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 PresuppositionsThe 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 LogicIn 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 sub-

formula 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 t-

relevant 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) (2.15)

7

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--------------------tFigure 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| . . . . . . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . . . . . .--------------------------------------------------xFigure 4

The presupposition is false; ~(Ey)((Ex)Fxy & Gy) therefore (Ex)(Ey)(Fxy & Gy) is neither true nor falsey|| . . . . . . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . . . . . .| . . . 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| . . . . . . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . . . . . .-------------------------------------------------xFigure 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 ParadoxBas 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 ValuesWe 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."

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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"

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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

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(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

A v B 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

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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.

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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 P v Q

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

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Bibliography[1] Philosophy in the Twentieth CenturyA.J. AyerVintage Books, New York, 1984

[2] Presupposition, Implication, and Self-ReferenceBas C. van FraassenThe Journal of Philosophy, Vol. 65, No. 5 (Mar. 7, 1968), pp. 136-152

[3] Topics in the Logic of RelevanceM. Richard DiazPhilosophia Verlag, 1981

[4] Mr. Strawson on ReferringBertrand RussellMind, New Series, Vol. 66, No. 263. (Jul., 1957), pp. 385-389

[5] On ReferringP. F. StrawsonMind, New Series, Vol. 59, No. 235 (Jul., 1950), pp. 320-344

[6] Introduction to Logical TheoryStrawson, P.F.Methuen, London, 1952

[7] Tractatus Logico-PhilosophicusLudwig Wittgenstein, Routledge & Kegan Paul, 1978

[8] 'This Statement Is Not True' Is Not TrueLaurence 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, USAhttp://www.columbia.edu/~hg17/gaifman6.pdf

Copyright © X.Y. Newberry 2008

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