In Defense of Symbolic Logic: A Response to Peter Kreeft

Peter Kreeft has written an article for Touchstone called ” Clashing Symbols: The Loss of Aristotelian Logic & the Social, Moral, & Sexual Consequences .” The thinking goes as follows: Symbolic logic has eclipsed Aristotelian logic in nearly all philosophy textbooks. This is bad news, because symbolic logic undermines metaphysical and epistemological realism, creating a nominalist culture. A nominalist culture can’t grasp true sexual ethics; in particular, it loses the ability to speak of the nature of sex and marriage.

Now, I’m not a philosopher or a logician, but I do use symbolic logic almost daily to write little bits of computer code. I also find the story of symbolic logic in the twentieth century entrancing: It’s an intellectual adventure on a level with rocket science or genetics, and you can read about it in  Logicomix . So at the outset I’d like to say that Kreeft drastically understates the aesthetic value of mathematics (“the more important the subject matter, the less useful mathematics seems to be”) and logic (“a computer can do symbolic logic”). Symbolic logic and mathematics speak of important things at least in the same manner as the great fugues, or anything else formally elegant. But the main weakness of the essay is that the mechanism by which symbolic logic undermines metaphysical realism and thereby destroys our culture is not made clear.

For one thing, it’s not clear to me what exactly Kreeft means by “symbolic logic.” Sometimes, Kreeft seems to be indicting the whole of analytic philosophy: “By the 1970s, most of the English-speaking establishment had cast in its lot with ‘analytic philosophy’ and the symbolic logic that was its methodological component.” At other times, it sounds like Kreeft is just talking about truth-functional propositional logic. “Symbolic logic is also called ‘propositional logic’ because it begins with propositions, not with terms.” “Symbolic logic has no way of knowing, and prevents us from saying,  what  anything is!” Now, this is clearly true for the very simple version of symbolic logic often taught to college freshmen, but it’s not at all clear to me that it’s true of symbolic logic as a whole. In particular, it seems that the basics of Aristotelian logic are simple to symbolize in a predicate calculus, as I’ll try to demonstrate below. If Aristotelian syllogisms are indeed expressible by means of modern symbolic methods, we may just have a pedagogical problem, which is to say that our current logic textbooks don’t do a good job of teaching everyday reasoning.

Let’s turn to something that Kreeft neglects to provide, namely, an example:

1. Aristotelian syllogism:

Premise 1: “All humans are mortal.”
Premise 2: “Adam is a human.”
Conclusion: “Adam is mortal.”
    Formal premises: “All S is P.” “A is S.”
Formal conclusion: “A is P.”

2. Propositional calculus:

Premise 1: If “Adam is a human”, then “Adam is a mortal.”
Premise 2: “Adam is a human.”
Conclusion: “Adam is mortal.”
Formal premises: P->Q, P
Formal conclusion - modus ponens: Q

3. Predicate calculus:

Premise 1: “If anything is a human, then it is mortal.”
Premise 2: “Adam is a human.”
Conclusion: “Adam is mortal.”
    Formal premises: For all x, H(x)->M(x). H(A).
Intermediate step - universal instantiation: H(A)->M(A)
Formal conclusion - modus ponens: M(A)

Notice that the propositional version (2) has all the weaknesses Kreeft mentions: We can’t get at the content of the propositions. We can’t say much about how or why they’re true; we just have symbols that we evaluate as true or false. But predicate calculus (3) seems to solve that problem: We have a way of representing some terms (“A,” or “Adam”) and things we say about those terms (“H(x),” or “x is a human”). However, perhaps you can see that the predicate calculus does draw the attention away from content and toward logical form, so the tool really might be too powerful for everyday reasoning. It does look to me as if the premises of the Aristotelian version (1) and those of the predicate calculus version (3) express much the same thing, in nearly the same level of detail. I would love to know if I’m missing something here.

How else could symbolic logic undermine metaphysical realism? It’s not clear to me whether Kreeft is arguing that metaphysical realism is inexpressible in symbolic logic or merely easy to avoid. “[The power of abstracting and understanding universals] is the thing that symbolic logic ignores or denies.” Well, does it ignore realism, or does it deny it? This is an important point. There’s a problem with the timeline, too. Symbolic logic was extensively theorized in the early 1900s, but (as Kreeft mentions) nominalism appeared in the 1300s. So it must have been possible to express nominalism by means of Aristotelian logic. There’s a historical question here: Was it difficult for nominalists to make themselves understood? Or can the Aristotelian framework be modified to be anti-realist? We can also ask how modern realists—and they do exist—manage to do their work in the context of symbolic logic. Do they simply avoid symbolization? Or can symbolic logic be used by realists?

And if the symbols themselves aren’t the problem, but rather the presence or absence of metaphysical or epistemological realism, why drag the symbols into this discussion?