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?