## Saturday, February 12, 2011

### Classical Logic, how is it correct?

According to Stewart Shapiro, Classical Logic is first-order (and hence formal) predicate logic, which he describes in some detail (in that SEP entry), having first noted that:
Formal languages, deductive systems, and model-theoretic semantics are mathematical objects and, as such, the logician is interested in their mathematical properties and relations. Soundness, completeness, and most of the other results reported below [in that SEP entry] are typical examples. Philosophically, logic is the study of correct reasoning. Reasoning is an epistemic, mental activity. This raises questions concerning the philosophical relevance of the mathematical aspects of logic. How do deducibility and validity, as properties of formal languages--sets of strings on a fixed alphabet--relate to correct reasoning? What do the mathematical results reported below [in that SEP entry] have to do with the original philosophical issue?
Shapiro goes on to list some possibilities; e.g. perhaps "the components of a logic provide the underlying deep structure of correct reasoning."
......Another possibility is that "because natural languages are vague and ambiguous, they should be replaced by formal languages," or rather (since formal languages are all defined using natural languages) "regimented, cleaned up for serious scientific and metaphysical work." Now, scientists often do define their own scientific terms; but how could that process apply to logic? Informal logic must be good enough for us to work out the correct formal language to use, if there is one (otherwise our justifications would become circular). "Another view is that a formal language is a mathematical model of a natural language in roughly the same sense as, say, a collection of point masses is a model of a system of physical objects." Such a view makes sense, in view of the many logics studied by logicians; but what, then, can we say about logic in the sense of correct reasoning? We use such a logic when we use any mathematical model scientifically; we must reason correctly about the model. Is classical logic a good model of informal logic?
......Despite the various logics that logicians work on, most arguments are presented in a classical logical style (even those about other logics). So presumably we do think that such components cut our reasoning pretty close to its joints, so to speak. Is classical logic correct? Some philosophers say so (e.g. Alexander Pruss recently gave that as a reason for rejecting open future views, in comments on this post of his), but I wonder how it is. It's hard for me to specify my worries without having already resolved them; but let's look at some simple logical arguments, such as might be used to introduce classical logic, and see how they may fail to be examples of classical logic (without going too far into the related problems of metaphor and vagueness).
......Grass is green, and all flesh is grass, so, is all flesh green? That's clearly invalid, the second premise being metaphorical; but did we use classical logic to work that out? And suppose I was holding something that wasn't green; could I deduce that it wasn't grass? Again no, because it's not true that all grass is always green. But what if all Blurps were always green; could I deduce that I wasn't holding a Blurp? I don't see why not. And surely I could put that argument in classical logical form. And yet if "is green" is a classical predicate, then not only what I'm holding, but anything and everything is either green or else not green (LEM); whereas there's clearly a shading from green, through greenish and vaguely greenish colours, to those that aren't green. Is it that classical logic is a good model, but that informal logic (real logic) must be slightly different? If so, what's the point of all that maths that Shapiro introduces (rather well)? I'm not saying there's no point to it. (I'm hoping it's not turning philosophy into a pseudo-science.) I'm rather asking whoever's reading this post, what do you think about classical logic?

Sam Alexander said...

I see logic as a tool for obtaining mathematical or even scientific results (c.f. Matiyasevich's Theorem, Hindman's Theorem, Goodstein's Theorem, nonstandard analysis, etc.) To me, the meter stick determining whether a logic is useful or not, is what it can tell us that we would not otherwise have been able to deduce. I guess I'm a pragmatic sort of guy.

Peter said...

I think classical logic is just about consistency. If you follow these rules, you're guaranteed never to contradict yourself. It just turns out that as a handy by-product of the usual rules (you can have more restricted ones if you want), we sometimes find that only one conclusion is consistent with certain premises, etc.

Classical logic certainly doesn't include all sound methods of reasoning.

Alexander R Pruss said...

Great question.

Let me stick to propositional logic for simplicity.

I think the rules of inference of classical propositional logic are correct rules of inference for the logic of propositions, in something like the following example:

If we have a proposition P, and if Q is a proposition that is a negation of P, and if R is a disjunction of P with Q, then R is true. (Excluded Middle)

We can think of a proof as an n-tuple of pairs ((a1,P1),...,(an,Pn)), where in each (ak,Pk), Pk is a proposition, and ak is a marker for the kind of role that pk plays in the proof (for instance: introduction of nth level subproof, step in nth level subproof or premise), and where the the sequence follows the rules of inference of any of the systems of classical logic, etc. For instance, there may be a rule that says that if you have accessible at line k the propositions Pi and Pj, then you are allowed to have at line k any proposition that is a conjunction of Pi with Pj (conjunction introduction). And there may be a rule that the last line must be a 0th level subproof step.

Validity is a semantic notion: an argument from propositions P1,...,Pk to a proposition Q is valid iff every possible world at which all of P1,...,Pk are true is a possible world at which Q is true.

The right version of the soundness theorem then tells us that if you have a proof, then an argument from the proof's premises to the proposition in the last line of the proof is valid. You can perhaps come up with a completeness result, but I don't think it is going to be very philosophically interesting (it'll have something to do with isomorphisms between propositions).

Formal first-order sentential logic, of the sort that involves a special language with vees and horseshoes and the like, is then a kind of simplified model of a subset of the above propositional logic, with sentences in the place of propositions, and some simplifying assumptions that the propositional logic doesn't need (such as that every pair of sentences has a unique conjunction). Natural language also embodies a subset of the propositional logic via such rules as:
If you have as premises sentences u and v, and if w is any sentence that expresses a proposition that is a conjunction of the propositions expressed by u and by v respectively, then you get to conclude w from u and v.

Natural language logic does not have much in the way of syntactic characterizations. (And that formal language logic has syntactic characterizations is just a side-effect of the fact that it has a semantics stipulated to run more neatly parallel to the syntax.)

To extend this to quantified logic, you need what logicians used to call "propositional functions" or what I like to call "schemata" (these correspond to wffs in the artificial language). These might turn out just to be be relations (abundantly conceived).

enigMan said...

I think I should've asked about Propositional Logic, for simplicity (e.g. via that IEP entry). But anyway, my problem is (I think) that while there are interesting psychological models of how we reason, logic is presumably about how we should reason; and I don't see how we could use formal models to find out how we should be reasoning, e.g. about those very models.

Furthermore, since we shouldn't think that Grelling's paradox had been resolved if we simply avoided using the word "heterological" (even though it would be easy not to use that word), so I'm unattracted by non-classical logics (even if they do model aspects of how we reason quite well); so classical logic does seem to me to be how we should reason. (E.g. even if the logic of science is probability, we still need to reason about the various possible worlds, and with the various statements of probabilities.)

However, let's say that "The light is red" is expressing a proposition (e.g. as we approach a traffic-light). Then Excluded Middle gives us that either the light is red or else it isn't. But in some possible worlds the light is not very well described either as red or as not red, but is rather vaguely reddish. Should we say that "The light is red" isn't expressing a proposition? But clearly it is, as shown by its use in such reasoning as Since the light is red (and since one should always stop at a red traffic-light), we should slow to a stop here (which is clearly classical).

Should we say that it's not expressing a proposition in some possible worlds, but does in this world? That lets us keep bivalence (for propositions), but it stops us building possible worlds out of propositions. And what of such propositions as This table is flat? This table is flat, and so I've put this keyboard and such (e.g. a printer) on it, and my reasoning was valid as I decided to do so (e.g. were it not flat, I wouldn't have put such things on it); but in other ways this table isn't flat (e.g. it's not flat enough to write on), or is only somewhat flat.

So I'm wondering, is it that classical logic applies to the elements of truth or falsity in what we say? Are such elements what logicians mean by "propositions"? But then what do we mean by "truth"? I suspect that we mean descriptions so accurate that classical logic applies. But how then could we use classical logic in philosophically difficult cases? We will only know that our premises were true enough (to count as true), for example, if (at least) the conclusion of our valid argument was true.

Of course, I may just be wrong about what classical logic is, or what it's for. But we seem able to reason from a contradiction to some problem with the premises of a valid argument. The argument is classically logical, and so we conclude, perhaps, that the premises were not simple truths. And yet in other contexts they might be true enough (for classical logic to apply to them).

Alexander R Pruss said...

I am more and more inclining to views on which (a) vagueness and ambiguity are the product of a one-many expression relation between sentences and propositions, (b) the primary truthbearers are propositions, not sentences and (c) the rules of inference for the primary truthbearers are classical. I suppose this is a version of supervaluationism.

I also think that persons adept at using the language can in practice tell when the onemanyness of the expression relation is not a problem because in the relevant range of cases under discussion there is no truth-value variation between the expressed propositions and so truth and super-truth coincide. In those cases, we have a local isomorphism between the logic of our language and the logic of propositions. Moreover, in some of these cases, the surface grammar of our language matches the logical grammar of our language, and then classical logic can be safely used. Where either the local isomorphism breaks down or the surface grammar and the logical grammar part ways, the ordinary competent speaker feels puzzled and disoriented, e.g., in Sorites cases and Liar Paradoxes or just in thinking about other silly argument like "Nothing is nothing, and hence nothing is something"; however, she knows that one shouldn't press such things too far, and that one should instead move away from them verbally (e.g., she does not conclude from the Liar that honey cures cancer).

enigMan said...

Many thanks for your input, Alex. I suppose the most obvious (but perhaps the least deep) problem for propositional logic (the real logic of the elements of truth, as I see it) being classical is that classical logic may have no real If A then B, just the so-called material conditional, Not-A or B.

When we reason in an if-then fashion we look at some disjunction of possibilities one term at a time. So "If A then B" often means something like: A is possible, and were A true, the truthmaker for A would also make B true (although that's only a paradigm case, subject to Frankfurt-style counter-examples).

So basically, A has to be possible. (E.g. I get around Curry's paradox by concluding that A was necessarily not true and then revising my first logical steps accordingly.) (In some ways, I suppose that may connect with what I take to be the deeper questions, about what propositions are.) Classical rules of inference may stop that being too much of a problem though.

Alexander R Pruss said...

I don't know that the logic of propositions has any conditional besides the material conditional. I don't think there is any other plausible mind-independent indicative propositional conditional.

Most of the time when we use indicative conditionals, there isn't any truthmaker relationship. "If the sky tonight is clear, I'll be out with my telescope." "If the queen invites me for dinner tonight, I'll be surprised."

enigMan said...

If the sky tonight is clear, I'll be out with my telescope.

So you're inclined to star-gaze if the weather allows, and so the sky being clear lets you act on that inclination? (I do think my suggestion about truth-makers was too simplistic though.)