by Corry Shores

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[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Graham Priest

*In Contradiction: A Study of the Transconsistent*

Part I. The Logical Paradoxes

Ch.1. Semantic Paradoxes

1.2 The Semantic Paradoxes

Brief Summary:

Tarski tried to give a semantic definition of truth for formal languages, and these formal languages would approximate natural languages. There are certain conditions in his system for doing this that lead to a dialetheia (a true contradiction). This is a sort of liar paradox. Priest here shows how Tarski’s conditions necessarily lead to the inconsistency. In the next section, he will argue that they apply to English, thus dialetheias are inherent to both natural and formal languages.

Summary

Previously Priest looked at the logical paradoxes of self-reference, and he distinguished semantic ones from set theoretical ones. Now he will focus on the semantic ones. We noted how these paradoxes generate dialetheias, or true contradictions. Many people are against saying there exists true contradictions. But to maintain that stance, they must show what is invalid in the arguments producing the paradoxes. Priest’s “aim here is to defend the view that the semantic paradoxes are *bona fide* sound arguments.” (10) Priest will first “state a set of conditions sufficient for contradiction and then defend against all comers the view that natural language satisfies these conditions (or if not these, then others which have the same effect).” (10) One benefit from the following is that we will better see that solutions to the paradox fail, and we will also learn a bit as to why they do.

Priest explains that “Tarski (1936) located the root of the semantic paradoxes” in closure conditions. [Recall that a language is semantically closed when within it are contained all its components and generative rules. Such semantically closed languages prove inconsistent, as seen in the liar paradox.] Priest will now give the three Tarski conditions for a formal theory (and for simplicity we concern ourselves just with formulas with a single free variable).

(1) For every formula,

α, there is a term of the language,, its name.α

(Priest, 11)

[Recall again the (T) scheme:

(T)

Xis true if, and only if,p

with the example:

“snow is white” if and only if snow is white

As we can see, the ‘meta-language’ on the right side of the equation contains the formula: snow is white, or instead, *p*. Its name, given in quotes “snow is white” or symbolized as *X*, is the name for that formula, articulated in the object language. Priest uses underline for quotation and α instead of p.]

(2) There is a formula of two free variables, Sat(

x y),such that every instance of the schemeSat(

t) ↔αα(v/t) (*)is a theorem, where

tis any term,αis any formula of one free variable,v, andα(v/t) isαwith all free occurrences of ‘v’replaced by‘t’(with the usual precautions concerning the binding of variables free int).

(Priest, 11)

[Recall the formula and our discussion from the prior section:

x satisfies

α↔ α(y/x)

We said that means,* x* satisfies the formula named α (that is, makes its value true when substituted into that sentential function) if and only if we can rightly substitute* x* in for the free variable *y* in the formula having that name. Let us look again at Priest’s formulation: “There is a formula of two free variables, Sat(*x y*)”. In the above, we said there was one free variable in α, which is *y. *But given the construction, which seemingly means, *x* satisfies *y*, this might be equivalent to: *x* satisfies __α __. And thus the ‘y’ here might stand for the formula name __α __. So again:

Sat(

t) ↔αα(v/t)

Here, *α *is a formula with one free variable, *v*. Then *t *is the term that substitutes for *v*. So again our formulation says, *t *satisfies the formula named * α *(that is to say, the formulation with that name is true), if and only if

*t*can be rightly substituted into the actual formulation.]

(3) The rule of inference {α ↔ ¬α} ⊢ α ∧ ¬α is valid in the logic underlying the theory.

(Priest 11)

[It seems simply that the third condition is that there is a particular rule of inference involved. Consider if we have a formula if and only if we have its negation. From this it is provable (we may infer) that we have both that formula and its negation. I am not sure how better to understand this. Using logical equivalences (as I find them on this wiki page) we might go about it like this:

α ↔ ¬α ≡ (α → ¬α) ∧ (¬α → α)

{from the first logical equivalence of biconditionals:

p↔q≡(p→q)∧(q→p)}

α ↔ ¬α ≡ (¬α ∨ ¬α) ∧ (¬¬α ∨ α)

{from the first logical equivalence of conditionals:

p→q≡¬p∨q}

α ↔ ¬α ≡ (¬α ∨ ¬α) ∧ (α ∨ α)

{double negation law: ¬(¬p)≡p}

α ↔ ¬α ≡ ¬α ∧ α

{from the idempotent laws: p∨p≡p}

]

Now Priest will show that “any theory which satisfies the Tarski closure conditions is inconsistent.” (11)

[So recall again our formulation in step 2:

Sat(

t) ↔αα(v/t) (*)

It seems we will make a complicated sort of substitution. So we first consider a formulation like

vdoes not satisfy itself

or perhaps,* *

vdoes not satisfy formulav

We would write it:

¬Sat(

v v)

This would be the *α *formulation. Then *t *would be the name for this

“vdoes not satisfy formulav”

And we would write this:

¬Sat(v v)

Then we make our substitions into

Sat(

t) ↔αα(v/t)

and we obtain

Sat(

¬Sat(v v)¬Sat() ↔ ¬Sat(v v)¬Sat(v v) ¬Sat(v v))

I am not sure how to make this more conceivable with an example. I would think that it should be something like:

“This sentence is false” is true if and only if this sentence is false.

Let us begin with:

“

vis white” is true if and only ifvis white.

Now, we use satisfaction.

tsatisfies “vis white” if and only iftcan be substituted forvin the formula with this name (vis white).

Instead of ‘*v *is white’, we have ‘*v *does not satisfy *v*’. So

tsatisfies “vdoes not satisfyv” if and only iftcan be substituted forvin the formula:vdoes not satisfyv.

Then we substitute “*v *does not satisfy *v” *for *t.*

“

vdoes not satisfyv” satisfies “vdoes not satisfyv” if and only if “vdoes not satisfyv” can be substituted forvin the formula:vdoes not satisfyv(hence making: “vdoes not satisfyv” does not satisfy “vdoes not satisfyv”)

Thus we have

“

vdoes not satisfyv” satisfies “vdoes not satisfyv” if and only if “vdoes not satisfyv” does not satisfy “vdoes not satisfyv”.

Now consider again:

This sentence is false.

Since it is self referential, it is equivalent to saying in our formulations,

This sentence does not satisfy itself.

When we put the liar sentence into the (T) scheme using our different term for satisfaction, we then get:

“This sentence is false” is true if and only if “this sentence is false” is false.

]

Priest then discusses some other issues that complicate the matter [see page 11], but concludes: “the point is shown: these closure conditions give rise to contradiction.” (11)

Now consider the English language, and ask if the three conditions hold for it. It would seem they do. However, since English is not a formal language, the jargon “formula”, “term”, “theorem” and so on will not apply to it, since they only apply to formal languages. Yet,

Still, it is easy enough to rephrase the conditions while retaining their spirit. A natural language satisfies the Tarski conditions iff:

(1) For every phraseα, there is a noun phrase, its name.α(2) There is a phrase Sat, requiring two noun phrases to be inserted to make a sentence, such that every sentence of the form

Sat(

t) iffαα(t)is true, where a is any phrase requiring a noun phrase,

t, to be inserted to make a sentence, and parentheses mark insertion.(3) The following rule of inference is truth preserving:

αiff it is not the case that aα:

Hence,αand it is not the case thatα.

(Priest, 12)

Thus “We can now proceed, exactly as before, to establish that any natural language that satisfies the Tarski conditions contains true sentences of the form ‘*α* and it is not the case that *α*.’ ” (11) And so “A natural language which satisfies the Tarski conditions therefore contains true contradictions.” (11) Next Priest will turn to the issue of whether or not natural languages can satisfy these Tarski conditions.

Citations from:

Priest, Graham.

*In Contradiction: A Study of the Transconsistent.*Oxford/New York: Clarendon/Oxford University, 2006 [first published 1987].

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