Archive for the ‘pragmatic’ 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…)


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…)


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…”.

Saving Grice’s theory of ‘and’ (with Kratzer lumps!)

June 11, 2007

I’ve always considered Grice’s theory of conversational implicature to be one of the most beautiful theories around. But nowhere is beauty so tightly yoked to truth as in the sciences, where beauty, in the form of simplicity, will decide the truth of two otherwise equally powerful theories. (It’s kind of remarkable when you think about it — truth and simplicity seem not only distinct, but unrelated, unlike say, truth and accuracy or consistency. A complex theory will cause more complexity in its relation to other theories, but if it’s still true, why should complexity ever matter? Is preference for simplicity just a bias?) Truth seems to be a necessary condition for the beauty of a theory in science, so if Grice’s theory isn’t true, its beauty all is lost. The application of conversational co-operation gets messy at and, impugning its truth. I’ve got an idea on how to clean up the mess and restore the symmetry of the structure.

Grice’s analysis of “and” goes like this:

Sometimes “and” is interpreted as simple logical conjunction

1. I brought cheese and bread and wine.

The order of conjuncts doesn’t change the meaning: I brought bread and cheese and wine; wine and cheese and bread; bread and wine and cheese; wine and bread and cheese; it’s all the same. This use of and is symmetric, exactly like the logical conjunction &: A&B<=>B&A

But sometimes and carries the sense of temporal order, “and then”

2. I took off my boots and climbed into bed.

(I think I got this example from Ed Bendix some years ago)

This conjunction is not symmetric: taking off your boots and then climbing into bed is not the same as climbing into bed and then taking off your boots, and the proof of the difference, you might say, comes out in the wash.

The difference in meaning, according to Grice, arises from the assumption that the speaker would not withhold relevant information or present it in a confusing form. If the order of events matters, the order of presentation will follow the order events, unless otherwise specifically indicated. So if I said

I climbed into bed and took off my boots

you’d be justified in surmising that I’d come home very late and very drunk.

The theory of conversational implicature avoids the undesirable circumstance that there might actually be two homonymic “and”s in English, one meaning “&” and the other meaning “and then.”

A problem for Grice was observed long ago by Bar-Lev and Palacas (1980, “Semantic command over pragmatic priority,” Lingua 51). They noted this wonderful minimal pair:

3. I stayed home. I got sick.
4. I stayed home and got sick.

If Grice is right, (3) should mean

3′. I stayed home and then got sick.

But it doesn’t. It means

3″. I got sick and therefore stayed home.

Now unless we are willing to say that the sentential boundary is a morpheme with meaning, we are compelled to drop Grice. Worse still, even though (3) means (3″), the sense of “and then” returns immediately we add “and” between the sentences. (4) means

4″. I stayed home and then I got sick.

even though that’s semantically unexpected. So it’s not about semantic bias, this violation of Grice’s principle. It’s a very real problem that Bar-Lev and Palacas pointed out.

So what’s with “and”?

Here’s my suggestion.

a. In order to use “and” you’ve got to be introducing something new. Think of Angelika Kratzer’s lumps of thought: you’d never say “I painted a portrait and my sister” if you’d only painted one portrait and it was of your sister. Information is structured in clumps of truths that the logical connectives don’t respect. Yes, a portrait was painted and a sister was painted, but if these two things were accomplished in the same act of painting a portrait of one’s sister, then they are in some sense the same fact, though two truths. Now notice the difference between :

“I painted a portrait. I painted my sister.”

Could be the same event. Not so easy to get the same-event interpretation from

“I painted a portrait and I painted my sister.”

The and implies a distinct, newly introduced fact not lumpable with the antecedent event.

b. Causal relations are internal to an event.

Put (a) and (b) together and you have an explanation for (3) and (4). I have a good deal more to say about this, but it’s really nice out, and I’ve been in all day. (more…)