Tag: abstraction operator

The Curry Paradox, Existence, and Bivalence (5) intuition restored

Bas van FraassenLeave a comment

The basic intuition about how talk about properties should be understood, expressed in our symbolism, would be with all of the following holding:

(b ε λxFx) is true if and only if Fb is true,

(b ε λxFx) is true if and only if the referent of b is a member of the referent of λxFx

Fb is true if and only if the referent of b satisfies the formula Fx

But the constraints on our classical valuations guarantee only that the referents of b and of λxFx are either members or subsets of the domain of discourse. For the first language, L, is not designed to discuss properties, though its syntax has the resources to start doing that, which is then utilized in the super valuational language LS, so as to achieve expressive completeness.

Can we do more than this, and show that when the assignment of truth values is by a supervaluation, it is possible to have (b ε λxFx) true only if the referent of b is a member of the referent of λxFx? That is required, for success on both the semantic and proof-theoretic levels.

I will now argue that the answer is yes, if the super valuation is induced by an ideal set constructed as in the preceding post.

Let us now consider a chain C of sets of sentences X1, X2, …., XK, …. where:

X1 contains no sentences with λ or ε in them

For each number n > 1, Xn+1 is a direct descendant of Xn

We know from the preceding that the union of this chain, UC, is satisfied by some classical valuation φ. We can now go through the chain, and alter φ at each stage, without affecting the assigned truth values, as follows:

There is a first member of the chain in which (b ε λxFx) appears, namely when it is added as a new cognate that enters in the transition from Xn to Xn+1 (the only way it can come in). And that is the case only if Fb is a member of Xn and hence assigned T by φ. Thus φ(b) satisfies the formula Fx. Let Y be the set of members of the domain of discourse that satisfy Fx, and change the assignment φ((λxFx)) to Y. That will not alter anything in the truth values which φ has assigned so far, since that term λxFx has not appeared at any preceding stage, and there are no general constraints on what φ can assign to a term of that sort.

With all these changes made to φ, and all the stages in the construction of good chain C, the result is a classical valuation φ* which satisfies UC and is such that for every formula (b ε λxFx) in UC — which is, every such formula that is assigned T by the induced supervaluation — φ*(b) is a member of φ*((λxFx)).

There are certainly more questions, more nuanced considerations, about the super valuational language LS and its use for talk about properties, that deserve to be taken up.

But we can claim this: just as Curry would have liked, LS is expressively complete (he would have said “combinatorially complete”) in the sense that

(t ε   λxFx) ╞  Ft

Ft  ╞  (t ε   λxFx) 

and deductively adequate, in the sense that all the classical tautologies, and the validity of modus ponies, are preserved. And we have added to this that for the super valuations whose construction we could make explicit, this was not just a matter of truth values of sentences but that

(b ε λxFx) is true only if the referent of b is a member of the referent of λxFx

Basta! è sufficiente per ora …

NOTE. These posts are to be superseded by an article, in which the gaps and shortcomings of this informal reasoning are redeemed by careful definitions and proofs. I will add notes here when an update is available

The Curry Paradox, Existence, and Bivalence (2) The argument

Bas van FraassenLeave a comment

Curry’s Paradox showed the incompatibility of two desiderata for a language, expressive completeness and deductive classicality.  (These are my terms, and correspond fairly closely to Curry’s “combinatorial completeness” and “deductive completeness” in his 1942.)  I’ll define these desiderata as minimally as is possible while preserving Curry’s argument. 

The prevalent reaction to Curry’s Paradox has been to be accept a lack of expressive completeness.  I’ll present a language for which expressive completeness is saved, with very little sacrifice of classical logic.  All classically valid arguments will remain valid, though certain natural deduction rules do not carry over.

(My idea is to introduce a super valuational language, in which it is typical for classical tautologies and classical valid arguments to be preserved, while ‘meta-rules’ like conditional proof are blocked. The importance of super valuations is not merely that they show the non-equivalence of the principles of Excluded Middle and Bivalence, but that they show the non-equivalence of axiomatic formulations and natural deduction or Gentzen formulations of classical logic.)

The language L

This is a language I’m setting up as a stepping stone. Its syntax will include expressions for properties, but its semantics will not follow suit. The idea will be to draw on this language to construct a better language.

Let L be a language with formulas, constants, predicates, and variables.  Terms are constants, variables, and expressions that are formed from formulas by means of the term-forming symbol λ (if A is a formula then λxA is a term). We call λ an abstraction operator. L has a binary connective  ⇒. Moreover there is a special relation symbol ε, read as “has”; if t and t’ are terms then (t ε t’) is a formula.  In the next post I’ll give a few more details, but this is all that we need for our present purpose, which is to present a formalized version of Curry’s Paradox argument.

Note about the symbolism: such an expression as λxλyRxy is not well-formed: λx turns sentences into terms, it does not turn terms into terms. But there is something in the neighborhood: the following are intuitively speaking equivalent:

Rab

a ε λxRxb

b ε λy(a ε λxRxy)

I say ‘intuitively equivalent’: Curry’s paradox shows that we can’t just blindly say they are all equivalent without running into a danger of self-contradiction. But these expressions are well-formed and show how many-place predicates can be handled in the ‘copula’ notation, in principle.

L has a family of admissible interpretations, which I will specify below. Their details are not needed for the argument, but we rely on the usual notion of argument validity:

X ╞ B in L iff all admissible interpretations of L that satisfy all of the members of X also satisfy B

(we include as limiting case X empty). 

The putative desiderata for valid deductions

L is expressively complete iff both

(t ε   λxFx) ╞  Ft

Ft  ╞  (t ε   λxFx) 

We introduce the notion of deductive classicality in two steps, since we want to see what can be salvaged.  Let L have a binary connective .

L is deductively adequate iff   modus ponens is valid, i. e.

  X, A, A⇒ B ╞ B

L is deductively classical iff it is deductively adequate and in addition conditional proof is valid, i.e. 

if  X, B  ╞ C then X ╞ B ⇒ C

Curry’s argument

Curry’s Paradox argument shows that L cannot be both expressively complete and deductively classical. 

For brevity  let π be the expression λx[( x ε x ) ⇒ A],  where A is any closed formula you like.

  1.          Assume π ε π
  2.          π ε λx[( x ε x ) ⇒ A]   from 1. given what “π” abbreviates
  3.          (π ε π) ⇒ A                   from 2. by expressive completeness
  4.          A                                  1., 3.  modus ponens
  5. (π ε π) ⇒ A                            1. – 4. conditional proof
  6. π ε λx[( x ε x ) ⇒ A]             5. by expressive completeness
  7. π ε π                                       from 6.  given what “π” abbreviates
  8. A 5., 7. modus ponens

This constitutes a proof of formula A, for any closed formula A, hence this language has a completely trivial ‘valid argument’ relationship, with every formula a tautology if it is deductively classical.

First task for the next post in this blog: to say precisely what are the admissible valuations of language L. I will do so in a way that makes L deductively classical.

Given Curry’s argument, it is clear that, whatever these admissible valuations are, if L is to be a viable language at all, then L will not be both expressively complete and deductively classical.

Indeed, we had better keep L as minimalist as possible, before the desired finale — which is to use it as stepping stone to a more satisfactory language which is, as Curry wished, expressively complete.