Fallacy_2008_09_29

April 22, 2009

The Fallacy of the Strengthened Liar Paradox by X.Y. Newberry

Introduction Numerous papers have been written on the topic of the Liar paradox, the sentence that asserts its own falsehood: X = This sentence is false The thesis of this paper is that the Liar sentence is meaningless. This idea is controversial but in section 2 we develop a theory of meaning where this view is fully justified. In section 3 we discuss Gaifman's semantics proposed in Gaifman (2000: Pointers to Propositions, http://www.columbia.edu/~hg17/gaifman6.pdf, Circularity, Definition, and Truth, pp. 79-121), and we observe that it is isomorphic with the theory of meaning and truth outlined in section 2. Section 4 is a critical analysis of Van Fraassen (1968: Presupposition, Implication, and Self-Reference, http://www.jstor.org/pss/2024557, The Journal of Philosophy, pp. 148.) Van Fraassen attempted to solve the paradox using Strawson's theory of presuppositions. We find that the latter is fully compatible with the theory of meaning in section 2.

When all is said and done we conclude that “this sentence is not true” is meaningless, and “ 'this sentence is not true' is not true” is true.

1. The Fallacy The Strengthened Liar paradox is the sentence Y = This sentence is not true If, so the reasoning goes, Y is neither true nor false then it is not true. But if it is not true then it is exactly what it says. Therefore it is true after all. “... there is a broad consensus that the 'gap' option fails to come to terms with the paradox of the 'Strengthened Liar' – where a statement says of itself that it is not true.” Goldstein(1992: 'This Statement Is Not True' Is Not True, http://www.jstor.org/pss/3328873, Analysis, p. 1) Firstly this so called “revenge” argument is not specific to the Strengthened Liar. Haim Gaifman pointed out that The contradiction returns also if the sentence says of itself that it is false: If it lacks a truth value, then it is not false; but it says of itself that it is false, hence it is false after all. Gaifman (2000: http://www.columbia.edu/~hg17/gaifman6.pdf, p. 16.)

Secondly, we will see below that the argument is fallacious. For example Richard Kirkham claims that theories that deny bivalence do not solve the Strengthened Liar. Kirkham (1992: Theories of Truth, URL, MIT Press, p. 293) The argument is repeated below.. Again, let the Strengthened Liar sentence be "This sentence is not true." 1. If the Strengthened Liar sentence is true, then it is just the opposite of what it

claims to be, so it is false (and thus not true). 2. So if it is true, it is both true and not true. 3. If it is false (and thus not true), then it is precisely what it claims to be, so it is true. 4. If it is neither true nor false (and thus not true), then it is precisely what it claims to be, so it is true. [Italics added] 5. If it is not true it has to be either false or neither true nor false, and either way it is true (from (3) and (4)). So if it is not true it is true. 6. So if it is true, then it is both true and not true. 7. But it has to be either true or not true, and either way it is both true and not true (from (2) and (6)). The error is in step 4. If the Strengthened Liar sentence is meaningless (and thus neither true nor false) then it does not claim anything. Step 4 is based on the following principle: If p is the case and a sentence s says that p is the case, then s is true. The fallacy here is the ambiguous use of "says." There are two possible interpretations: 1) "s says p" means that s contains the string p 2) "s says p" means that s conveys p But if s is meaningless it does contain the string p yet it does not covey p. "This sentence is not true" contains the string "this sentence is not true" but it does not convey that it is not true.

This will become apparent in a formalized version of the argument. 1

T('Y')

assumption

2

Y

Tarski's principle: If "Y" is true, then Y

3

~T('Y')

substitution: Y ::= ~T('Y') Proof 2.1

4

~T('Y')

assumption

5

Y

substitution: ~T('Y') ::= Y [ERROR!!]

6

T('Y')

Tarski's principle: "If Y, then Y is true Proof 2.2

Step 5 is incorrect. Let's look at it in greater detail. 4

~T('Y')

assumption

5

F('Y') v M('Y')

trivalence

Case 6a

F('Y')

7a

~Y

Tarski's principle

8a

~[~T('Y')]

substitution: Y ::= ~T('Y')

9a.

T('Y')

double negation (so far so good)

Case 6b

M('Y')

7b

~Y ?

ERROR!! Nothing follows from M('Y') Proof 2.3

It is apparent that nothing follows from M('Y') except perhaps T(M('Y')). In particular Y does not follow from M('Y'). The following clearly does not hold M('Y') |- Y "This sentence is not true" does not follow from " 'this sentence is not true' is not true" because if 'this sentence is not true' is not true then 'this sentence is not true'

cannot be the case. This is a proof by contradiction that “this sentence is not true” is meaningless. We assume that it is meaningful and derive a contradiction. As formulated above the revenge argument is clearly incorrect. If the liar sentence is not true, it could be meaningless, in which case it does not “say” that it is not true. Properly stated the argument should list among its premises that the sentence is meaningful or that it expresses a proposition. One could argue that the Liar is not or cannot be meaningless. But that is a different argument, and we will address it below. But the Liar paradox can be formulated differently. It can refer to itself by its own name: C: The sentence C is not true or formally C: ~T(C) We will address this variant in section 3. But first we will elaborate on our notion of meaningfulness.

2. Theory of Meaning We will utilize the following three definitions: Definition 1: A sentence is meaningful iff it expresses a possible state of affairs. Definition 2: A sentence is true iff the possible state of affairs it expresses corresponds to an actual state of affairs. Definition 3: A sentence is false iff the possible state of affairs expressed by its negation corresponds to an actual state of affairs. Definition 1 is Ayer's interpretation of Wittgenstein. "A genuine proposition pictures a possible state of affairs." Ayer(1984: Philosophy in the Twentieth Century, Vintage Books, p.112). Whether Ayer intended it or not, definition 1 is not the same as the Verification Principle. A state of affairs is possible if we can picture it to ourselves. Definition 2 is very similar to "In order to tell whether a picture is true or false we must compare it with reality." [TLP 2.223] By negation we mean the negation of the predicate. For example when we negate “the apple in the basket is red” we are asserting that the apple has a color other than red. By negation we do not mean that the state of affairs described by the sentence does not exist. Negation in the latter sense would mean that either the apple had some color other than red or that there were no apples in the basket at

all. The possible state of affairs the sentence pictures is its sense. The actual state of affairs we compare it with is its reference. Suppose I have a basket with a red apple. I can then formulate various sentences about it such as “the apple [in the basket] is blue” and compare these sentences with the reality in the basket. Let's take a look at a few examples. T stands for true, F for false, N for neither true nor false, M for meaningless.

Sentence A

"The apple is red"

Possible state of affairs

The apple is red

Actual state of affairs

The apple is red

Truth value

T Table A

Sentence B

"The apple is blue"

Possible state of affairs

The apple is blue

Actual state of affairs

The apple is red

Truth value

F Table B

Sentence C

"The apple has a quadratic solution"

Possible state of affairs

[blank]

Actual state of affairs

N/A

Truth value

N&M Table C

The sentence C is meaningless because it does not express a possible state of

affairs. Suppose now that there is no apple in the basket: Sentence D

"The apple is red"

Possible state of affairs

The apple is red

Actual state of affairs

[blank]

Truth value

N & ~M Table D

If we enumerate all the red things and all the things that are not red, the apple in the basket will not appear on either list. Therefore D is neither true nor false. D nevertheless expresses a possible state of affairs and is thus meaningful. If a sentence does not have a truth value there has to be a good reason for it. This happens when the possible state of affairs cannot be compared with an actual state of affairs. Being meaningless is but one reason why it can happen; therefore M is a proper subset of N. Sentences themselves are facts and we can form other sentences about them: Sentence E

"A is true"

Possible state of affairs

"The apple is red" = The apple is red

Actual state of affairs

"The apple is red" = The apple is red

Truth value

T Table E

We compare the sentence E with the table A. If there is 'T' in the fourth row of A then E is true. If there is 'F' or 'N' then E is false.

Sentence G

"B is false"

Possible state of affairs

"The apple is blue" ≠ The apple is red

Actual state of affairs

"The apple is blue" ≠ The apple is red

Truth value

T Table G

If there is 'F' in the fourth row of B then G is true. If there is 'T' or 'N' then G is false. Sentence H

"C is meaningless"

Possible state of affairs

"The apple has a solution" = [blank]

Actual state of affairs

"The apple has a solution" = [blank]

Truth value

T Table H

If there is 'M' in the fourth row of C then H is true. Now consider Y = This sentence is not true We determine the truth value of Y analogically to E: We compare the sentence Y with the table Y. If there is 'T' in the fourth row then Y is true. If there is 'F' or 'N' then Y is false. Sentence Y

"This sentence is not true"

Possible state of affairs

"Y" ≠ {"Y" ≠ {"Y" ≠ ... }}

Actual state of affairs

N/A

Truth value Table Y

There is nothing yet in the fourth row of the table Y. Now like Sisyphus we have to evaluate Y over and over again. We do not have to go through the cycle too many times to realize that we will never reach the bottom. The comparison with reality cannot be done; the truth value cannot be determined. Furthermore the infinite recursion "Y" ≠ {"Y" ≠ {"Y" ≠ ... }} does not express any possible state of affairs. Y is meaningless. Table Y is effectively equivalent to: Sentence Y

"This sentence is not true"

Possible state of affairs

[blank]

Actual state of affairs

N/A

Truth value

N&M Table Y1

Sentence H1

"Y is meaningless"

Possible state of affairs

"This sentence is not true" ≠ [blank]

Actual state of affairs

"This sentence is not true" ≠ [blank]

Truth value

T Table H1

Table Y1 has 'M' in the fourth row, therefore the sentence H1 is true. Sentence H2

"Y is not true"

Possible state of affairs

"This sentence is not true" ≠ [blank]

Actual state of affairs

"This sentence is not true" ≠ [blank]

Truth value

T Table H2

Table Y1 does not have 'T' in the fourth row, therefore the sentence H2 is true.

The two sentences below are meaningless. "This sentence is meaningless." “This sentence is true” When we try to determine their truth or falsehood we enter an infinite loop just like in case of Y. These sentences cannot be compared wit reality and they do not express any possible state of affairs.

3. Referring to A Sentence by Its Name 3.1 Introduction The Liar sentence can be formulated such that it refers to itself by its own name. We obtain the two-line puzzle. C: The sentence C is not true D: The sentence C is not true or formally C: ~T(C) D: ~T(C) (Note that no quotation marks are required around “C” because “C” already is a name.) Assume that the sentence C is meaningless. Then the sentence C is not true. This is what we have written just below C as D. But C and D are the same. Hence we have a contradiction. Before attempting to solve this paradox let us observe the differences between This sentence is not true and C: The sentence C is not rue.

In the first case every sentence-token of “This sentence is not true” is selfreferential. In the second case only the token C is self-referential, the token D is not. This opens the question of who are the truth bearers, sentence-tokens or sentence-types? Sentence-types are strings of symbols (characters or phonemes), sentence-tokens are the instances of the strings of symbols. Sentence-types are the stamps that produce those instances. Another difference will emerge when we compare the two hierarchies below. As above, Y = this sentence is not true. Anonymous Liar

Named Liar

Value

Y

~T(C)

M

~T('Y')

~T('~T(C)')

T

~T('~T('Y')')

~T('~T('~T(C)'))

~T

Table 3.1 If we substitute “C” for “ '~T(C)' “ in “~T('~T(C)')” we obtain “~T(C)”. By repeating the substitution we could reduce the entire hierarchy to “~T(C)”. It seems that the quickest way to fix the rapidly collapsing pyramid would be to prevent the inter-substitutability of “ '~Tr(C)' “ and “C”. The first impressions are often correct, and we will see in the following paragraphs that this indeed is the right solution.

3.2. Gaifman's Semantics Gaifman developed formal semantics that assigns different truth values to C and D. It resembles the semantics of the programming languages. Gaifman (2000:

Pointers to Propositions, http://www.columbia.edu/~hg17/gaifman6.pdf, p.3, Circularity, Definition, and Truth, pp. 79-121.) “The sentence x is not true” is evaluated as follows: 1) Go to the label “x” 2) Evaluate the sentence next to it 3) If the sentence is not true then “The sentence x is not true” is true else “The sentence x is not true” is false We see that if we substitute “C” for “x” and we evaluate the sentence-token C, the “program” will go into an infinite recursion. In this case we assign the value GAP to C. Let's try to evaluate D. We go to the label “C”. We evaluate the sentence next to it. It has the value GAP, i.e. it is not true. Then according to 3), D is true. Gaifman's position was that the evaluation procedure was the meaning of the sentence, and that if a sentence evaluated as GAP then it did not express a proposition. But it is also apparent that Gaifman's technique is isomorphic with the theory of meaning presented in section 2. We will demonstrate this by the way of the example below. J: Snow is purple K: J is not true J expresses the possible state of affairs that snow is purple. We compare it with the actual state of affairs, which is that snow is white. The possible state of affairs expressed by J and the actual state of affairs differ. Hence J is false. K expresses the possible state of affairs that J does not match reality. Therefore in order to evaluate K we have to first evaluate J, that is, we have to

1a) Go to the label “J” 2a) Compare the sentence next to it with reality 3a) If the sentence is not true then K is true else K is false

If we want to evaluate C then we have to 1b) Go to the label “C” 2b) Compare the sentence next to it with reality 3b) If the sentence is not true then C is true else C is false Step 2b) directs us back to 1b.) Clearly we will be stuck in the loop for ever. We are unable to compare C with reality or with anything else for that matter. C does not express a possible state of affairs. We are not able to picture to ourselves what C expresses. (It does not express anything.) Therefore according to our definition 1, C is meaningless. According to Gaifman, it gets GAP. The view that the Liar sentence is meaningless is highly controversial. Part of the problem is the belief that if the Liar sentence were meaningless we would not be able to reason about it at all. For example we would not be able to tell that it was self-referential. This concern is misplaced. A sentence can refer to itself and yet be meaningless! It was Gaifman himself who pointed out that his semantics was non-compositional. Even if each component of a sentence has a definite meaning the sentence as a whole does not have to have a meaning. We understand perfectly well what “C”, or “this sentence” mean. This understanding enables us to execute the “algorithm” to evaluate the sentence … and during this

process we realize that we are not able to picture to ourselves what the sentence expresses. And yes, we are able to translate the sentence into French. Furthermore the grammatical subject of C has the same referent as the subject of D, and the predicates of C and D have the same extent. But the meaning of a sentence is not the sum of the meanings of its parts. The evaluation procedure yields different results for C and D. Similar to this is the argument that we have to understand the sentence in order to reason with it to a contradiction. But we do not need to do so! We only have to observe that “This sentence is not true” is meaningless. Nothing more is required. However, we can assume that C has the same meaning as D. Then we derive a contradiction! Last but not least let us not forget that if two different tokens of the same type can have different meanings, so can two different tokens of two different types. In the example below: C: ~T('C') D: ~T('C') E: ~T('~T('C')')

-M -T -T

the subject of E has the same referent as the subject of C, and the predicates of C and E have the same extent. Yet C evaluates as M and E evaluates as T.

3.3. Quantifying over Sentence-tokens But tokenism appears to be vulnerable to another paradox. The author of this paradox is Martin Cooke (The Reasoner 3(3), http://www.thereasoner.org, p.7.) Definition: “C*” is the name of the sentence-type of the following sentence-token: “All the sentence-tokens of the sentence-type C* are not true.” Let c1: All the sentence-tokens of the sentence-type C* are not true be a token of C*. Assume that c1 is true. Then all the sentence-tokens of the sentence-type C* are not true. It includes c1. Hence c1 is not true. Assume that c1 is false. Then at least one sentence token of C* is true. Let's say it the token c2. But c2 says that “All the sentence-tokens of the sentence-type C* are not true”, which contradicts that at least one sentence token is true. So c1 can be neither true nor false. Perhaps c1 is meaningless. Let us apply Gaifman's procedure to it. We will start with a sentence token c3 All the sentence-tokens of the sentence-type C* are not true 1) Go to the label “c3” 2) Evaluate the sentence next to it According to Gaifman's semantics if v(α(a)) = T for all α in the range of ‘x’, then v((x)α(a)) = T. Gaifman (2000: p. 17) It means that we have to evaluate all the sentencetokens of the type C*. 3.1) Evaluate the sentence token c1 3.2) Evaluate the sentence token c2

3.3) Evaluate the sentence token c3 Sooner or later we will run into the token c3 – and we will be in an infinite loop again. Hence we assign GAP to c3. We can therefore say “All the sentence-tokens of the

sentence-type C* are not true” is not true. We could have done the same thing with the strengthened Liar: This sentence is not true “This sentence is not true” is not true

-M -T

where 'M' means meaningless and 'T' means true. But the problem can be reformulated as a two line puzzle. C: The sentence C is not true D: The sentence C is not true

-M -T

which led us to the conclusion that sentence-token C had a different meaning than the sentence-token D. If we did the same thing with C*, e.g. C*: “All the sentence-tokens of the sentence-type C* are not true” - M D*: “All the sentence-tokens of the sentence-type C* are not true” - T D* would cause a contradiction. But we cannot do so because C* does not refer to a token. Nevertheless there is a problem. We have concluded that “All the sentence-tokens of the sentence-type C* are not true” is not true. But the sentence has many tokens and we ought to be able to generalize our conclusion to all the tokens. We might be tempted to say: c4: All the sentence-tokens of the sentence-type C* are not true But we have already concluded that c4 is meaningless. What this means is not that we have a contradiction, but that we cannot use c4 to express the state of affairs

we had in mind. We need to find another way of expressing it, e.g. P: All the sentence-tokens of the same type as “All the sentence-tokens of the sentence-type C* are not true” are not true. The sentence-token P has different meaning than the sentence-token c4. Can we form a paradox with P just like we did with c1? That does not seem to be possible. Instead of using the name of the sentence-type we quote a sentence token. P is not self-referential because P is not a sentence-type of C*. We could attempt: Q* = is the sentence-type of the following sentence-token: “All the sentence-tokens of the same type as 'All the sentence-tokens of the sentence-type Q* are not true' are not true.” But then we can of course conclude that All the sentence tokens of the same type as “All the sentencetokens of the same type as 'All the sentence-tokens of the sentence-type Q* are not true' are not true” are not true. If we attempted to avoid using any name altogether we would have to write an infinite string: All the sentence-tokens of the same-type as “all the sentence-tokens of the same type as “all the sentence-tokens of the same type as “ … “ “ “. But if P does not have the same meaning as c4 it means that we cannot substitute C* for “All the sentence-tokens of the sentence-type C* are not true”, i.e. generally we cannot substitute the name of a self-referential sentence for its quotation. Assume now that the sentence-types are the truth bearers, that is, all the sentencetokens of the same type have the same truth value:

C: The sentence C is not true D: The sentence C is not true E: “The sentence C is not true” is not true

-M -M -T

We conclude that C is not true and write the conclusion below as D. But D must have the same value as C. Do we have a contradiction? No! What this means is that we cannot use D to express the state of affairs we had in mind. We have to use E instead. We can express this formally: C: ~T(C) D: ~T(C) E: ~T('~T(C)')

-M -M -T

We observe again that we cannot substitute “C” for “ '~T('C')”. By adopting the convention that tokens of the same type have the same truth value we obtain a cleaner system.

3.4 Conclusion We can use Gaifman's evaluation procedure regardless if sentence-types or sentence-tokens are the truth bearers. In either case the application of the procedure reveals that C and '~T('C') have different meaning. Therefore our syntactical rules must prohibit the inter-substitutability of “C” and “ 'T('C')”. This is the solution of the Liar paradox. It is rather simple.

3. Presuppositions 3.1 Introduction Bas C. van Fraassen analyzed the Liar paradox using the Logic of Presuppositions. Van Fraassen (1968: Presupposition, Implication, and SelfReference, http://www.jstor.org/pss/2024557, The Journal of Philosophy, pp. 136151.) This is of particular interest because the Logic of Presuppositions is fully compatible with out theory of meaning presented in section 2. The theory of presuppositions is the view that a sentence is neither true nor false if its presupposition is not true. Its originator is P. F. Strawson, who contended that a property cannot be attributed to what does not exist. Strawson (1950: On Referring, http://www.jstor.org/pss/2251176, Mind, pp. 320-344), Strawson (1952: Introduction to Logical Theory, Methuen, pp. 173-179.) For example the sentence R = All John's children are asleep is neither true nor false if John has no children. "All John's children are asleep" presupposes “John has children.” Intuitively 'R' is meaningful. Strawson rejected the trichotomy: true, false, meaningless, and attempted to justify the view that sentences such as 'R' could be neither true nor false yet meaningful. For us such a justification is hardly a problem. 'R' expresses a possible state of affairs. It is an exact analogy of “the apple is red” when there is no apple in the basket. We have seen that this sentence is meaningful yet does not have a truth value. But there are also sentences that are meaningless. For example "all the round

squares are large” or the Liar sentence. "All the round squares are large” does not express a possible state of affairs. It is neither true nor false because its presupposition, "there are some round squares", is not rue.

3.2 Negation, Truth, Necessitation Let A = All John's children are asleep B = John does have children We observe that presupposition has the following property: (a) if A is true then B is true (b) if ~A is true then B is true We can extract this property of presupposition and define: Definition 3.1: A presupposes B if and only if (a) if A is true then B is true (b) if ~A is true then B is true Presupposition thus defined in not synonymous with the presupposition in Strawson's sense, but we can nevertheless utilize it as an analytic tool. Perhaps we should use a different term. But none comes to mind, so for the rest of this chapter we will use the term presupposition in the sense of Definition 3.1. According to this definition if B is universally valid then every sentence presupposes it. For example every sentence presupposes that 2 + 2 = 4. But for our investigation this will not be relevant. A sentence can have multiple presuppositions. For the time being, “if ... then” in our definition will be the

classical material implication. We will assume a non-bivalent logic and define negation as follows: A

~A

T

F

N

N

F

T Table 3.1 - Negation

'N' stands for neither true nor false. Then we define truth as a second order predicate:. P

T(P)

F(P)

T

T

F

N

F

F

F

F

T

Table 3.2 - Tarski's principle If a sentence does not have a truth value then it is not true and not false. The table expresses Tarski's principle for our non-bivalent system. However bivalence holds for the second level sentences. Our definitions of negation and truth follow Van Fraassen's convention. We will also need the concept of necessitation or semantic entailment. Definition 3.2: A necessitates B iff whenever A is true then B is true. From definitions 1 and 2 we obtain: Lemma 3.1: A presupposes B if and only if (a) A necessitates B (b) ~A necessitates B

By combining the tables 3.1 and 3.2 we obtain: P

~P

T(P)

T(~P)

F(P)

~T(P)

T

F

T

F

F

F

N

N

F

F

F

T

F

T

F

T

T

T

Table 3.3 We note that T(~P) = F(P), P and T(P) necessitate each other, ~P necessitates ~T(P) but not vice versa. The analogue of modus ponens holds for necessitation and Tarski's principle: If P then T(P) but the analogue of modus tollens does not If ~T(P) then ~P [ERROR !!] The reason is the second row, which does not exist in the bivalent system. Van Fraassen defined a second order language with syntactic distinctions between first and second order sentences. X and Y are first order sentences. The paradox is modeled by a relation of co-necessitation N* between X and T(~X) as well as between Y and ~T(Y). “N* imposes trans-level semantic relations among the sentences.” Van Fraassen (1968, p. 149.) However we will dispute that such modeling is correct.

Van Fraassen obtained the following result for the Plain Liar paradox: X presupposes T(~X)

Since T(~X) cannot be true, X is neither true nor false. The paradox is thus

resolved. He proposed to relax the bivalence of the second order sentences in order to solve the strengthened version. But this is implausible since it violates Tarski's principle. The relation N* of co-necessitation between Y, ~T(Y) and X, T(~X) that van Fraassen suggested appears to be based on the silent assumption that Y and ~T(Y) [X and T(~X)] have the same meaning. When we make this assumption explicit it turns out that the Plain paradox is not solvable either. But even if this assumption is discarded the result X presupposes T(~X) still holds. In addition we obtain Y presupposes T(Y) for the Strengthened paradox.

3.2 The Plain Liar Paradox Using non-bivalence and interpreting Tarski's principle as co-necessitation seems to solve the Plain Liar paradox:

a

b

c

d

e

f

X

~X

T(X)

T(~X)

T(~X)

~T(~X)

1

T

F

T

F

T

F

2

N

N

F

F

F

T

3

F

T

F

T

F

T

Table 3.4 The columns c, d represent Tarski's principle (according to table 3.2), columns e, f represents what the Liar said (modeled by the relation N*.) The table reveals that a) if ~X is true then T(~X) is true [b3, d3] b) if X is true then T(~X) is true [a1, e1] Therefore according to lemma 3.1, X presupposes T(~X). But T(~X) cannot be true. Hence X does not have a truth value. We can read the solution from the table. Only row 2 does not contain a contradiction.

3.3 The Strengthened Liar Paradox The Strengthened Liar paradox is next: a

b

c

d

e

f

Y

~Y

T(Y)

~T(Y)

T(Y)

~T(Y)

1

T

F

T

F

F

T

2

N

N

F

T

T

F

3

F

T

F

T

T

F

Table 3.5 Columns c, d represent Tarski's principle (according to table 3.2), columns e, f represent what Y “says” (modeled by N*.) Column f encodes the postulate that ~T(Y) necessitates Y. Hence the 'F' in f2 (and corresponding T in e2.) We have a contradiction as e.g. c2 contradicts e2. Here again we have some doubts about the relation N*. We already know that the sequent ~T(Y) |- Y is problematic. Furthermore, how did the 'F' get in f2? What does Y “say” about its own truth value when it does not have a truth value? Clearly if Y is true then what it says must be the case hence ~T(Y) is the case [f1]. If Y is false then what it says is not the case hence T(Y) is the case [e3.] But it is not clear what to conclude when Y is neither true nor false. We can actually prove that ~T(Y) necessitates Y by a relatively simple argument using the principle that if two different sentences mean the same thing and one is true then the other one is also true.

Principle 1a

Principle 1b

Assumption

A means the same as B

A means the same as B

Premise

T(B)

T(~B)

Conclusion

T(A)

T(~A)

Let us assume that Y and ~T(Y) have the same meaning. ~T(Y) necessitates Y T(Y) necessitates ~Y 1

Assumption

Y means ~T('Y')

Y means ~T('Y')

2

Premise

T~(T('Y'))

T(T('Y'))

3

Tarski's principle from 2

~T('Y')

T('Y')

4

Principle 1 from 1,2

T('Y')

T('~Y')

[f1]

[e3]

We have proven that ~T(Y) necessitates Y, but we have also proven a contradiction in the process (lines 3 and 4.) In addition we have proven that T(Y) necessitates ~Y. The two results contradict each other if the second order sentences are bivalent. By the definition of necessitation if Y is N then ~T(Y) must not be T [f2.] Analogically if T(Y) necessitates ~Y and Y is N then T(Y) must not be T [e2.]

The paradox is represented by the table below.

a

b

c

d

e

f

Y

~Y

T(Y)

~T(Y)

T(Y)

~T(Y)

1

T

F

T

F

F

T

2

N

N

F

T

F

F

3

F

T

F

T

T

F

Table 3.6 e2 and f2 conflict with each other, d2 and f2 conflict with each other. Van Fraassen argues that “The Strengthened Liar paradox is averted if we hold that T(Y) and T(~Y) are themselves neither true nor false. From this it follows immediately that the sentence ~T(Y) & ~T(~Y) also is neither true nor false.” van Fraassen (1968: p.149.) In a tabular form this would perhaps look as follows:

a

b

c

d

e

f

Y

~Y

T(Y)

~T(Y)

T(Y)

~T(Y)

1

T

F

T

F

F

T

2

N

N

N

N

N

N

3

F

T

F

T

T

F

Table 3.7 But this is implausible. If it is the case that Y is neither true nor false then by Tarski’s principle ~T(Y) & ~T(~Y) is true.

3.3 The Plain Liar Again It is hardly surprising that we have reached a contradiction. After all we already knew that sooner or later we would run into one. What is perhaps more surprising is that the same argument can can be used against the Plain Liar as well. Note that T(~X) = F(X). T(~X) necessitates X

~T(~X) necessitates ~X

1 Assumption

X means T(~X)

X means T(~X)

2 Premise

T(T(~X))

T(~T(~X))

3 Tarski's principle from 2

T(~X)

~T(~X)

4 Principle 1 from 1, 2

T(X)

T(~X)

[e1]

[f3]

The result is in table below. f2 conflicts with d2 and e2. a

b

c

d

e

f

X

~X

T(X)

T(~X)

T(~X)

~T(~X)

1

T

F

T

F

T

F

2

N

N

F

F

F

F

3

F

T

F

T

F

T

Table 3.8

3.4 Resolution What is of course wrong with the argument is the assumption that Y has the same meaning as ~T(Y). What we have in front of us is a proof by contradiction that Y does not mean ~T(Y). And if ~T(Y) is the only thing Y could possibly mean then Y does not mean anything. Thus row 2 in the table below is the solution. If Y has

no truth value it could be because it is meaningless. Therefore we cannot draw any conclusions from val(Y) = N, and e2, f2 are undetermined:

a

b

c

d

e

f

Y

~Y

T(Y)

~T(Y)

T(Y)

~T(Y)

1

T

F

T

F

F

T

2

N

N

F

T

TvF

FvT

3

F

T

F

T

T

F

Table 3.9 The table also tells us that a) if Y is true then T(Y) is true [c1] b) if ~Y is true then T(Y) is true [e3] Therefore according to lemma 3.1, Y presupposes T(Y). Let us look at this paradigm again more closely. Instead of assuming that Y and ~T(Y) mean the same thing we will assume that if Y is meaningful then Y and T(~Y) mean the same thing. This is plausible: both Y and ~T(Y) refer to the same object [Y] and both attribute the same property [T()] to it. (Of course the assumption that Y is meaningful eventually results in a contradiction.) Row 1 in table 3.9 represents the assumption that Y is true. Since true entails meaningful then Y means ~T(Y), i.e. ~T(Y) is true [f1]. Row 3 represents the assumption that Y is false. False also implies meaningful, i.e. ~T(Y) is false [f3], and T(Y) is true [e3]. So ~Y necessitates T(Y). Obviously Y necessitates T(Y) by Tarski's principle [c1].

Note that step 3 in proof 2.1 was correct. It was done under the assumption that Y was true. In our table we can trace the steps as c1 → a1 → f1. But step 5 in proof 2.2 was incorrect because from the assumption that Y is not true we cannot conclude that it is meaningful. In our table we can trace the steps as f1/f2 → a1/a2 → c1. The first step f1/f2 → a1/2 is incorrect since our table does not warrant f2 → a2. The new Plain Liar table is below: a

b

c

d

e

f

X

~X

T(X)

T(~X)

T(~X)

~T(~X)

1

T

F

T

F

T

F

2

N

N

F

F

TvF

FvT

3

F

T

F

T

F

T

Table 3.10 The result that X presupposes T(~X) still holds. The relation N* should not be modeled as X necessitates T('~X') & T('~X') necessitates X Y necessitates ~T('Y') & ~T('Y') necessitates Y but as X necessitates T('~X') & ~X necessitates ~T('~X') Y necessitates ~T('Y') & ~Y necessitates T('Y')

3.4 Sentence Names and Propositional Constants The reason van Fraassen used the former model probably is that the Liar

sentence can be expressed as C: ~T(C) and we may be tempted to postulate that C and ~T(C) necessitate each other, and by analogy that Y has the same meaning as ~T('Y'). But “C” is not a counterpart of “Y”; the former is a name and the latter is a propositional constant. “C” does not make an assertion. We will illustrate by the way of example. We can write a sentence Jack is short and give it the name “Jack.” But Jack is not a proposition. Kripke has shown that it is quite natural for sentences to refer to themselves by their own name. Kripke(1975: Outline of a Theory of Truth, http://philo.ruc.edu.cn/logic/reading/Kripke_%20Theory%20of %20Truth.pdf, The Journal of Philosophy, p. 693.) However impredicative definitions of propositional constants are not legitimate. Here is an example: L: ~T('L')

(3.4.1)

(with quotation marks or corner quotes around “C”), Beall-Glanzberg(2009: The Liar Paradox, http://philosophy.ucdavis.edu/glanzberg/sep-liar-submit.pdf, Stanford Encyclopedia of Philosophy, p.1.) According to our convention this would be written as L = ~T('L') First of all, (3.4.1) does not have any counterpart in the natural language.

Secondly, propositional constants stand for well formed formulae such as grammatically correct sentences of English. (3.4.1) will not produce a grammatically correct sentence of English. Good comparison would be with the macros of the C programming language: #define J Jack is short The preprocessor will replace every occurrence of “J” with “Jack is short.” But if we define “L” recursively: #define L ~T('L') then “L” will not expand into anything sensible.

Conclusion Symbol

Interpretation

Value

Y

This sentence is not true

meaningless

F(Y)

"This sentence is not true" is false

F

T(Y)

"This sentence is not true" is true

F

~T(Y)

"This sentence is not true" is not true

T

Y presupposes T(Y). T(Y) cannot be true. Hence Y does not have a truth value.

Copyright © X.Y. Newberry 2009