Presuppositions 2008-10-08

Revised October 12, 2008

Presuppositions and Occurrence Relevance Logic by X.Y. Newberry The objective of this paper is to explore the relationship between the logic of presupposition and occurrence relevance logic. In section 1 we recapitulate the logic of presuppositions as proposed by P.F. Strawson. In section 2 we note the similarities between the logic of presuppositions and occurrence relevance logic developed by Richard Diaz. In section 3 we recall van Fraassen's solution of Liar's paradox based on the logic of presuppositions. In section 4 we ponder the similarities between Liar's paradox and Gödel's sentence. Finally in section 5 we justify the view that not every sentence has to have a truth value.

1. The Logic of Presuppositions The idea of presuppositions was championed by P.F. Strawson. According to this view a property cannot be either truly or falsely attributed to what does not exist. Examples are S1 = "The present king of France is wise" S2 = "All John's children are asleep." S2 is neither true nor false if John has no children. In this case the subject class, John's children, is empty. There are alternative views. Bertrand Russell proposed the Theory of Descriptions. It holds that in all the cases when a property is attributed to a subject there is an implicit assertion that an entity corresponding to the subject exists. Let's take S1 as an example. According to Russell it is equivalent to (Ex)(K(x) & (y)(K(y) -> x=y) & W(x))----------------(1.1)

1

Since (Ex)Kx is false, (1.1) is also false. S2 can be written as (x)(Cx -> Sx)----------------(1.2) if John does not have any children then according to classical logic (1.2) is true. But we could also apply Russell's analysis to S2 1) and write it as (Ex)(Cx) & (x)(Cx -> Sx)----------------(1.3) then S2 would be false if the subject class were empty. According to these two alternative views, the non-existence of any Children of John's is sufficient to determine the the truth (1.2) or falsity (1.3) of S2. This is not the case with the logic of presuppositions. The existence of John's children determines that S2 is either true or false and their non-existence means that S2 is neither true nor false.

The presupposition analysis can be applied to many sentences, which have the form ' All ...', ' All the ...', ' No...',' None of the ...', ' Some ...', ' Some of the...', ' At least one ...', ' At least one of the ...'. "the existence of members of the subject-class is to be regarded as presupposed (in the special sense described) by statements made by the use of these sentences; to be regarded as necessary condition, not of the truth simply but of the truth or falsity, of such sentences."2)

1 Compare also Strawson, 1957, p. 169 2 Strawson, 1957, p.176

2

Strawson proposed that the four Aristotelian forms should be interpreted as forms of statement of this kind. The four Aristotelian forms and their modern interpretations are A --------~(Ex)(Fx & ~Gx) or (x)(Fx -> Gx) or (x)(~Fx v Gx) E --------~(Ex)(Fx & Gx) or (x)(Fx -> ~Gx) or (x)(~Fx v ~Gx) I --------(Ex)(Fx & Gx) or ~(x)(Fx -> ~Gx) or ~(x)(~Fx v ~Gx) O --------(Ex)(Fx & ~Gx) or ~(x)(Fx -> Gx) or ~(x)(~Fx v Gx) Strawson concluded that the presumption that the subject class has members preserves all the laws of traditional syllogism except the simple conversion of E and of I. For these rules hold in traditional Aristotelian Logic xEy --> yEx ---------------(1.4) xIy --> yEx----------------(1.5) When xEy is true and the subject class has members it does not guarantee that the predicate class has members, hence it does not guarantee that yEx is true. Analogically when xIy is false. Strawson points out that when both xEy and yEx are either true or false then (1.4) holds. For our purposes we will simplify the matter. One glance at (Ex) (Fx & Gx) convinces us that the formula is completely symmetrical with respect to F and G and we will require that both (Ex)Fx and (Ex)Gx be true for (Ex)(Fx & Gx) to be either true or false.

We observe that E can be obtained from A by substituting H for ~G. In this sense O can be also reduced to I. We end up with only E and I, which differ merely in the negation sign.

3

We can now consolidate our presupposition rules: 1. Any sentence of the form [~](Ex)(Ax * Bx) is either true or false iff (Ex)Ax and (Ex)Bx are both true. It follows that 2. For any sentence of the form [~](Ex)(Ax * Bx ) a) If |= ~(Ex)Ax then |≠ [~](Ex)(Ax * Bx ) b) If |= ~(Ex)Bx then |≠ [~](Ex)(Ax * Bx ) Some theorems of classical logic are at variance with the above stated principles. For example (x)((Px & ~Px) -> Qx) is a theorem of classical logic. However, the subject class is empty. That is ~(Ex)(Px & ~Px) A new, non-classical logic is required. A related question is if the logic of presuppositions can be applied to purely deductive systems such as formalized arithmetic. In such systems there are no contingent propositions. All the true sentences are necessarily so, and all the false sentences are necessarily false. For example, according to Strawson's analysis ~(Ex)(x < x & x > 6) is neither true nor false because the subject class is empty, that is because ~(Ex)(x < x)

4

2. Occurrence Relevance Logic In 1981 Richard Diaz published a monograph Topics in the Logic of Relevance, in which he presented two closely related logical systems, truth relevance logic and occurrence relevance logic. His aim was to provide an alternative to the relevance logics of Anderson and Belnap, which he found lacking. It turns out that both Diaz's logics satisfy the criterion 2 above.

We will illustrate the workings of truth-relevant (t-relevant) logic by the way of examples. Let us compare P v ~P -----------------------(2.1) with (P v ~P) v Q ----------------(2.2) Both are Boolean tautologies but only the former is t-relevant. We observe that a subformula of 2.2 is itself a tautology. Furthermore we see that when we construct a truth table for (2.2) P | Q | (P v ~P) v Q ---+---+-------------T | x | ...T.F...T T | x | ...T.F...T F | x | ...T.T...T F | x | ...T.T...T we can determine that it is a tautology without evaluating Q. We say that Q is not a trelevant variable and (2.2) is not a t-relevant tautology. Similar observations apply to ~((P & ~P) & Q) -------------(2.3) (P & ~P) -> Q ----------------(2.4)

5

The reason that (2.3) and (2.4) are not t-relevant is that (2.5) below is a tautology. ~(P & ~P) --------------------(2.5) Generally speaking F v G ------- ---------(2.6) will not be a tautology if either F or G are necessarily true (i.e. are tautologies.) Similarly ~(F & G) ----- -----------(2.7) will not be a tautology if either F or G are necessarily false.

The non t-relevant tautologies are the propositional equivalents of the "vacuously true" formulas. By quantification of (2.3), (2.4), (2.5) we obtain ~(Ex)((Px & ~Px) & Qx) --------(2.3') (x)((Px & ~Px) -> Qx) ----------(2.4') ~(Ex)(Px & ~Px) ----------------(2.5') (2.3') and (2.4') are not t-relevant because (2.5') is a tautology. T-relevant logic obeys rule 2. We can now generalize our result and conclude that any formula of the form [~](Ex)(Fx & Gx) ----------------(2.6) is not a t-relevant tautology if either ~(Ex)Fx or ~(Ex)Gx -------------(2.7) are tautologies. This is in conformance with the logic of presuppositions; (2.7)

contradict the presuppositions of (2.6) O-relevant logic differs from t-relevant logic by considering the relevance of occurrences of variable instead of just variables. 6

***** The case of more than one variable is little more complicated. Let us now consider for example: [~](Ex)(Ey)(Fxy & Gxy)

(2.10)

Let us replace y with an arbitrary individual b: [~](Ex)(Fxb & Gxb)

(2.11)

According to the logic of presuppositions (2.11) can have a truth value only for such b's that (Ex)Fxb & (Ex)Gxb

(2.12)

Analogically [~](Ey)(Fay & Gay)

(2.13)

can have a truth value only for such a's (Ey)Fay & (Ey)Gay

(2.14)

It means that (2.10) can be true only if there is a b such that (2.12) and there is an a such that (2.14). In other words (2.10) has a truth value only if: (Ex)((Ey)Fxy & (Ey)Gxy) & (Ey)((Ex)Fxy & (Ex)Gxy)

7

(2.15)

The situation is depicted in Figure 1 below y | | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . F F F F F . . . . . . . . . . . . | . . F F F F F F F . . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F . . . . . . . . . . . | . . . F F F F F . . . G G G G G G . . . | . . . F F F F . . . G G G G G G G . . . | . . . . . . . . . G G G G G G G G . . . | . . . . . . . . G G G G G G G G G . . . | . . . . . . . . G G G G G G G G G . . . | . . . . . . . . G G G G G G G G G . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . ------------------------------------------------x Figure 1 The F's indicate (x,y) pairs for which Fxy holds and the G's indicate the (x,y) pairs for which Gxy holds. The conditions (2.15) means that the F and G regions have to overlap along both axes. Figure 1 illustrates the case where (Ex)(Ey)(Fxy & Gxy) is false because F and G do not overlap. Figure 2 below depicts a situation where (Ex)(Ey)(Fxy & Gxy) is true.

8

y | | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . F F F F F . . . . . . . . . . . . | . . F F F F F F F . . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F * * G G G G G G . . . . | . . . F F F F F * * G G G G G G G . . . | . . . F F F F F * * G G G G G G G . . . | . . . . . . . . G G G G G G G G G . . . | . . . . . . . . G G G G G G G G G . . . | . . . . . . . . G G G G G G G G G . . . | . . . . . . . . G G G G G G G G G . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . -------------------------------------------------x Figure 2 y | | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . F F F F F . . . . . . . . . . . . | . . F F F F F F F . . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . . F F F F F F . . . . . . . . . . . | . . . . F F F F . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . G G G G G . . . . . | . . . . . . . . . G G G G G G G . . . . | . . . . . . . . G G G G G G G G G . . . | . . . . . . . . . G G G G G G G . . . . | . . . . . . . . . . G G G G G . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . ----------------------------------------------- x --------------------t Figure 3 9

Figure 3 illustrates the case where (Ex)(Ey)(Fxy & Gxy) does not have a truth value because there is no line y such that both F and G are on it. When we select x = t we obtain (Ey)(Fty & Gty) ------------------(2.16) On the face of it this formula has a truth value because (Ey)Fty & (Ey)Gty ----------------(2.17) However since (2.10) does not have a truth value no instance of it should have a truth value. So we will stipulate that the presupposition condition (2.15) applies to any instance of (2.10), hence to (2.16). ***** Let us now study a special case of (2.10), namely: (Ex)(Ey)(Fxy & Gy) ----------------(2.18) The situation is depicted below. Here there are only two cases. Either the two regions overlap or they do not. In case of (Ex)(Ey)(Fxy & Gxy) when the two regions did not overlap there were two further sub-cases: either the formula was false or it was meaningless. Here the presupposition condition becomes (Ey)((Ex)Fxy & Gy) ----------------(2.19) This is equivalent to (2.18). (Ex)(Ey)(Fxy & Gy) presupposes itself, and ~(Ex)(Ey)(Fxy & Gy) presupposes its own negation.

10

y | | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . F F F F F . . . . . . . . . . . . | . . F F F F F F F . . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F . . . . . . . . . . . | . . . F F F F F . . . . . . . . . . . . | . . . F F F F . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | G G G G G G G G G G G G G G G G G G G G | G G G G G G G G G G G G G G G G G G G G | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . --------------------------------------------------x Figure 4 The presupposition is false; ~(Ey)((Ex)Fxy & Gy) therefore (Ex)(Ey)(Fxy & Gy) is neither true nor false y | | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . F F F F F . . . . . . . . . . . . | . . F F F F F F F . . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F F . . . . . . . . . . | . . F F F F F F F . . . . . . . . . . . | . . . F F F F F . . . . . . . . . . . . | . . . F F F F F . . . . . . . . . . . . | . . . . F F F . . . . . . . . . . . . . | . . . . F F F . . . . . . . . . . . . . | G G G G * * G G G G G G G G G G G G G G | G G G G G G G G G G G G G G G G G G G G | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . -------------------------------------------------x Figure 5 11

The presupposition is true: ~(Ey)((Ex)Fxy & Gy), therefore (Ex)(Ey)(Fxy & Gy) is true and ~(Ex)(Ey)(Fxy & Gy) is false. We have concluded that (Ex)(Ey)(Fxy & Gy) cannot be false and ~(Ex)(Ey)(Fxy & Gy) cannot be true. This conforms to the observation made by van Fraassen that if A presupposes A then it is never false and if A presupposes ~A then it is never true.3) (p. 142)

3. Liar's Paradox Bas C. van Fraassen found a solution of Liar's paradox using the logic of presuppositions. A proposition A can be true or false only if its presupposition B is true. That is A presupposes B iff (a) if A is true then B is true (b) if ~A is true then B is true Liar's paradox is the sentence X = "This sentence is false" When we assume that it is true then what is says is the case. But it says that it is false, which contradicts the original assumption. When we assume that it is false then what it says is not the case. But is says that it is false, so it must be true, which contradicts the original assumption. We have concluded that X is neither true nor false. This is a paradox only when we assume bivalence. This is not the case with the logic of presuppositions.

3 van Fraassen, p.142

12

We can argue as follows: a) X "says" F(X). so if X is true then F(X) is true. But F(X) = T(~X) and hence if X is true then T(~X) is true b) According to Tarski's principle if ~X is true then T(~X) is true. The conclusion from a) and b) is that X presupposes T(~X). But T(~X) cannot be true. For suppose it is. then F(X) is true and a contradiction results. The presupposition fails, a hence X is neither true nor false.

That was easy. More problematic is Y = "This sentence is not true." It is said that if Y is neither true nor false then it is not true, which is exactly what it says. There is an error in this reasoning. If Y is neither true nor false then it does not say anything. So we cannot argue that "it is exactly what it says."

Similarly Van Frassen suggests that if Y is true then ~T(Y) is true and vice versa. Then the only way out of the paradox is to drop the requirement that the assertions of truth are themselves bivalent. This is perhaps based on an analogy with the plain liar. We can say that if X is true then T(~X) is true and vice versa, so it seems that we should be able say the same thing about Y and ~T(Y). But this is not correct. If Y does not have a truth value then ~T(Y) is true. So it is not the case than if ~T(Y) is true then Y is true.

We can argue as follows: a) According to Tarski's principle if Y is true then T(Y) is true b) Y says ~T(Y), so if Y is false then T(Y) is true, but Y is false is equivalent to ~Y is

13

true, so if ~Y is true then T(Y) is true The conclusion from a) and b) is Y presupposes T(Y). T(Y) cannot be true. For if it is then what Y says is the case. It says that ~T(Y), which contradicts T(Y). The presupposition fails therefore Y is neither true nor false.

Symbol

Interpretation

Value

Y

This sentence is not true

Neither

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

Table 3.1

4. Gödel's Formula Gödel's sentence has the form ~(Ex)(Ey)(Pxy & Qy) ----------------(4.1) where Pxy means x is the proof of y and Q has been constructed such that only one y = m satisfies it, and m is the Gödel number of (4.1) itself. Based on the conclusion we reached towards the end of section 2 (4.1) has (Ex)(Ey)(Pxy & Qy) ----------------(4.2) as a presupposition. But there is exactly one y = m that satisfies Q, so (4.2) reduces to (Ex)Pxm ----------------(4.3) Hence (4.1) presupposes (4.3)p.

14

We now have an almost complete analogy of Liar's paradox

Symbol

Interpretation

Arithmetic

Value

Y

This sentence is not true

~(Ex)(Ey)(Pxy & Qy), G

neither

F(Y)

Y is false

(Ex)Px#[~G]

F

T(Y)

Y is true

(Ex)Px#G

F

~T(Y)

Y is not true

~(Ex)Px#G

T

Y presupposes T(Y)

Presupposition

G presupposes (Ex)Pxm

N/A

T(Y) iff Y

Tarski's principle

(Ex)Px#G |- G

N/A

Table 4.1 G = "~(Ex)(Ey)(Pxy & Qy)", #A stands for Gödel number of "A", m = #G If we postulate (Ex)Px#G |- G We can prove ~(Ex)Pm: Proof 4.1: (by contradiction) (1)----(Ex)Pxm ----------------------Assumption (2)----~(Ex)(Ey)(Pxy & Qy) -------from (1), m is the Gödel number of (2) (3)----(Ex)(Ey)(Pxy & Qy) ----------equivalent to (1) (4)----~(Ex)(Pxm) ------------------reductio ad absurdum There is no further contradiction because (4) negates the presupposition of (2). ***** Probably the easiest way to see that (4.1) does not have a truth value is to convert it to an expression with free variables: ~(Pxy & Qy) ----------------(4.4) ---------------There are two cases Case A: y = v # m ~(Pxv & Qv) ----------------(4.5) Then for any x Qv is false and (4.5) does not have a truth value.

15

Case B: y = m ~(Pxm & Qm) ----------------(4.6) Then for any x Pxm is false and (4.6) does not have a truth value. The reason that Pxm is false is that there is no proof of m. We have the same case as in Figure 4. ***** Arithmetic based on o-relevant logic will be omega-complete. To see this we will construct the following hierarchy: ~P1m ~P2m ~P3m ....... ~(Ex)Pxm All the sentences Ptm are true. There is no proof of m, where m is the Gödel number of (4.1). In order for the system to be omega-complete none of the expressions in the hierarchy below must be true. ~(Ex)(Px1 & Q1) ~(Ex)(Px2 & Q2) ~(Ex)(Px3 & Q3) ..................... ~(Ex)(Pxm & Qm) ----------------(4.7) Because ~(Ex)(Ey)(Pxy & Qy) is not true. This is indeed the case. Q1, Q2, Q3 etc. are all false except Qm. That is no member in the hierarchy has a truth value except perhaps ~(Ex)(Pxm & Qm) But in this case ~(Ex)Pxm. To see that (4.7) indeed does not have a truth value we will break ~(Ex)Pxm down further into yet another infinite hierarchy. ~(P1m & Qm) ~(P2m & Qm) ~(P2m & Qm) .....................

16

Clearly for any member of the hierarchy ~(Ptm & Qm) The first term Ptm is always necessarily false. Therefore no member of the hierarchy has a truth value. Therefore ~(Ex)(Pxm & Qm) does not have a truth value. The following hierarchy is a little bit more complicated. ~(Ey)(P1y & Qy) ~(Ey)(P2y & Qy) ~(Ey)(P3y & Qy) ..................... ~(Ex)(Ey)(Pxy & Qy) First we note that (Ey)Qy. And it is not obvious that ~(Ey)Pty for some t. So if ~(Ey)(Pty & Qy) is true for all t the system would be omega-incomplete. However, we recall from section 2 that we stipulated that the presupposition conditions apply to any instance of a quantified sentence. All the formulae ~(Ey)(Pty & Qy) are instances of (4.1) and since (4.1) lacks a truth value so do all its instances. Therefore the hierarchy above is not a case of omega-incompleteness..

*****

Let us assume that a sound derivation system exists. There is no reason to suspect that there will be any true but underivable formulae. The equivalent of Gödel's sentence in our system is not true as in the classical system, but rather lacks a truth value. The system will still be syntactically incomplete; there will be formulae F such that |/-F and |/-~F. But it will be much more well behaved. Unlike in the system based on classical logic we are positive that the system is syntactically incomplete; we do not need the assumption that the system is consistent. To prove the unprovability of consistency one 17

uses the result that if the system is consistent then ~(Ex)(Ey)(Pxy & Qy) is unprovable.

That is, if consistency is provable then ~(Ex)Pm, which results in a contradiction. But we have already seen that in a system based on o-relevant logic we can prove ~(Ex)Pm and no contradiction results. Therefore there is no reason to suspect that the consistency of such a system is unprovable. We are now in a position to summarize our conjectures. A formal system of arithmetic based on o-relevant logic will be a) semantically complete b) omega complete c) able to prove its own consistency It will certainly not suffer from the principle of explosion. Many classical formulae will not be provable, e.g. (Ax)(x < x --> x > 36) (Ax)(x < x --> x < 36) If anything I would regard this as a plus. The endless proliferation of the "vacuously true" sentences does not enhance our knowledge of arithmetic by one iota. All such sentences will be "undecidable." We have traded the decidability of (Ax)(x < x --> x > 36) for the decidability of ~(Ex)Pxm. While the former is useless the later is significant.

5. Why Some Sentences Lack Truth Values We have concluded after a long and painful process that "This sentence is false" is neither true nor false. This seems rather odd. In the course of everyday life when we 18

communicate with others we do not spend hours on each sentence looking for a proof by contradiction that it does not have a truth value. There must be a more direct method.

"In order to tell whether a picture is true or false we must compare it with reality." [TLP 2.223] Sentences are pictures and therefore to find if "This sentence is false" is true we have to compare it with reality. What does it say about reality? Nothing. There is nothing to compare it with. It is meaningless. "This sentence is false" does not have a truth value because it is meaningless.

We will utilize the following three definitions. Definition 1: A sentence is meaningful if and only if it or its negation is a picture of a possible state of affairs. 4) Definition 2: A sentence is true if and only if it corresponds to an actual state of affairs. Definition 3: A sentence is false if its negation corresponds to an actual state of affairs.

To further characterize possible we can say that it means imaginable.. Ludwig Wittgenstein says something very similar. He alludes to imaginability a few times: If I can imagine objects combined in states of affairs, I cannot imagine them excluded from the possibility of such combinations. [TLP 2.0121], We picture facts to ourselves. [TLP 2.1] A picture is a model of reality. [TLP 2.12] Finally he says: A picture represents a possible situation in a logical space [TLP 2.202] But what does "possible" mean? What is thinkable is possible too. [TLP 3.02] So we have: A picture represents a thinkable situation in a logical space. What is thinkable? 'A state of affairs is thinkable': what this means is that we can picture it to ourselves. [TLP 3.001] So finally we have: A situation 4 A similar definition appears in A.J. Ayer, Philosophy in the Twentieth Century, Vintage Books, New York, 1984, p. 112: "A genuine proposition pictures a possible state of affairs."

19

in a logical space is a state of affairs that we can picture to ourselves.

Definition 1 is similar to the Verification Principle of the Neopositivists. But the verification principle is too restrictive. For example according to our definition "All unicorns are white" is perfectly meaningful although not verifiable. Another major class of meaningful but unverifiable sentences are the statements about other minds. We can "verify" such sentences by observing the overt behavior of other people. But the behavior associated with pain is not the feeling of pain. ***** Consider now the sentence S1 = "The present King of France is wise" and its negation ~S1 = "The present King of France is not wise." If we enumerate all the wise things and all the things that are not wise, the King of France will not be on either list. It means that neither S1 nor ~S1 corresponds to an actual state of affairs. S1 is neither true nor false. The reason is that the present King of France does not exist. However S1 is a picture of a possible state of affairs, therefore S1 is meaningful.

What meaning can be given to the sentence S2 = "Round squares do not exist"? If it does not exist how can we possibly say anything about it? Can a non-entity be a subject of a sentence? Do round squares perhaps somehow subsist before they exist? There are two equally good answers: 1) S2 = "All squares are non-round" 2) S2 = " 'round square' does not have a denotatum" If "round square" does not have a denotatum, it follows that S3 = "All the round squares are large" does not have a meaning. "Round square" does not have a denotatum in a strong sense - we cannot even imagine a round square. "The present King of France"

20

does not have a denotatum in a weaker sense that he does not actually exist. It still does have a denotatum in the broader sense that the present King of France is imaginable. For

example a fictional movie could be made in which the present King of France would be depicted in all his glory.

Needles to say a Venn diagram is not a picture of a state of affairs. But we can nevertheless illustrate our theory by the way of Venn diagrams. S3 is depicted on Fig. 5.1. We see that ~(Ex)(Rx & Sx) can be translated as (x)(Rx -> ~Sx), but ~(Ex)[(Rx & Sx) & Lx] cannot be translated as (x)[(Rx & Sx) -> ~Lx] because the intersection of R and S does not exist. Therefore the non-entity "round large square" cannot be a subject of a proposition.

***** The formalization of "all the round squares are large" and similar sentences would be (x)[(Px & ~Px) -> Qx----------------(5.2) This formula is neither true nor false and the same should hold for its propositional counterpart 21

(P & ~P) -> Q-----------------------(5.3) It is the paradox of material implication and nobody will regret if it is gone. What is even more problematic about (5.3) is that just like the predicate version it attempts to

say what would be the case if the impossible happened. It ought not to be derivable. But (5.3) is equivalent to P v ~P v Q--------------------------(5.4) Isn't this a perfectly valid tautology? Upon analysis it turns out that it is not. Let's look at it from the point of view of Information Theory. Assume that somebody in the other room has one nickel and one quarter and periodically flips them. The probability that the result of flipping one coin will be heads is 0.5. Let Pa be the probability that the flipped nickel will show heads, P~a the probability that result will be tails. If your friend in the other room tells you that the result of flipping the nickel is heads, you have received the information of 1 bit. Ia = Pa * lg(Pa) + P~a * lg(P~a) = 2 * 0.5 * (lg 0.5) = 1 where lg is a logarithm of base 2. Let A be the message "flipping the nickel resulted in heads", and B the message "flipping the quarter resulted in heads." If Pa, Pb are the a priori probabilities then the a posteriori probability P'b after receiving the message AvB can be calculated as P'b = Pb/(1 - (1-Pa)(1-Pb)) The information the message "A v B' carries about B depends on the a priori probability of A. One extreme case is such that that the probability of A is zero. Let Pa = 0, Pb = 0.5, then P'b = 0.5/(1 - (1-0)(1-0.5)) = 0.5(1 - 1*0.5) - 0.5/0.5 = 1

22

E.g. if the message is "I won one million dollars or flipping the quarter resulted in a head" you have received almost 1 bit of information about the quarter The other extreme case is when Pa = 1: P'b = 0.5/(1 - (1-1)(1-0.5)) = 0.5(1 - 0*0.5) = 0.5*1 = 0.5 If you already know that the result of flipping the nicked was heads and then you receive the message "A v B" you have not received any information about the quarter. The information received about the nickel can be expressed as Ia = 1 + lg(P'b) + lg(1-P'b) The dependency of the information about B depending on the a priori probability of A is shown in the graph below Pa

Ib Figure 5.2 The information the message "A v B" carries about B rapidly approaches zero as the a priori probability of A increases. 23

Logic arguably is more fundamental science than information theory, and we are not attempting to base the former on the later, but the preceding paragraphs help to illustrate our point. One particular example when A has probability 1 is the case of A being

necessarily true. The message "today is not tomorrow or the quarter came up heads" will be true whether the flip results in heads or tails. Therefore the sentence transmits no information no about the quarter. The string "the quarter came up heads" alone carries the information of 1 bit. When it is concatenated with "today is not tomorrow" with the 'or' connective it carries zero information, and hence is meaningless. We observe that in (P v ~P) v B--------------------------((5.5) the term "P v ~P" completely masks the truth value of B. If we further use table 5.1 below we conclude that the entire formula () is meaningless, which is what t-relevant logic predicts.

P

Q

P &Q

PvQ

F

F

F

F

F

M

M

M

F

T

F

T

M

F

M

M

M

M

M

M

M

T

M

M

T

F

F

T

T

M

M

M

T

T

T

T

Table 5.1 24

Bibliography [1] Philosophy in the Twentieth Century A.J. Ayer Vintage Books, New York, 1984 [2] Presupposition, Implication, and Self-Reference Bas C. van Fraassen The Journal of Philosophy, Vol. 65, No. 5 (Mar. 7, 1968), pp. 136-152 [3] Topics in the Logic of Relevance M. Richard Diaz Philosophia Verlag, 1981 [4] Mr. Strawson on Referring Bertrand Russell Mind, New Series, Vol. 66, No. 263. (Jul., 1957), pp. 385-389 [5] On Referring P. F. Strawson Mind, New Series, Vol. 59, No. 235 (Jul., 1950), pp. 320-344 [6] Introduction to Logical Theory Strawson, P.F. Methuen, London, 1952 [7] Tractatus Logico-Philosophicus Ludwig Wittgenstein, Routledge & Kegan Paul, 1978 [8] 'This Statement Is Not True' Is Not True Laurence Goldstein Analysis, Vol. 52, No. 1 (Jan., 1992) http://www.jstor.org/pss/3328873 [9] Pointers to Propositions Haim Gaifman Department of Philosophy, Columbia University, New York, NY 10027, USA http://www.columbia.edu/~hg17/gaifman6.pdf

Copyright © X.Y. Newberry 2008

25