mathematical logics vs philosophical logics

[gusl]

From a comment I just wrote to jcreed.

Another thing I asked Pfenning about was the proper interpretation of sentences in Linear Logics (it might as well have been about non-monotonic logics, or para-consistent logics (thanks to quale for the reminder)). My inclination would be to say that these are "logics" only in the mathematical sense, not in the philosophical sense (i.e. In what I call "philosophical logics", sentences are about real truth. In particular, excluded middle and monotonicity hold.). But when we talk about agents' beliefs, we are in an intensional context, and these two no longer need to hold.

If we claim that sentences of such logics are meaningful, then we should be able to translate them into sentences in philosophical logics, e.g. temporal logics, by jumping out of the agent, and into an outsider's "objective perspective". But I don't see anyone bothering to do this.

For an illustration of what I mean:

While non-monotonic logic can model an agent's belief revision, we know that sentences in this logic are not to be judged as modeling truth. When we see a pair of sentences like:
X |- Z
X, Y |/- Z

we know that |- can't possibly refer to truth (afterall, truth is monotonic). Instead, |- must refer to the agent's beliefs and reasoning processes. Furthermore, this formalism is vague about what refers to the agent's beliefs about facts, what refers to the agent's beliefs about what inferences are valid, or whether the agent's inferences follow this logic blindly, without reflection.

Therefore, if we want to use a true philosophical logic, we should write something like:
B( B(X) ||- B(Z) ) (agent believes that: belief in X, in the default case, justifies belief in Z)

B( B(X) /\ B(Y) ||/- B(Z) ) (agent believes that: belief in X, when accompanied by belief in Y, in the default case, does not justify belief in Z)

Real reasoning involves reflection. Logicians often don't care enough about reflection.

Comments

[jcreed.livejournal.com]

Here is a basic question that I've wrestled with: why do we know that excluded middle holds for "real" logic? If the argument is something like "that's just part of our definition of true and false" then what makes bivalent logic so canonical if it arises out of human language?

Changing my opion a bit.

[quale.livejournal.com]

So as I already said I mostly agree with you. I think the general notion of a logic is something that given a set of truths or justified beliefs generates other beliefs that are true or justified.

Thinking more about it I can't fully rule out para-consistant logics on this ground as they claim to generate true statements from true statements, these statements just sometimes turn out to be false as well. Since I think false just means meaningful and not true, i.e., describes the way things aren't, I think this notion is silly but of course if you buy the suppositions of para-consistant logic it is a logic in the philosophical sense.

Also I think I might grant non-monotonic logics as genuine philosophical logics as well. I think it is reasonable to call an inductive logic a logic, that is a logic that determines what you should deem to be true given certain observations. It would of course be non-monotonic as you can later get evidence that undermines a previous conclusion.

I think including things like inductive logic as logics is a perfectly reasonable use of the word and it is just an uninteresting semantic debate as to whether a logic should only refer to things that are guarnateed to preserve truth or also includes things that let you draw justified conclusions from true premises.

I still agree about your point about these linear logics if they work like I think they do. I don't really know enough about them but I understand them to be something that models non-logically omniscent belief. If on the other hand it really is supposed to capture valid inductive reasoning I think my conclusion would have to change.

Also I don't quite understand your point about reflection. What diffrentiates logicians from mathematicians is exactly that logicians tend to care about reflection. The heart of our subject is diagnolization arguments and considerations about interpreting the system in itself. However, being a logic doesn't require reflection. propositional logic doesn't do reflection but is surely a logic.

[quale.livejournal.com]

Yes that is just part of our definition of true and false.

A statement is true (roughly) if it describes things the way they are. A statement is false if it describes things the way they are not.

As to why this choice of definition is so useful and pervasive it is just because we want to know how the world is and isn't. Given a statement it is important to know if that statement accurately describes the way things are or if it doesn't and from that the concepts of true and false spring naturally.

Or to put it another way you are just asking why is it really useful to have a concept that means 'is not the way things are' or 'meaningful but not true'. This seems totally obvious to me but I'm not sure it can be further justified without begging the question.

Re: Changing my opion a bit.

[gustavolacerda.livejournal.com]

I think it is reasonable to call an inductive logic a logic,

My point is to formally distinguish defeasible from non-defeasible entailment. This is not always made very clear. I am not very fond of logics missing a non-defeasible entailment.

Linear logics model resources. jcreed knows a lot about them.

The sentence about reflection is unrelated to the rest of the post. Bad style on my part.

[jcreed.livejournal.com]

But applying this to mathematics buys into Platonism pretty severely, that not only is there a world where, say, the natural numbers have properties or not, but also that we have succeeded in naming a definite thing in that world when we say "the natural numbers".

Besides which, the idea that language is even centrally about clear propositions is something I'm finding increasingly discredited. Take normative sentences like "cheese is delicious". This certainly isn't a proposition in the bivalent logic sense at least until you qualify it with something like "according to bob". And perhaps further with "on september 1, 2006". And perhaps further with "if you don't remind him of gouda, specifically, which he hates". I'm not sure this process of framing (much like the process of arbitrarily selecting one of the goedel sentence G or its negation ~G in finding a complete (but not R.E.) axiomatization of some extention of PA) ever ends in a finite amount of time.

[easwaran.livejournal.com]

All these features might be dealt with by the appropriate theories of context-sensitivity in semantics. The sentence "cheese is delicious" doesn't express a proposition by itself any more than "he is John" does - they both only express propositions when uttered in a context. Somehow the context of utterance (and, if you believe John MacFarlane, Brian Weatherson, Andy Egan, and others, the context of assessment) help specify the proposition, which is then capable of being true or false.

[jcreed.livejournal.com]

So when I see a string like "there exists a natural number such that property P holds of x":

with a more charitable attitude, I don't know what proposition this expresses.
with a less charitable attitute, I don't believe it expresses a proposition.

compare this to "there exists a group element x such that property P holds of x". Certainly you can see here that it matters which group I am talking about, and it's not a proposition until I tell you which one. I (am at least arguing as if I) feel the same way about the natural numbers N as I do about the concept of group.

Re: Changing my opion a bit.

[jcreed.livejournal.com]

can you explain how you are using "defeasible"?

I should point out that linear logic usually also includes ordinary entailment, so it is better described as a logic plus an ill-behaved (linear) implication, rather than a logic lacking a well-behaved (ordinary) implication.

Re: Changing my opion a bit.

[gustavolacerda.livejournal.com]

I'm using "defeasible" in the ordinary way, in the context of criticizing non-monotonic logics, NOT linear logics.

I don't know anything about linear logics.

To me, the RightThing would be to use a philosophical logic (with ordinary entailment) for talking about this instrumental logic (linear entailment). When you have them in the same language, it would seem that you are mixing meta- and object- languages.

What are your thoughts?

[easwaran.livejournal.com]

Then how do you count things?

To me it seems plausible that counting uses of numbers have meanings expressed by numerical quantifiers, but that all existential quantification over numbers (or other mathematical objects) expresses a proposition that turns out to be false, because there are no such things.

I don't quite know what proposition most mathematical claims will end up expressing. But there are so many gaps left in our understanding of semantics for so much of language that this is unsurprising. We don't really quite know what proposition _any_ sentence expresses, but we have good ideas (and they generally turn out much more complicated than one would have thought, once all the tense markers and stuff are included).

Re: Changing my opion a bit.

[jcreed.livejournal.com]

I don't know what the "ordinary" way is. I tried looking up the definition, but all I found was "not logically valid, but temptingly plausible", which doesn't make sense to me given how you used it. Linear logic is nonmonotonic, by the way. A |- A, but not A, B |- A.

I'm not inclined to believe in the distinction between "philosophical logics" which are actually correct in describing the real world, as against "instrumental logics" that somehow don't.

[jcreed.livejournal.com]

Hm, that is a good question. I believe that whenever I encounter a quantifier-free statement in PA with no free variables together with a proof, and I investigate the proof and believe it, then the corresponding set of actions involving (say) arranging little lines rectangles of pebbles will lead to equally big piles of pebbles.

I might ask you, how do you analyze permutations? to which the answer is "using the axioms of groups". You don't need a complete theory to deduce useful things. PA doesn't tell you everything about sets of pebbles (whether they are gray or blue) but it does tell you many things about counting them.

[quale.livejournal.com]

But applying this to mathematics buys into Platonism pretty severely, that not only is there a world where, say, the natural numbers have properties or not, but also that we have succeeded in naming a definite thing in that world when we say "the natural numbers".

Not at all. The exact same story works for a fictionalist. We tell a story about numbers and math is about working out what is true of them in the story. The notions of true and false work here for exactly the same reason the notions of true (in the story) and false (in the story) are applicable in a fictional detective story.

Taking an if-thenism approach to mathematics also justifies the use of true and false in the regular way. On this account mathematical statements are all just a shorthand way of saying 'If the axioms were true then blah would also be true' Thus these statements can be literally true or false even without any platonism about the mathematical objects themselves.

Heck we don't have a problem even if we are formalists. As a formalist we are just working in a syntatic system of our choosing and all that needs to be explained is why choose this particular system. Well since we are interested in applying the results of math to the actual world (counting, physics what have you) we should use a syntactic system that has a nice correspondance to the things we want to say about the external world.

--

As to your other point I think this is just trading on an ambiguity about what 'about' means. Besides, I'm not very convinced by any argument of the form 'x is increasingly discredited.' Unfortunatly in philosophy this often means x is simple, well understood and doesn't lead to interesting papers. So it's possible there is some convincing reason to believe this but I will withhold judgement until I hear a compelling argument.

However, I don't need the claim that language is primarily about propositions. Only that true and false are primarily about propositions. This distinction is readily illustrated by asking questions like, "Is it true that the moon is made of green cheese or you are a human." People will tend to answer slowly, "Well I guess it's technically true" even though they would never assert it.

Your argument merely shows that their is no map between strings of words and propositions. Of course this is true, context matters as well. Nothing in this does anything to suggest that a particular instance of the statement "cheese is delicious" in a paritcular context doesn't express a proposition. Also generalizations like this don't correspond to the proposition Ax (cheese(x)->delicious(x)) so the fact that he doesn't like gouda or doesn't like rotten cheese isn't a problem.

Also I don't see a process here at all. There is a correspondence between assertions and propositions (sometimes not fully determined when people don't have something clear in mind but good enough for our purposes). Even if no finite list of information about the context would be enough to fully settle what proposition was uttered wouldn't show there wasn't some particular proposition uttered.

Since the issue of context creates confusion we try and create our mathematical language so as to minimize context dependence. Since mathematical sentences correspond to unique propositions (by whatever is the right philosophy of math) once the formal system they are being deployed in is understood we can talk about mathematical sentences being true or false without creating confusion about what propositions we are endorsing as we would in natural language where we must call things true or false inside a context.

[jcreed.livejournal.com]

what? If we "tell a story" by providing some axioms, then some statements neither follow from nor are refuted by the axioms. Bivalent truth and falsity goes immediately out the window. In a detective story, what status does the sentence "the detective's hat was red" have if the story never says what color it was?

Without platonism, I can see how things can be true, and how they can be false, if true and false have to do with proofs in a formal system, but this doesn't get you that every sentence is true or false.

The "language is or isn't about propositions" bit was very poorly worded, sorry. I mean just to say that the idea that "the One True Logic of the real world (i.e. "philosophical logics") is founded on propositional truth", something that I feel gustavo is arguing for (assuming I haven't misinterpreted him) is something that I used to believe, and now I think not only do I not believe it, but I suspect that his distinction between philosophical and instrumental logics is not workable. I don't have a good argument for this, just a hunch.

[easwaran.livejournal.com]

Why do you need all of PA? Won't the quantifier-free fragment suffice? Or, Robinson's Q, which is sufficient for Godel's theorems.