Logic And Reality
Gil Sanders of Walking Christian has responded to my article on logic, math, and presuppositionalism. While his objections are far more well thought out than the non-responses of presuppositionalists, there are a few points I need to clarify about logic.
First, he takes classical logic as something metaphysical. This is fine in a general sense, but as I have mentioned in my previous article, classical logic doesn’t always match up perfectly with reality (here I am taking metaphysics as a discipline to be something which attempts to describe the features of reality). I cited relevant logic as an example of this – material implication just does not capture reality all the time* (hardly an obscure problem!).
Gil writes, “A language could even create a contradictory syntax and strangely arrive at some ”coherent” function but at that point, it just gets ridiculous. If that’s all “conditional logics, relevant logics, paraconsistent logics, free logics, quantum logics, fuzzy logics” do then I think they’re useless in telling us what reality is.”
He is actually touching on an important point here – not all logics are useful. Elsewhere on my blog I’ve mentioned Douglas Hofstadter’s MU puzzle, and said that it’s possible to define a logic with only one theorem. While we can do this, there’s really no utility in doing so. But there are some nonclassical logics that are highly useful, and do seem to say something about reality in a sense. The relevant logic example is only one – there is also ternary logic. The database language SQL implements ternary logic, and we’ve even built ternary computers a few times in the past.
A third example is fuzzy logic. I’ll simply say this – I challenge Gil to solve the Sorites paradox using only classical logic.
Now, as for classical logic, I do think it is useful at describing reality – most of the time. I see classical logic as analogous to regular, “everyday” English, with nonclassical logics being analogous to the various technical jargons of different fields. This doesn’t seem exactly right to me, but it’s the best analogy I can think of. One might say that technical jargons are merely extensions of everyday English, but I don’t think this is the case, because everyday English includes many slang terms not acceptable in technical fields; thus there is some exclusivity.
Now, this article is admittedly a bit short, and doesn’t touch on every point. This is intentional. I could write pages and pages and pages about logic, with hundreds of citations, but that really won’t get us anywhere. Rather, this is just to continue a back and forth discussion about logic with Gil, and I would suggest that we focus specifically on the paradox of material implication and the Sorites paradox, and the failure of classical logic to explain these things well.
*addendum: this can be seen in the truth table for p → q (http://en.wikipedia.org/wiki/Material_conditional#Truth_table). When p = True and q = True, p → q = true. This leads to natural language problems where “If the sun is a star, then cheese is made from milk” is true. But this obviously doesn’t follow. For this to make sense, there needs to be a causal connection that classical logic just can’t capture.