Tag: supervaluations

Moore’s Paradox’ Revenge?

Bas van FraassenLeave a comment

A solution or dissolution of a paradox will typically encounter, a little ways down the road, a new version specially designed to flummox it in turn. Paradoxes live, change, move, and quickly grow twice as many heads as we cut off. It seemed to me that Moore’s Paradox too might engender a new version of itself. I want to see whether my ‘self-transparent believer’ (of earlier posts) can deal with it.

The original paradox

Moore’s original version appeared in his ”Russell’s theory of descriptions”. in the Schilpp volume, The Philosophy of Bertrand Russell (1942, p. 204):
“To say such a thing as ‘I believe he has gone out, but he has not’ is absurd.”

This is explicitly about saying, asserting. Today we address it as pertaining to belief, due to Wittgenstein’s rendering of it in his Philosophical Investigations:

“Moore’s paradox can be put like this: the expression “I believe that this is the case” is used like the assertion “This is the case”; and yet the hypothesis that I believe that this is the case is not used like the hypothesis that this is the case.”

The first part of Wittgenstein’s remark is as important as the second. Someone who asserts (without qualification) that something is the case is thereby also committed to the assertion (or to its veridicality) that s/he believes that it is the case, and vice versa. This is a point about the logic of belief.

From this it follows that the assertion of “I believe he has gone out, but he has not” commits one to the further assertion of “I believe that (I believe he has gone out, but he has not)”. And then we reflect that if someone holds that s/he believes a conjunction then s/he must believe both conjuncts. Relying then on the equation in Wittgenstein’s first part we arrive, in three steps, at the conclusion that s/he is committed to both the assertions “He has gone out” and “He has not gone out”, which is a contradiction.
Thus a person in the situation Moore describes is in a belief state that includes belief in two mutually contradictory propositions, a belief state which is not coherent.

In phrasing the conclusion in this way, I am being very careful, I am only attributing incoherence to the person’s state of opinion, and am careful not to use the more familiar, but more biting, term “inconsistent”. The reason for this is that the statement “I believe he has gone out, but he has not” could be true, there are possible situations in which it is true, although the speaker cannot believe it, on pain of incoherence. There are various technical uses of “inconsistent”, but as normally used, the term implies “not possibly true”. It appears then that we must distinguish incoherence from logical inconsistency.

A previous dissolution of the paradox

In previous posts, called “Study of a self-transparent believer”, I proposed that someone who is entirely correct in his beliefs about what his beliefs are (and whose beliefs are consistent), is automatically precluded from believing anything of form “I believe that A, and ~A”, or “I do not believe that A, and A”. This had much to do with the difference between two consequence relations:

  1. X entails A on pain of inconsistency (X ├ A) if and only if A is true in all possible worlds in which all of X is true.
  2. X entails A on pain of incoherence (X → A) if and only if A is true in all possible worlds compatible with the agent’s beliefs, in which all of X is true.

Here the possible worlds model is taken to have a designated agent, and for each world w what is specified is the set of worlds which are compatible with the beliefs that this agent has in w.

Point One: if A is any sentence and BA the corresponding sentence to the effect that I (the agent) believe that to A, then it is clear that A and BA do not entail each other on pain of inconsistency. But they do entail each other on pain of incoherence: if my beliefs include either, then they must include the other as well:

A → BA and BA → A

Point two: the two entailment relations have something in common: if (A ⊃ B) is a classical tautology then it is entailed, in either sense, by any or all premises. And the rule of Modus Ponens holds for both entailment relations. Tautologies plus Modus Ponens are in effect all there is to classical sentential logic, so:

If X, A ├ B then X, A → B

In view of these two points it follows that if a self-transparent believer had (BA &~A) it its beliefs, it would also have both A and ~A in its beliefs, which is not possible.

So, at least in this limited domain of believers who are right about what they believe (surely a reasonable ideal, as ideals go), Moore’s Paradox is completely avoided.

A disturbing thought: revenge?

Imagine that Peter lives in a house quite far away, and I have no idea about whether he has gone out just now. Neither of the following expresses a belief of mine:

  1. Peter has gone out, but I do not believe that he has gone out.
  2. Peter has not gone out, but I do not believe that he has not gone out.

But of course I realize that either he has gone out or he has not gone out. So clearly, one of 1. and 2. must be true. That I realize full well, and I am ready to say so:

(1 or 2). Either (Peter has gone out, but I do not believe that he has gone out) or (Peter has not gone out, but I do not believe that he has not gone out)

But how can I believe (1 or 2) without absurdity, when I cannot believe either 1 or 2, on pain of incoherence? Have I not landed in an absurdity equally great? I have to admit that it would be absurd to believe either 1 or 2, but do believe, in effect, that one of them is true! This new, apparently paradox-producing argument I will call Moore’s Paradox’ Revenge.

Exploring a way to respond

At first sight we can dissolve the difficulty quite quickly. If I believe A then I will believe A & (B v ~B) because that is entailed in either sense of “entail”. And that in turn entails, in either sense, by several classical logical moves, (A & B) v (A & ~B). So what is the puzzle?

(See Appendix for a little clarification.)

But this response assumes that the explication of “entails, on pain of incoherence” was correct — and that is precisely what is being challenged, if Moore’s Paradox is to have its revenge.

To feel the thrust of the disturbing thought about disjunctions, let’s take an example. A physicist in the 19th century– perhaps it was Phoebe Sarah Hertha Ayrton (see Appendix) — believes a disjunction:

light can only be either a particle stream or a wave in a material medium.

This belief derives from a more basic one: there are only two possible ways of transmitting energy, namely material transport and action by contact. Now we enter the troubled period that runs from the Michelson-Morley experiments to just before Einstein’s solution in 1905. During that period this same physicist comes to believe that the experimental outcomes were impossible if light is a particle stream, and equally impossible if light is a wave in a material medium.

Can she now assert

“Either (light is a particle stream, and I do not believe that) or (light is a wave in a material medium, and I do not believe that)”?

Certainly not! That is what she would believe if she retained the belief in the original disjunction. But how could she retain that belief, and stay coherent? To put it the other way around: she cannot believe that sort of a disjunction (one in which each disjunct is thus about belief) while also denying, of each disjunct, that she believes it.

A diagnosis

What I propose is that Moore’s Paradox does not get its revenge after all. But the putative revenge challenges us to understand the reason why the thought of this disjunctive belief should be so disturbing. The disjunction (1. or 2.) displayed above is part of my belief about Peter. I want to concede that there is an apparent paradox. That disjunctive belief creates the appearance of paradox, and that appearance is well founded in something important to understand, but finally no real paradox.

There is a pattern of inference, involving disjunction, that is valid for entailment on pain of inconsistency, but not valid for entailment on pain of incoherence.

I can put this in two different ways. The first is that if X is consistent, then either X plus A, or else X plus ~A, is also consistent. That thesis is correct if “X is consistent” means that X does not entail a contradiction on pain of inconsistency. It is not correct for entailment on pain of incoherence.

The second way, really equivalent, is to point to the meta-rule

if X, A├ C and X, B├ C then X, (A or B)├ C

which is correct, if ├ stands for the relation of entailment on pain of inconsistency, but is not correct for →.

To see why this pattern of inference is not correct for entailment on pain of incoherence, remember the first part of Wittgenstein’s remark. For the self-transparent believer it is certainly correct that he believes that A if and only if he believes that BA. So, using → to symbolize entailment on pain of incoherence, the following are correct; recall:

A → BA BA → A

If the above pattern of inference were correct for → then we could argue as follows:

  1. A → BA
  2. A → (BA or B~A) from 1 by tautology plus modus ponens
  3. ~A → B~A
  4. ~A → (BA or B~A) from 1 by tautology plus modus ponens
  5. (A or ~A) → (BA or B~A)
  6. (A or ~A) tautology
  7. (BA or B~A) 5, 6 modus ponens

But 7. is not a tautology, nor is it something that every self-transparent believer must believe — there will always be many sentences A such that the agent does not believe either that A or that ~A.

This shows that the move in this putative proof, from 2. and 4. to 5. is an invalid move. It follows by the meta-rule of Disjunctive Syllogism, it does not follow by any sequence of classical tautologies with help of Modus Ponens alone.

So, to put the diagnosis in a nutshell:

We may have the impression that Moore’s Paradox gets its revenge when we start looking at disjunctions that a believer has to believe. That impression will be strengthened by our familiarity with a certain inference pattern which is perfectly valid for entailment on pain of inconsistency. But that pattern is not one that governs, or sanctions rules, for entailment on pain of incoherence. And for a persons’ beliefs, avoiding incoherence is the only constraint to which their belief is subject.

APPENDIX

That A implies (A & B) v (A &~B) is clear enough. But in that passage I implicitly argued from “I do not believe that C and I do not believe that ~C” to “Either (C and I do not believe that C, or ~C and I do not believe that ~C)”. This inference has the more complex form of moving from (A & B) to [(C & A) v ( ~C & B)].

It is important to show that this inference is valid both for ├ and for →, and does not rely on the meta-rule for the Disjunctive Syllogism. This we can do by checking the inference with classical truth-tables. If the conclusion is false then both (C & A) and (~C & B) are false. But since either C or ~C is true, that means that either A or B must be false — and hence that (A & B) is false.

(Phoebe Sarah) Hertha Ayrton born in 1854 was one of the few woman physicists of 19th century Britain, and lived through that heady period that saw the Michelson-Morley experimental anomaly for classical electrodynamics (her field) and the eventual victory of Einstein’s ‘electrodynamics of moving bodies’, as well as his photon theory of light.

By the way, those previous posts “Study of a self-transparent believer” have now been superseded by my article “Logic of a self-transparent believer” Filosofiska Notiser, Årgång 8 (2021) Nr 1: 11–25.  Online in 2020: <http://www.filosofiskanotiser.com/aktuelltnummer.htm&gt;

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.