Archive for the ‘logic and modality’ Category

Liar paradoxes, a problem with reductio proof and speech acts

June 20, 2013

It’s easy to mistake paradoxical sentences for liar paradoxes. “If this sentence is true, then it is false,” is a liar paradox. If the sentence is true, then the antecedent is true. If the antecedent is true, then the consequent must be false, the implication as a whole is false, so the sentence must be false. So if the sentence is true, then it is a contradiction and a falsehood. So the antecedent must not be true. If the sentence is false, antecedent is false, and the implication as a whole is true.

“If this sentence is false, then it is true,” however, is not a liar paradox. If it is false, then the antecedent is true and the implication fails, and the whole is false. If the sentence is true, then the antecedent is false, the implication holds, and the sentence is true. That’s not a paradox, it’s just a sentence the truth of which cannot be determined. It’s like the sentence, “This sentence is true.” Is it true or false? How could you tell?

Similarly, “The sentence I am now writing is true,” is indeterminate. “The sentence I am now writing is false” is provably a liar paradox, athough one could ask of these two sentences “true or false of what?” The deductive proof that yields a liar paradox of the latter, is a reductio: assume the sentence is true, you deduce that it is false; assume it’s false, you deduce it’s true. So if it’s true, it’s false and vice versa. But if you ask “true of what?” then you’re asking for an empirical answer — does the sentence corresponds to something, in this case to its own truth. Is truth a thing that can be pointed to? If it’s a correspondence with something, we’re stuck in an infinite recursion. So these sentences, on the one hand, lead to a questioning of the correspondence theory. But they also lead to questioning of the validity of deductive reductio argumentation, not unlike that questioning of the reductios that led Cantor to multiple levels of infinities, and the intuitionist rejection of the reductio in favor of proof by demonstration. Several directions from here: you can say these sentences don’t correspond to anything; or correspondence is not complete; or correspondence, even with its incompleteness is a better option than reductios that lead to liar paradoxes.  (more…)


non epistemic possibility and the lay of the land

May 31, 2012

The peculiarity of classical notion of possibility is that it has a relation to the actual world as well as a relation to the irreal world of conditions counter to the actual and the epistemic world of certainty and uncertainty. Lukasiewicz’ notion of possibility seems to apply only to uncertainty — it seems essentially epistemic.

So here’s the lay of the land, as I see these two modal programs: (more…)

possible, but not necessary

May 30, 2012

[This post has been updated for clarity.] Łukasiewicz supposed that the actual is necessary (if I have no coins in my pocket, then it is not possible that I do have a coin there) and that possible implies possibly not. I want to contest both of these. There is good reason to distinguish the actual from the necessary — the earth revolves around the sun is an actual fact, but that the sun is the center of the solar system is necessary (on the grounds that “solar” means “sun”). But if the earth does revolve around the sun, and it’s not possible therefore that it doesn’t revolve around the sun, then isn’t the earth’s revolution around the sun necessary (Łukasiewicz’ not-possibly-not)? So hasn’t he leveled a useful distinction? (more…)

third place

May 13, 2012

Is there a place for a third truth value?

There are two uses for a third truth value: one is the ontological trash bin; the other uncertainty. They are worlds apart.

The trashbin is filled with sentences like “The present king of France is bald” (Russell’s example), “Colorless green ideas sleep furiously ” (Chomsky’s), “The a book left,”  “Did you stop beating your wife?” asked of someone who never beat his wife or doesn’t have a wife. Generally, these sentences fail in some presupposition: there is no king of France, nothing can be both green and not, a thing can’t be designated as the particular and not particular,  you can’t spot beating your wife if you never did.

Dumping these sentences into a third category of neither true nor false seems harmless to me. It doesn’t have any adverse consequences for logic and has the advantage of dealing simply with presupposition failures. If such a sentence is neither true nor false, its negation with also be false. If there were an operator (like negation which takes a true sentence into the other truth value: false) that took true sentences into meaningless sentences, then logic would have to be restructured with far greater complexity. But since there is no such operator — there are too many ways for a presupposition to fail — there’s no reason to worry about this third value. It is logically inert.

The other strategy — using the third value for uncertainty — seems to me appealing in some ways, but has many adverse consequences. It levels necessary truth to mere truth. It’s also not clear whether the category of uncertainty means simply possibility or epistemic possibility. And it leaves the imagination bereft of conceptual fancy — possible worlds like ours, but different.

That’s not to say that there aren’t problems with conceptual possibilities — Nelson Goodman and David Lewis troubled at length about how such worlds can be consistently imagined. Bivalent modality in possibly worlds has a gross failure in that all necessary truths seem to have the same possible world meaning — true in all possible worlds. Possible worlds are too coarse-grained to distinguish those truths. (But leveling all necessary truths is not as bad as leveling all necessary truths and mere truths together.)

Looking at Kratzer’s lumping problem, or the ill fit of the material conditional with natural language, I get the impression that logic is in its infancy. Beginning with Frege there has been rapid innovation in logic. Unlike technology, which responds with increasing rapidity to a fiercely competitive market, logic hasn’t found its market value, so its progress will seem slow in comparison — unless someone can discover a logical structure that solves AI challenges better than the Aristotelean models.

possible or not

May 10, 2012

Is it possible to swim the Atlantic?
An ex neighbor points out that if “possibly” doesn’t imply also “possibly not” then how is “possibly” different from necessity? Doesn’t “Life on Mars is possible” mean “It’s also possible that there’s no life on Mars”? And doesn’t it also mean “Life on Mars isn’t necessary”?
Grice gave an answer to this question, and I’ve written about it elsewhere in this blog, but I think there’s more to be said and I want to try to sort all of them out.
Suppose Goldbach’s postulate is possibly true. Suppose someone proves it. Now it’s necessarily true. Is it no longer possibly true?
My neighbor says no. I think Lukasciewicz agreed with him. (more…)

language and logic, English and Aymara

January 5, 2012

Following up on the last post —

If English were incapable of expressing ternary logic, then Aymara could not be described in English. It does not seem too contentious, then, to conclude that any natural language, like English, Spanish (Rojas’ original) or Aymara, can express any known logic. The means will no doubt differ: analytic languages would express notions of uncertainty with word-morphemes (like “might”); agglutinative and inflectional languages should express them with affixal morphemes.

The notions seem to drive the languages, not the other way around. (more…)


January 4, 2012

I see that Wikipedia’s article on ternary logic references Aymara “a Bolivian language famous for using ternary rather than binary logic.” Aside from the vaunted adjective “famous,” I am skeptical of the claim that Aymara uses a ternary logic rather than binary, skeptical also of the presupposition that natural languages use a particular logic rather than another logic, and skeptical as well of the implicature that other languages use only binary logic, infamously or otherwise.

Much of the excitement over Aymara derives from a monograph written  in the 1980’s by an engineer and machine translation pioneer, Ivan Guzman de Rojas, who observed that Aymara indicated in its inflections the degree of certitude of its respective assertions. He takes these as logical operators, just as “not” can be taken in English as a negation operator: in English, “not” takes a true statement into a false one, and a false one into a true one. E.g.,

snow is white =>True;  snow is not white =>False

snow is green =>False; snow is not green =>True

Aymara, however, also uses an inflection that takes a true statement into a neither-true-nor-false statement. This shows, he claims, that Aymara uses a third truth value, neither-true-nor-false, which is used for uncertainty.

He says, further, that the ternary logic allows the Aymara people to derive logical conclusions that are not available in binary logic, and that the Aymara people think differently from people who are limited to binary logic.

It may already have occurred to the reader that English does have exactly such an operator, “might”:

snow will fall;  snow will not fall; snow might fall

that is, “might” takes an assertion or its negation into an uncertainty.

Does this mean that English has a third truth value? Well, yes and no. (more…)

mixing speech and non contextual logic

October 29, 2011

Geach and Horwich both criticise Strawson’s performative analysis of the truth predicate in English on grounds that seem to me not only mistaken, but a common mistake, really a category error, conflating the speech situation with logic uncontextualized — the logic you study in college. Not just a common mistake, it’s just about everywhere. Here’s their argument:

If Strawson is right that the truth predicate is some kind of gesture of agreement, then accepted deductive arguments will fail, e.g.,

1. Phil’s claim is true

2. Phil’s claim is that snow is white

3. conclusion: snow is white.

If (1) means that the speaker is agreeing with Phil’s claim, then there’s no deduction from the speaker’s agreement to the fact in the conlcusion. At best, the argument concludes that the speaker agrees that snow is white. But even that wouldn’t hold, since agreement, like belief, is probably an opaque context — imagine a speaker who always agrees with Phil whether he knows the details of what Phil thinks, not knowing that Phil has been deceived by someone and believes something that the speaker knows is false.

The Horwich argument is itself a fallacy of equivocation. Assertions in logic are distinct from logical assertions in speech. Speech is always contextualized, so no facts about the world can ever be deduced from it, except the facts of the speakings and believings and, following Strawson, the gestures indicating attitude. To get from speech to facts, you have to move on up to a meta-assertion like, <If S said x and x is true, then what S says is in fact true of the world> where the words “and x is true” is not spoken by any speaker, but sits in a noncontextual world of assertions about the world.

The proper form of the argument in speech goes like this:

1. (I assert/believe) Phil’s claim is true = (I assert/believe) I agree with Phil’s claim

2. Phil’s claim = snow is white

3. (I assert/believe) snow is white is true = (I assert/believe) I agree that snow is white

Now (1) is still an opaque context, so the conclusion might actually not fit with the speaker’s actual prima facie beliefs, but this is exactly the right argument to show why the context is opaque. It’s armed with this argument that you confront the speaker who doesn’t believe that snow is white but who does believe whatever Phil claims, and explain that there’s something wrong with his assumptions about Phil’s claims. That is to say, this argument shouldn’t arrive at any better conclusion than the one it does.

There is, I think, vast confusion about this. You simply can’t shift from speech to non contextual logic. One is necessarily opaque, the other transparent. To derive facts from speech beyond the facts of the speech and it’s speaker, you need a metalanguage.

Now, you might ask whether there is such a thing as non contextual logic — aren’t all statements spoken or written by some speaker or writer. The answer is, that’s solipsism and should be assumed and forgiven. Alternatively, add a hypothetical meta-metalanguage beyond the solipsistic language. It’ll be speculative — “suppose there is a noncontextual logic and it says…”.

Wallace’s solution

April 9, 2011

I’m a little uncomfortable reading Wallace’s book since it was a youthful work not intended for publication, was never published while he lived, and is being published now without his permission. And he left no later comments on it and can’t respond to critics.

Wallace’s solution depends on a scope difference in an alethic tensed modality. Using Taylor’s example: “if the battle occurred, then the admiral must have ordered it” is ambiguous between

1. if the battle occurred, then yesterday it was the case that the admiral must have ordered it

2. if the battle occurred, then it must have been the case that the admiral ordered it

Wallace admits that (1) entails fatalism, but points out that (2) doesn’t. According to (1), the admiral must have ordered the battle, and so had no choice. In (2) the admiral ordered it, but not under duress, as it were, of necessity (must). He had a choice — he might have contemplated several possible worlds in which he orders and several in which he declines to order. It’s just that none of those possible worlds turn out to have been real. That is, yesterday’s world in which the battle was ordered, turns out to have been the only possible world.

But if that’s the only possible world, why is it the only? Wallace seems to show successfully that the answer cannot be the logic alone.

Suppose you are at the moment of ordering. That moment excludes any moment in which you decline to order. That moment includes only moments in which you order the battle. The difference seems to be between whether you have free will or whether you have freedom. Wallace’s draws a nice distinction between fatalism and a kind of post hoc determinism.

Is this a difference without a distinction? If the admiral knows that the moment determines his order (he has no freedom), what does it serve him to have free will? Nothing in the real world. But that accords with our experience: no matter how we plan for the world, the consequences are beyond our ability to control.

The utilitarian/consequentialist effects of determinism and fatalism are equally discouraging. But the entailments for (Kantian)  moral sensibility are completely distinct. Determinism is consistent with holding moral sentiments. Not a fatalist, and that’s why even philosophers spurn it.

On the other hand, while Wallace has found a distinction, I’m not sure that it is telling against Taylor’s view. Relations of necessity among physical effects depend on circumstances, and these are explicit in Taylor’s assumptions. If there is no world in which the order for battle is not given, then the only possible worlds are those in which he chooses to order it. That entails a strange paradox: he is free to choose, but he can only choose one option, not the alternate choice.

How the logic of implication works entirely depends on how you set up the modal system — its axiomata or its inferential rules or both — and its consistency. What makes a system meaningful, assuming it’s consistent, is its usefulness or accuracy. Wallace uses our natural language notions of “it couldn’t have happened” and “it can’t have happened.” That’s good for his system, but not telling against Taylor, since Taylor is specifically using logic against natural language notions which, he is attempting to show, are wrong. And on the other hand,  Wallace’s distinction seems to violate our linguistic, and maybe real-world, understanding of “free.” It may be that the logical syntax should include an inference from

must yesterday order


yesterday must order

or it may be that the inference should be dealt with in the semantics, in the model — in any world in which there is only one option, there is no free choice.