self-evident. However, our refutation of Foundationalism that appeals to a self-evident foundation does not refute the notion of self-evidence in logic. If we have noted that logical principles or propositions are of equal status, it may be argued that there are no self-evident first principles because there are no such principles. But it may be argued that there are self-evident principles of logic; and since they are of equal status, one may argue that they are all self-evident. This leads us to enquiring on the essential condition for something to be worthy of being called "self-evident" in logic - the necessity of logic. Being necessary means being true in all possible worlds. Those that are arguing for the necessity of logic argues that logic cannot be thought of as not true in all possible worlds even in a world where nothing exists. But as we have noted, there is no logic in a world where only one object exists. There is no logical proposition that could be true where only one object exists because there is no relation among objects in the world (and even a relation of the object to itself) that could yield the assertion of logical principles. Logic, then, is not possible in a world where no relation among objects exists. Likewise, logical principles are also not possible in a world where there is no rational mind that could perceive and interpret the relations among objects. A world where there is no rational mind that has no predisposition (or intellectual framework) that could be used in interpreting the relations among objects, will only be a world of scattered objects. This means that logical principles are not evident-in-themselves even if they are evident-to-us.

68

This being evident-to-us of logical principles is not anchored on their being evidentin-themselves. They are not necessary and hence, their truthfulness is dependent on some conditions. Likewise, their truthfulness is dependent on a rational mind that recognizes the relations in the world. The system/s of logic that we are studying in school, on the other hand, is just a product of systematization as we have abstracted the logical principles from how we are predisposed to think. We assert "A is A" not because A is indeed A but because we are framed to think that A is A. Hence, logic is not essentially normative as traditionally argued. Logic does not speak of how the world (necessarily is) and thus does not speak that we ought to reason in accordance to how the world necessarily is (or about how we ought to view reality as how reality necessarily is). Logic merely shows how we think or reason. But it does not mean that it has no normative character. It has, but it is due to the objectivity that we give to logic after we have systematized it. It is normative in the sense that we cannot (or should not) go against it because it is an abstraction and systematization of how we, in our background, think. It may be the case that A is indeed A. But since it transcends our intellectual framework and our background, we could not know if such is really the case. Our assertion that "A is A" is only due to the fact that our intellectual framework is framed to think (or predisposed to think) that A is A. Hence, we could not recognize that the proposition "A is A" is evident-in-itself. All that we could recognize is that it is evidentin-us because such is how we think. We have then proven our second claim that there is no warrant to claim for the self-evidence of anything in logic. This course of enquiry does not only provide proofs for the two claims. It also provides an alternative conception of logic. Logic is generally viewed not only as

69

absolutely objective, but also as necessary, that is, true in all possible worlds. But we have shown that it is not necessary, and its objectivity is only caused by its systematization (which gives logic an outside existence). We have also shown logic's analyticity and non-necessity. Logic is analytic, but is not necessarily true (true in all possible worlds). Logic is, therefore, analytic and objective but never necessary. In addition to this, logic is a priori. But we must qualify its being a priori. It is not a priori in the sense that we use Chisholm's intuitive induction to know that a logical principle is necessary because the sort of intuitive induction that occurs is merely an occasion for us to merely think and postulate what a thing necessarily is. It is a priori because it is wholly analytic, and is analytic by virtue of the use of language. With a prioricity, we are only talking about a thing's being known independently of sense experience. Our intellectual framework works or understands the relations among objects by positing tautologies (we posit a tautology "A is A" in our understanding of A and its relation to itself and the notA's) that we express through language. We do not need to verify through sense perception that A is indeed A because our minds already works tautologously, which actually makes us think that A is itself. It is our intellectual framework that yields a priori insight about logical propositions. Because logic is about how we are predisposed to think, it is about how we understand the world for the reason that we use our intellectual framework (predisposition of the mind) to understand the world. As Epstein argues "whether a notion is called logical is a measure of how fundamental it is to our reasoning, our communication, our view of reality, not how close it is to reality."34

34

Ibid., 404.

70

We can still, however, apply the terms "unavoidability," "rationality" and "unprovability" in logical propositions or principles, but only in a qualified manner. There is unavoidability in believing in a logical proposition because it is how (or abstracted from how) we think and is a product of our background, not because it is necessary and evident-in-itself. There is rationality in believing in a logical proposition because its systematization has led to its objectivity, which makes us appear to have a faulty reasoning (if not insane or mad), not because it is necessarily true and that we ought to reason in accordance to what necessarily is the case. There is unprovability because a logical proposition is how we think, and is evidentto-us, not because it is evident-in-itself, which we could not know because this transcends our intellectual framework. The evidence-to-us of a logical proposition is not a mark of its necessary truthfulness and evidence-in-itself, but of its dependence on how we think. We have now thus solved the Problem of First Principles of Logic by arguing that there are no such first principles. Those that Aristotle calls as prior or better known are only those with the most obvious tautological form, but are not evident-in-themselves. Foundationalism does not hold true in logic, and its appeal to self-evidence is warrantless.

71

BIBLIOGRAPHY Alvira, Tomas, Luis Clavell, and Tomas Melendo, Metaphysics, trans. Fr. Luis Sapan, ed. Fr. M. Guzman. Manila: Sinag-tala Publishers, Inc., 1991. Aristotle. "Posterior Analytics." In Thales to Ockham, ed. Walter Kaufmann, 287297. New Jersey, USA: Prentice Hall Inc., 1968. Bass, Robert. 11 Dec 2006. E-mail to the author. Accessed 4 Dec 2006. Bass, Robert. 9 Dec 2006. E-mail to the author. Accessed 10 Dec 2006. Burge, Tyler, "Frege on knowing the foundation." Gottlob Frege, philosophy of mathematics. Internet. Available from http://findarticles.com. Chisholm, Roderick. "The Problem of Criterion." In The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman, 9-17. Canada: Wadsworth, Thomson Learning Inc., 2003. Chisholm, Roderick. "Truths of Reason." In The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman, 409-421. Canada: Wadsworth, Thomson Learning Inc., 2003. Copi, Irving M. Symbolic Logic, 5th ed. United Stated of America: MacMillan Publishing Co., Inc., 1979. Epstein, Richard, The Semantic Foundations of Logic, Propositional Logics, 2d ed. New York: Oxford University Press, 1995. Ewing, A.C. "In Defense of A Priori Knowledge." In The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman, 385-390. Canada: Wadsworth, Thomson Learning Inc., 2003. Godden, David M. "Psychologism, Semantics and the Subject Matter of Logic." Ph.D. diss., McMaster University, 2004. Kant, Immanuel. "Prolegomena to Any Future Metaphysic." In "A Priori Knowledge," The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman, 370-377. Canada: Wadsworth, Thomson Learning Inc., 2003. Kripke, Saul A. "A Priori Knowledge, Necessity and Contingency." In The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman, 422-429. Canada: Wadsworth, Thomson Learning Inc., 2003.

72

Leibniz, Gottfried Wilhelm. "The Monadology." In Bacon to Kant, ed. Walter Kaufmann, 205-214. New Jersey, USA: Prentice Hall Inc., 1968. Leibniz, Gottfried Wilhelm. New Essays on Human Understanding. Ed. and trans., Peter Remnant and Jonathan Bennett. Cambridge: Cambridge University Press, 1981. Poincaré, Henri. "Chapter 1, On The Nature of Mathematical Reasoning." In Science and Hypothesis. Pojman, Louis J. "A Priori Knowledge." In The Theory of Knowledge, Classical and Contemporary Readings, 3d ed., ed. Louis J. Pojman. Canada: Wadsworth, Thomson Learning Inc., 2003. Russell, Bertrand. Human Knowledge: Its Scope and Limits. Wikipedia. "Axiom." Internet. Available from http://en.wikipedia.org/wiki/Axiom. Wittgenstein, Ludwig. Tractatus Logico-Philosophicus. Internet. Available from http://www.kfs.org/~jonathan/witt/tlph.html.