Metaphysics and Formal Logic

I find formal logic very interesting and powerful. I have enjoyed the times I’ve been able to teach it in my courses and have benefited from it numerous times in analyzing the validity of arguments. But as a Thomist who is an enthusiast of Henry Babcock Veatch and others, I’ve come to see more and more the shortcomings that the formalization (or schematization) of arguments have in capturing the metaphysics that might underlie any given argument. (In addition to the article cited below, I would recommend Veatch’s Two Logics: The Conflict Between Classical and Neo-Analytic Philosophy (Evanstan: Northwestern University Press, 1969) and Intentional Logic: A Logic Based on Philosophical Realism (New Haven: Yale University Press, 1952. Republished, n.c.: Archon Books, 1970)). Schematizations of philosophical arguments usually only capture the form of an argument, but are unable to capture the matter of the argument (to use Aristotelian categories).

Two quick examples might help. Everyone I know accepts the standard categorizations from Kant of analytic and synthetic propositions. Being an excluded middle, the categorization logically exhausts the options. Either the predicate is included in the subject (All bachelors are unmarried) or it is not (It is raining in Atlanta). There can be no denying the truth of the excluded middle. But what this simple categorization is unable to capture is that, metaphysically, there is more than one way that a predicate may be included or excluded in a subject. In fact, in Aristotle, there are five ways, viz., the predicate may be related to its subject as its genus, its differentia, its definition, its property, or its accident.  [Henry Veatch, “St. Thomas and the Question, ‘How Are Synthetic Judgments A Priori Possible?'” Modern Schoolman 42 (March 1965): 240] Taking advantage of how the excluded middle or certain other logical relationships overlook the metaphysical relationships can be the basis of many jokes. “The temperature is 93 degrees. The temperature will rise this afternoon. Therefore 93 degrees will rise this afternoon.” [John Allen Paulos, I Think, Therefore, I Laugh: An Alternative Approach to Philosophy (New York: Columbia University Press, 1985), 27] It’s funny because the “is” in the first premise connects the subject and predicate in a different way than the (implied) “is” of the second premise. But, (and this is the key) the logical formalization (or schematization) cannot easily capture that difference in its symbols. To be sure, one can introduce more symbols to try to clarify the differences, but then what can happen is that the schematization can become so complicated and cumbersome, that one loses the whole point of schematizing it in the first place, viz., to conveniently analyze the argument. Indeed, I submit that, for complex philosophical arguments, it would be humanly impossible to sufficiently schematize the argument such that all the metaphysics is captured.

A more substantive example would be: consider hearing someone exclaim that he saw his pastor with a woman who was not his wife. Such an exclamation might give rise to scandal. Logically, it is necessarily true that the woman either is or is not his wife (again, an excluded middle). However, there are any number of ways that a woman might fall into the category of “not his wife.” The woman could have been his mother, or sister, or daughter; none of which would give rise to scandal. But exactly how it is that the woman was not his wife cannot be captured in the excluded middle. It’s not that I’m opposed as such to formal schematizations of philosophical arguments. But very often an argument given in a context could be formalized in such a way so that it seems to commit a formal logical fallacy. Many otherwise legitimate scientific arguments, for example, when reproduced into formal, truth-functional arguments commit the fallacy of affirming the consequent (like, “If the substance is acidic, then the blue litmus paper will turn red. The blue litmus paper turned red. Therefore, the substance is acidic.”)

I was recently reminded of this issue when I had a pleasant exchange with a fellow Christian philosopher regarding his post here as to whether an argument by Norman Geisler committed a formal fallacy. Geisler argued:

1. God created all things.
2. Evil is not a thing.
3. Hence, God did not create evil. [Norman L. Geisler, If God, Why Evil? (Minneapolis: Bethany House, 2011), 19]

The critic pointed out (I think correctly, taken in one way) that Geisler’s argument, when cast into predicate or quantificational logic this way, commits the fallacy of denying the antecedent. Thus:

(x) (Tx ⊃ Cgx)
~Te / ∴ ~Cge

which reads

For any (or every) x, if x is a thing, then x is created by God.
Evil is not a thing. / Therefore, evil is not created by God.

Perhaps it is already easy for the reader to see (without showing the formal proof) that Geisler’s argument (again, taken in one way) does, indeed, commit the formal fallacy of denying the antecedent.

My response had to do with whether his first premise (cast in quantificational logic) adequately captured the metaphysics behind Geisler’s first premise. I argued that, in its broader context, Geisler was not merely saying that God created all things (allowing for the sake of brevity here that God did not create Himself, which, again, arises from a broader metaphysical context) but also that everything was created by God. Now, stated as such, it might seem that I’ve only repeated the statement. But, if one casts the premise truth-functionally, Geisler’s first premise is actually a bi-conditional. I argued that Geisler’s argument was this:

T = thing; C = created; g = God; E = evil

1. (x) ((Tx ⊃ Cgx) & (Cgx ⊃ Tx))           Premise
2. (x) (Ex ⊃ ~Tx) / ∴ (x) (Ex ⊃ ~Cgx) Premise/Conclusion
3. (Ta ⊃ Cga) & (Cga ⊃ Ta)                    U.I.
4. Ea ⊃ ~Ta                                               U.I.
5. Ea                                                          Assumption
6. ~Ta                                                       M.P. 4, 5 (5) [dependency on 5]
7. Cga ⊃ Ta                                              Simp. 3
8. ~Cga                                                     M.T. 7, 6 (5)
9. Ea ⊃ ~Cga                                           C.P. 5-8
10. (x) (Ex ⊃ ~Cgx)                                  U.G. 9

The argument would read: Premise 1. For every x, both, if x is a thing, then God created x and if God created x, then x is a thing. Premise 2. For every x, if x is evil, then x is not a thing. Therefore, for every x if x is evil then God did not create x. Premises 3 and 4 are universal instantiations of 1 and 2, respectively. Premises 5 through 9 are a conditional proof (assuming premise 5, deriving premise 8, and then concluding the conditional in premise 9). Premise 10 is a universal generalization of premise 9.

Now, who is to say whether my formalization of Geisler’s original argument is better than that of my philosopher friend? I should note, by way of clarification, that I’m not suggesting that any universal affirmative categorical proposition can be rightfully cast as a truth-functional bi-conditional. Counter-examples are easy to find. I’m only suggesting here that Geisler’s meaning in this particular proposition is better grasp as a bi-conditional (if one insists on casting it truth-functionally). The issue, then, arises as to whether (in any given instance) a formalization adequately captures a given premise. But such considerations can only be addressed by a look at the broader philosophical/metaphysical context in which the argument is given. So, strictly speaking, the former characterization might be fair when one only considers the bare argument in isolation. But that characterization might be inadequate when the argument is considered in its metaphysical context. (I’m not suggesting that the critic deliberately tried to ignore the argument’s context.)

One issue that underlies the problem here is whether a given philosophical argument is truth-functional (or quantificational) in the first place, which is to say, whether casting arguments in such a way would do violence to the argument. There may be something in formal, truth-functional or quantificational logic that is inherently inadequate (if not opposed to) sound metaphysics. In my experience, the almost universal assumption in contemporary, analytic philosophy is that every argument (in principle) can be construed as a simple truth-functional or a more sophisticated quantificational argument. I beg to differ. As Veatch argues in his Two Logics, the very nature of (what appears to be) a “neutral” accounting that formal logic thinks it affords, can be fraught with metaphysical assumptions that invariably affect what seems to follow from the schematizations (as Kant’s analytic/synthetic distinction, while logically exhaustive, masks important and real metaphysical distinctions). This is not to say that I (following Veatch) deny any usefulness of these logical systems. Instead, “a scientific type of knowledge, while impressive in its own right and certainly indispensable, is, or at least ought to be, subordinate to the properly architechtonic knowledge of the more humanistic and philosophical variety” (Veatch, Two Logics, p. 22).

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