share your logical intuitions

Consider the following two sentences:

A follows from 'not-not-A' -- but not intuitionistically.

B follows from 'A and not-A' -- but not relevantly.

Are they contradictory? Perfectly fine? Infelicitous?

(If you don't know why I'm asking, Google grice "cancellation test".)


Daniel Lindquist said...

Sounds fine to me. ("B follows from A and not-A" by itself would irritate me. DNE has never troubled me.)

Also, this post is the second hit for "grice 'cancellation test'".

Greg Frost-Arnold said...

Thanks Daniel.

(Stupid google echo chamber!)

Anonymous said...

I take it you are asking about (eg) the sentence "A follows from 'not-not-A' -- but not intuitionistically" rather than some proper part of that line. If so, then I think there must be some implicit modifier after 'follows' in order for it to make sense. Perhaps "A follows classically..." or even "A follows actually..." Without such a contrast, I don't get what the 'but' is doing.

So, I guess they are both perfectly fine.

Brad Weslake said...

Both sound fine to me.

Colin said...

I think P.D. summed up most of what I wanted to say... I would be very interested to hear what you think is funny or questionable about these sentences, Greg.

Greg Frost-Arnold said...

Since you asked...

I've been thinking (too much?) about Beall and Restall's version of logical pluralism. They claim that pluralism about logic is justified because 'follows from' (or 'entails' etc.) has some sort of indeterminacy. Pluralism is OK because there are multiple equally acceptable ways of making that indeterminacy determinate.

The question I've been thinking about is: what KIND of indeterminacy is this? B&R say it's not vagueness. In a few places, they say that it is ambiguity. However, linguists have developed several diagnostic tests to determine whether an expression really is ambiguous, and 'follows from' appears to fail some of the tests, and the other tests give equivocal results.

So if 'follows from' is (probably) not ambiguous, then what other sort of indeterminacy could it have? (If there's no other kind, then B&R's justification for pluralism would disappear.) One candidate is what we could call 'relativity', for example, 'finished'. Many people have claimed that there's no such thing as being finished simpliciter; rather, you're always finished with this or that. That is, without further information about context or whatever, the sentence 'Mary is finished' is indeterminate. Is she finished with her book, or her meal, or...? So the idea is that the kind of indeterminacy in 'follows from' might be of the same species as the indeterminacy in 'finished.'

Now, the reason I asked about those two sentences in the original post was that if they were contradictory, then 'follows from' could not be like 'finished'. since we can legitimately say things like 'Mercury is big -- but not for a planet.'

And more generally, if the two sentences in the original post were contradictory, then intuitionistic consequence would be part of the content of consequence simpliciter (which would be disastrous for logical pluralism).

Tomáš Sobek said...

I think, the ambiguity of "A follows from B" is a kind of purpose dependency. The logical pluralism is a consequence of the logical instrumentalism.