In Defence of the Finest of Presuppositionalists: A Response to Omni by Chris Matthew

Hello there. I’m writing in response to Omni’s video about classical logic and presuppositionalism. I’m a modest (Framean) presuppositionalist.

Now, I am not committed to classical logic myself. I think that P.F. Strawson (1950) is correct in rejecting Russell’s theory of definite descriptions in favour of an understanding of presuppositions that requires a three-valued logic, such as Bochvar semantics (minimally, bivalence and excluded middle collapses). This is applied to the matter of apologetics by the presuppositionalist scholar Don Collett (2009). I’m not even opposed to paraconsistency for that matter.

However, even so, I’m doubtful that Omni’s video presents a worry for those who maintain classical logic. Let me begin with some brief considerations. First, he’s using propositional logic to make a point when quantificational logic is more appropriate, given he is talking about existence and ∃x should be used. This might be for the sake of simplicity, so I’ll put that aside. Second, the consequents in both of the conditionals presented are oddly phrased and not reflective of the most sophisticated presuppositionalist literature. As such, I will adjust the conditionals in view to the following two propositions:

1. If God does not exist, rationality is possible.
2. If God does not exist, rationality is not possible.
1. ¬G → R
2. ¬G → ¬R

The argument, of course, is predicated on the denial of (1). I grant most presuppositionalists would deny (1) because, no doubt, most presuppositionalists’ knowledge of classical logic is comparable to that of Dillahunty. But as Graham Oppy is wont to say, the responsible practice is to interact with the best version of a particular tradition. The principle of charity would demand as much. Presuppositionalist scholars (Dr Greg Bahnsen, Dr John Frame, Dr Chris Bolt, Jared Oliphint, James Baird, and others) who subscribe to classical logic would say that (1), if interpreted as a material conditional, is vacuously true. So is (2). That’s all there is to it, frankly.

But hold on. Isn’t this a devastating problem for presuppositionalism? If (2) is only vacuously true, then the presuppositionalist’s apologetic contention falls short of being significant or meaningful, we are told.

Two points.

Firstly, Omni seems to be conflating vacuous truth in classical logic with some third truth value that denotes meaninglessness. But of course, classical logic is not a trivalent logic. A vacuously true conditional is a conditional that has been meaningfully determined as true. Classical logicians think that we derive conclusions from vacuously true conditionals with every modus tollens inference! Even the conditional, “If George Bush is a Martian, I am God.” Or are we to think that modus tollens is improper on classical logic? Unless we’re biting on the implausible, there’s evidently some confusion here. Vacuous truth is not meaninglessness.

Secondly, some of these paradoxes of material implication arise from a mistranslation between natural language and propositional calculus (or higher-order languages). Berit Brogaard at the University of Miami explains:

“[A]s many commentators on the web have pointed out, philosophy undergraduate students often mistakenly believe that if you adhere to classical logic, you are required to treat corresponding English expressions accordingly, for example, they believe that you are required to treat the indicative conditional as a material conditional. Sometimes their logic teachers are responsible for inducing this belief in them. In any event, this belief often motivates undergraduate students (and some graduate students) to reject classical logic, or at least it has motivated many of my undergraduate students.” (— “PhilPapers Survey: Classical or Non-Classical Logic?” [2009], Lemmings [blog])

How, then, should the presuppositionalist’s contention be interpreted? I suggest that we interpret (2) as a counterpossible conditional. We can capture this as follows:

2′. ¬G □→ ¬◇R

(2′) is an accurate representation of the presuppositionalist’s contention. (2) is not. For those unaware, □→ is the counterfactual operator. Some useful rules of inference (eg., modus tollens) work with □→, but contraposition does not hold for it. In natural language, (2′) could be read in the subjunctive mood: If God were not to exist, rationality would not (even) be possible. Robert Stern (2000, p. 58) uses this schema to formalise Descartes’ cogito as an example of a transcendental argument (viz., “I could not think were I not to exist”).

Does this reading damage classical logic? It’s not obvious that it does. We don’t need relevance logic, or any other subclassical logic, to make sense of (2′). Resisting a truth-functional analysis of counterfactuals is not a weakness in classical logic. Few philosophers think that our inability to capture imperatives, questions, or alethic modalities in truth-functional classical logic exposes inadequacy on the part of classical logic in doing what it does. Some recent literature explores proposals for how impossible worlds could provide a semantics for counterpossibles like (2′) without rejecting any of the principles of classical logic. See Berto (et al.) 2018; Jago (et al.) 2019; and Weiss (2017). Looking at older literature, some logicians think that we can provide context-sensitive strict analyses of counterfactuals, even if it must be integrated with a pragmatic explanation of how counterfactual antecedents are interpreted non-monotonically.

Does this reading damage the “absoluteness” of classical logic? This depends entirely on what’s meant by absoluteness. After all, even on (classical) logical monism, classical logic is the correct logic for reasoning only given a consistent premise set. (Otherwise, the principle of explosion means that it would be acceptable for anyone to infer any proposition whatsoever from their inconsistent beliefs.) But what’s going on here is that, again, we’re not recognising the best of the presuppositionalist tradition. Not Darth Dawkins or Matt Slick, but the presuppositionalist scholars I had mentioned earlier and more.

Should the absoluteness thereof be construed dialectically—as in, it would be unreasonable to deny the rules of classical logic? This isn’t right because Dr Bahnsen affirms that reasonable people can disagree over, say, the LEM in his article, “Revisionary Immunity.” Should the absoluteness thereof be construed in the sense that classical logic is applicable in every context? This isn’t right because, as I noted above, it seems hardly plausible that our inability to formalise all sorts of sentences in natural language exposes the inadequacy of classical logic. It’s possible to acknowledge this (even to the point of interpreting the ordinary usage of the indicative conditional, “if p, then q”, as being better represented as, “Pr(q | p) is high”) without rejecting any of the principles of classical logic as valid in their appropriate contexts. I even pointed to the fact that we can extend classical logic (non-truth-functionally) to formalise counterfactuals and counterpossibles, as we do with normal modal logic.

What exactly are the best presuppositionalists arguing, then? Here I am including both presuppositionalist scholars who endorse classical logic and those who don’t. The focus is one of either two things.

The first focus is that God is the only satisfactory explanation of the metaphysical character of necessary truths. Anderson & Welty (2011) is representative of this argument, albeit there are other ways to motivate it. There is nothing about this argument that requires a commitment to a particular logical system. In fact, Anderson & Welty make this point explicitly in footnote 10 in relation to the rejection of the LNC: “Furthermore, even though dialetheists reject classical logic, whatever logical laws they advocate in place of the classical laws are typically held to be necessary rather than contingent truths. Even a qualified Law of Non-Contradiction, or an alternative to it, would be taken to hold in every possible world as a law about truths qua truths. As stated earlier, our argument only assumes that there are logical laws; it doesn’t assume any particular specification of those laws, except to insist that some of those laws must be viewed as necessary truths.” These paradigm necessary truths could be propositions about propositions (Anderson’s approach) or, more ambitiously, the structural core of a metaphysics (similar to how Timothy Williamson approaches logic in his in his Modal Logic as Metaphysics).

The second focus is on epistemic norms, as my earlier point about inconsistent belief-sets and explosion hinted at. God is the only satisfactory explanation of epistemic normativity (or epistemic realism, as it is sometimes called). This is somewhat similar to the moral argument and it is defended in Harrison (2018) and Koons (p. 297ff in Koons, Bealer (eds.) 2010) to highlight just a couple of examples.

I hope to have shown, then, that Omni’s video does not present a worry for presuppositionalists who subscribe to classical logic. I exhort you, and this server, to engage with the very best of this tradition and not those apologists on the Internet who rarely put their thought to anything remotely philosophical.