Should Logic be Metaphysics-first or Metaphysics-free?

What exactly does logic do? (Is “do” the right word? Maybe “study”?) Do logicians study how propositions relate to each other, or what happens to truth under certain conditions? If the former, it seems we can carry out logic without any metaphysics, i.e. metaphysics-free. For instance, we can presumably do logic even if there are no actual propositions (a bizarre, but possible world). If the latter, we must say something about truth and how it is that logic can even say anything about logic at all, which can require some heavy-duty metaphysics, i.e. metaphysics-first. Note, that, it seems these approaches can either correspond or diverge. An introductory course in first-order propositional logic (i.e. “normal” logic) presents logic as talking about the truth of propositions. From a metaphysics-free perspective, we can carry out this kind of logic without really thinking about truth a whole lot. But from a metaphysics-first perspective, we first must say some things about truth: every proposition is only true or only false and nothing else (principle of exhaustion, excluded middle). Then, for a proposition C to follow from a set of propositions G means it is not possible for C to be false and for G to contain no false propositions. In this case, we see a sort of convergence between the metaphysics-first and metaphysics-free approaches. But this clearly need not be the case. Consider that the truth of some statements is ambiguous: This sentence is false. This classic example, the liar’s paradox, can either be answered by denying that this is really a proposition or by saying something different about truth or logic.

By my lights, the former is more popular. I won’t say this is a knockdown argument against it by any stretch of the imagination, but consider that it is probably far from obvious that there is a principled way to determine which statements fail to be propositions. Whether you hold the former or not, it doesn’t actually matter too much for the present discussion. For consider that the latter is a plausible response. We are then faced with another decision: do we change logic or truth? In other words, do we change the rules under which propositions relate to each other or do we change what the notion of truth of propositions is? These correspond to metaphysics-free and metaphysics-first respectively. The natural metaphysics-free thing to do here might be to reject the principle of explosion, i.e. that anything follows from a contradiction. This would change the rules sufficiently to avoid the liar’s paradox from being logically problematic, in that we don’t allow anything to follow if we assert the liar’s paradox. Then sense in which it is still problematic is just that we still want to know whether the proposition is true or false. But the metaphysics-free logician says it’s a problem for the epistemologists to tell us whether the liar’s paradox is true or false (or perhaps both or neither). We might even notice that the metaphysics-free logician can probably avoid saying anything at all so long as he asserts that the problem of the liar’s paradox has to do with truth and not (metaphysics-free) logic. Now a metaphysics-first approach would assign the truth values differently, that is change what it means for a proposition to be true or false. For instance, we might allow propositions to be neither true nor false, and so the liar’s paradox is one such “truth-value gap”. The rules of logic and how propositions relate to each then follow from this metaphysical claim (e.g. FDE). In this instance, metaphysics-first and metaphysics-free diverge: the metaphysics-free logician can maintain classical logic (logic minus principle of explosion), while the metaphysics-first logician adopts a many-valued non-classical logic.

Let us also take a moment to consider some philosophical implication. Metaphysics-free turns logic more or less into a sort of formal game about propositions. There are rules and propositions must follow those rules. There is nothing to ground (in the colloquial sense) logic. This can quickly slip into logical pluralism or nihilism (see Curtis Franks’ paper), which might be an adverse effect. On the other hand, metaphysics-first might “get the wrong idea” about logic: why do we have to do so much metaphysics before doing logic? We don’t have to understand the nature of a number to do mathematics, nor to understand that a certain mathematical claim is true, so why do we have to do that for logic? The clear upside is that logic then says something about truth and falsity, unlike metaphysics-free. Similarly, we might have to commit ourselves to logical monism, which also may be a good thing or bad thing.

Here’s the million-dollar question then: which approach is right? And a follow up: can we determine which approach is “right” without doing any metaphysics? That is: what does it mean for an approach to be right? One last follow up: can we hold both approaches as “right”, and be Carnapian and say which approach is right depends on the context? I’m not certain about the answers to any of these questions. Here’s where my instincts lie at the moment: 1) I lean toward metaphysics-first but I’m quite unsure, 2) we can’t determine which approach is “right” without doing metaphysics, and 3) we can probably be Carnapian about it. My motivation for (1) is related to why I think game formalism about mathematics is incorrect. It just is not the case that logic and mathematics is purely a game for it can say stuff about science and reality far too well to be a coincidence (we might invoke the Quine-Putnam indispensibility argument here). Mathematics and logic are privileged over other games like chess, but I do not believe we can articulate the grounds on which they are privileged. I have reservations because it does seem the case that we must do a lot of metaphysics before doing logic, which strikes me as unintuitive, especially if we compare it to mathematics. My motivation for (2) is that to determine which approach is right (if that’s possible), we must articulate the grounds on which an approach is right, which is already metaphysics. My motivation for (3) is that it seems like different sets of sentences (i.e. theories) can give rise to different logical systems, and in certain “descriptive” tasks, we might want to build up a logic that best describes that particular theory as opposed to fixing ourselves to what contemporary metaphysics says about truth. For example, in theology we might want a little more flexibility in how we allow propositions to relate to each other. We might also want to restrict meta-reasoning (about logic or otherwise) to classical logic not for any metaphysical reason but because otherwise we couldn’t even talk to each other about how we should conduct philosophical research (i.e. metaphysics-free approach to logic in meta-reasoning).

I’m definitely interested to hear what others think. Although I can’t say I’m the most well-read in philosophy of logic literature, I haven’t seen this distinction discussed in what I have read. I do believe it lurks in the background of discussions about logic since Russell.

Bristol, UK

Dec 11, 2025