u/StrangeGlaringEye

It is well known that every logic, i.e. a structural Tarskian consequence relation, is characterizable in terms of some family of matrices, namely its Lindenbaum matrices. Three years ago a user on Maths Stack Exchange asked whether there are logics that cannot be characterized by a single matrix rather than a family thereof. That is a natural question. Maybe by some artful combination of the matrices in some characteristic family?

Since nobody seems to have answered their question, I thought I’d do that here. The answer is that there indeed exist logics not characterizable by a single matrix. I shall give an example and show that that is the case for it.

First, recall that if a logic ⊢ is characterized by a single matrix, it is then uniform, meaning that whenever

(1) Γ U Δ ⊢ α,

(2) Δ is consistent w.r.t. ⊢ (meaning it doesn’t ⊢-imply all formulae), and

(3) Var(Δ) ∩ Var(Γ U {α}) = ∅,

then

(4) Γ ⊢ α.

Exercise for the reader: show this! Hint: suppose ⊢ has a characteristic matrix, and suppose (4) is false. Can you construct a Δ for which (1)-(3) are not all true?

Now let B be the 2-valued Boolean algebra on {T, F} with, say, infimum (conjunction) and complement (negation). Let B(T) be the matrix consisting of B plus {T} as its designated set, i.e. the natural matrix for classical logic, and B(F) mutatis mutandis with designated {F}. Let ⊢ be the logic characterized by the pair of these matrices.

(Notice that since the logic characterized by B(F) alone is, so to speak, the logic of preservation of falsehood, at least to the extent that classical logic is the logic of preservation of truth, then ⊢, as their intersection, is the logic of preservation of any uniform value the premises may have. It is the logic of equivalence!)

Observe that we have

(5) p & ~p ⊢ q & ~q

in this logic. But, {p & ~p} is actually consistent w.r.t. ⊢, since any valuation onto B(F) assigns it designated. Moreover, premise and conclusion have no variables in common. Hence, if ⊢ were uniform, we should have

(6) ⊢ q & ~q,

putting ∅ = Γ. But, we do not, since any valuation onto the classical matrix refutes (6) of course. Hence, by the above result, ⊢ has no characteristic matrix.

Logical truths are often held as paradigmatic necessary truths. But there’s a droll example of a very simple contingent but plausibly logical truth. (If we assume that logical truths are analytic, this also counts as a contingent analytic truth, and hence a counterexample to Kripke’s supposition that analytic truths, whatever they are, are at least necessary and a priori.)

The truth involves an operator of actuality. We evaluate “Actually P” as true in a world W just in case “P” is true in the actual world; its value in any world depends only on its value in the actual world, and nothing more. So even in a world where “P” is false, “Actually P” comes out true if “P” is actually true.

Now consider the material conditional “If actually P then P”. A simple informal derivation using the excluded middle shows this is true.

Proof. Either “P” is true in the actual world, or it is not. If “P” is true in the actual world, then any instance of “If Q then P” is true, by the rules for the material conditional. But “If actually P then P” is one such instance. And if “P” is not true in the actual world, “Actually P” is false in every world, and so in particular the actual world. Therefore, again by the rules for the material we have the truth of every instance of “If actually P then Q”, and hence “If actually P then P”. QED

But this conditional is contingent. For take “P” to be actually true, and move to a world W where “P” is false. (Consider, say “P” = “Socrates is wise”, and take a world W where Socrates or his counterpart is foolish.) By the rules of the material conditional, “If actually P then P” is false in W, given that by hypothesis “Actually P” is true in W but “P” is false.

The informal derivation we gave compels us, I think, to hold “If actually P then P” as a logical truth. Hence, it advises us to adopt a local, as opposed to global, possible worlds semantics, at least for statements involving the actuality operator. In the global semantics, a set of statements Γ entails a statement α just in case for any frame F and world W in F, if all of Γ are true in W then α is true in W. But in a local semantics, every frame has a distinguished element @, the actual world; and the semantics is that Γ entails a statement α just in case for any frame F, if all of Γ are true in F’s @ then α is true in @. The former semantics is easily seen to entail that logical truth requires necessity; whilst the local semantics does not.

In a fascinating essay called “The method of truth in metaphysics”, Davidson argues that we can do metaphysics by analyzing language. His argument is simple. It proceeds from the thesis that communication is only possible if interlocutors share a largely correct view of reality. This, we can concede, implies that we can read the general features of the world off the general features of language.

And how does the defense of that thesis, that communication requires a largely correct common view, go? Well, Davidson argues that communication requires at least a broad common view. We cannot interpret each other unless we ascribe certain background beliefs to one another, and we cannot do this ascription unless we square those background beliefs with our own. I can only make sense of what you’re saying against the backdrop of my own beliefs, of what I think there is and what it is like. Since communication is, we might also concede, successful mutual interpretation, it does seem to follow that communication requires a background of certain beliefs.

Then, in a curious move, Davidson claims there’s nothing incoherent in the idea of an omniscient interlocutor. I could be speaking right now to one, we might think. But by the first part of the argument, that interlocutor would have to share many of their beliefs with me; and by hypothesis all of their beliefs are true. Hence, so must be many of mine.

Arguments in philosophy often have less the goal of dispelling disagreement and more of refining it. Davidson probably will not convince the skeptic of armchair metaphysics, but it will bring attention to some unexpected commitments. The skeptic faces a dilemma: either communication can happen between two people in massive disagreement; or the very idea of an omniscient interlocutor is nonsensical.

It is worth pointing out, however, that perhaps there is something suspicious in Davidson’s strategy. It doesn’t make sense of deep metaphysical disagreement. Evidently idealists and materialists, say, can communicate with one another; but they offer radically different conceptions of reality. If Davidson is right, therefore, one of them doesn’t just have to be wrong; they must not sincerely believe what they claim. Metaphysicians, as the disagreeable bunch they are, largely consists of liars.

The Kalam argument runs thus:

1. Everything that begins to exist has a cause
2. The universe began to exist
3. Therefore, the universe has a cause

Now this is just the beginning of a long case for the ultimate conclusion that

4. There is a God

But the initial argument has generated enough debate on its own to be of interest. One line of defense for premise 2 runs more or less as follows:

5. There cannot be an actual infinite

6. If the universe never began to exist, there would be an actual infinite

From which 2 follows. But it does seem intuitive that 4 is inconsistent with 5; surely if there is a God there is an actual infinite. Surely if there is a God, She has an infinite mind of infinite wisdom and infinite goodness and infinite beauty etc. And in all these characterizations, “infinite” means actually infinite, not just potentially infinite.

I conclude that this particular strand of the Kalam case is self-refuting. It is a Kalamity, if you will, albeit a very small one.

Robert Sokolowski suggested that philosophy essentially deals with distinctions. Often by drawing them; but sometimes by obliterating distinctions, arguing that they are in some sense illusory or unhelpful.

Quine is a notorious distinction obliterator. The analytic-synthetic is not his only victim; he also argued that we shouldn’t distinguish between existence and “mere being” or “subsistence” in the way Meinongians did. Or, that we should not distinguish between different ways of existing. The way in which a table exists is not a different way in which an abstract object like the empty set exists, Quine says. The difference is just in their nature, the kind of object they are; not presumed “modes of being”.

And of course, Quine is not the only philosopher to have erased distinctions other philosophers have tried to draw. Many of them will have accused their opponents of having drawn ad hoc distinctions in an attempt to dodge counterexamples. Physicalists think we shouldn’t really distinguish the mental from the physical, because the former can be subsumed under the latter. Some people think we shouldn’t distinguish moral normativity from other kinds of purported normativity, like aesthetic or epistemic normativity: they think there’s one kind of normativity, and moral, aesthetic, epistemic norms etc. all express it. With regard to different subject matters of course; but the normative force is the same. Much like truth is the same across different domains (or so most philosophers would think). The truth of “Snow is white” is the same truth of “Blood is red”, although these statements concern different parts of reality. The list goes on.

Now the reason I’m talking about this is because there is a distinction many people often appeal to when thinking about free will which it may be best to discard. Namely, between incompatibilist and compatibilist kinds of free will. We can think of this, to start things off, as a distinction between two sorts of concepts or notions. There’s the compatibilist notion of free will and the incompatibilist notion of free will. Often this distinction comes with a shocking number of further assumptions. For instance, many people will not just distinguish a “compatibilist notion of free will” from an “incompatibilist” or “libertarian” notion; they will say something about what these notions are supposed to be. Say, that the compatibilist notion of free will consists in doing what one wants to do; and that this contrasts with the libertarian notion of free will, which consists in something like being an unmoved mover, or not behaving deterministically, or whatever.

My suggestion is that in the very least everyone should think carefully about whether they would really like to make this distinction. I’m not going to argue at length that they shouldn’t; partly because I’m kind of tired of arguing in this sub, but also because I’m not completely confident anymore that this distinction shouldn’t be drawn. But what I will do is suggest that the alternative is viable: we can think of the notion of free will as being a single notion, and that compatibilism and incompatibilism are theses about this very same notion.

Compatibilists say that the notion of free will is the notion of a kind of ability that can be had by agents living in a deterministic world; incompatibilists say that it is not, that there could not be agents endowed with free will in a deterministic world. In either case, the words “free will” mean the same thing; the compatibilist and the incompatibilist are not using these two little words in different senses. They are not saying, respectively, that human beings have free will of this or that kind; the compatibilist is not saying that humans have “compatibilist free will”, and the incompatibilist is not saying that humans have “incompatibilist free will”.

Indeed, it’s not clear at all what incompatibilists as such are saying, by the lights of the “two notions approach”, because some incompatibilists think we don’t have free will, whilst others, namely libertarians, think that we do. In what sense are they then united as incompatibilists, and opposed to compatibilists? Maybe not because of any particular view they defend, but because of what they’re worried about. Incompatibilists are people who are interested in whether or not they have a kind of free will, not necessarily people who endorse a specific position about this question. Or so that seems like the best definition of incompatibilism under the “two notions” approach.

These considerations tie in with the familiar charge that compatibilists are changing the subject or “redefining “free will”” somehow. I’ve argued at length in other posts that we are not, and one reason for that stems from the view that there is a single notion of free will in play. Compatibilists and incompatibilists disagree not necessarily about this notion, but the kind of ability it picks out in the world. Compatibilists think it picks out an ability that could be had by deterministic agents; incompatibilists think that it does not. Who is right doesn’t depend on what sense we choose to assign to the words “free will”. The hypothesis is that it already has a definite sense, and it has picked out a certain feature of agents; who is right depends on what that feature is.

Edit Here we have what seems like an example of the two-notions approach being taken, probably without realization

Let us suppose we’ve two stable distinctions, the a priori/a posteriori, or a priori/empirical distinction, and the analytic/synthetic distinction. Let us not assume any particular construal of these distinctions for now, e.g. that a statement is analytic iff it can be turned into a truth of some suitable logic by putting synonyms for synonyms.

Here’s a view we could call strong empiricism: all a priori truths are analytic.

And here’s an argument against strong empiricism: for each agent S, the truth that S exists is a priori for S. But, no analytic statement implies the existence of anything. Therefore, strong empiricism is false.

Let’s look at the first premise; we may call it Cartesianism. Here’s a further argument in favor of Cartesianism: it is a priori true that everything that has properties exists; for each agent, it is a priori for them that they have certain properties; and a prioricity is closed under entailment. Therefore, Cartesianism is true. (Here I assume the falsehood of hardcore nominalism for the sake of argument. Ideally I’d rephrase this argument in terms of predicates or concepts applying rather than having properties.)

And here’s an argument for the second premise of the above argument for Cartesianism. It is a priori true that everything has some properties; and the schema “If everything …, then I …” should hold in a sound logic of indexicals. Given the entailment closure of a prioricity, the second premise follows.

Now let’s look at the second premise of the first argument, that no analytic truth carries existential implications. Here’s two possible objections: every truth entails that it exists, and therefore carries existential implications. Moreover, it is a truth of first order logic that there exists something; and everything implies every truth of first order logic.

Regarding the second objection, you might guess that my move will be to require that the correct logic be a free logic, free indeed from any ontology whatsoever, even the minimal ontology of first order logic that there is something.

But your guess would be wrong. I am actually going to concede both objections, for the sake of argument, and will ad hoc-ly distinguish analytic from synthetic existential commitments.

Here’s the idea. First, we need a distinction between linguistic and extralinguistic entities; and we also need a distinction between general and particular existential commitments. I’m going to leave the first distinction unexplained because it seems, at least superficially, pretty self-explanatory, although no doubt an able metaphysician will poke some holes in it without having to think for long. But the second distinction can be explained as such: an existential commitment C is particular just in case there is an entity x such that C is a commitment that x, specifically, exists.

Take for example the statement that Socrates exists. Intuitively, this statement is particularly existentially committing to Socrates. Not so for the statement that humans exist. Intuitively, for each human (here we can use possibilistic quantifiers) h there is a world W(h) where h does not exist, and yet that statement is true in W(h), in virtue of the existence of some other human. And I take it that a statement is committed to some entities only if (although not necessarily iff) it is false when those entities don’t exist; here I mean it for both general and particular commitments.

I suggest that an existential commitment is synthetic when it is not particularly committing to any linguistic entities. Again not when and only when. I don’t need the second direction.

Then we get this revision of the first argument: for each agent S, the truth that S exists is a priori for S. The ontological commitment to S is synthetic. And no analytic truth carries synthetic ontological commitments. Therefore, strong empiricism is false.

Why do we love the people we love? It’s plausible that this answer has to have a question, and as a consequence all its instances: Why does Romeo love Juliet? Why does Achilles love Patroclus? Etc. A full-blown principle of sufficient reason is probably untenable; reality probably contains brute facts. But it would be very surprising if among these resided facts pertaining to as parochial a domain as our love lives.

Yet if we are the last romantics, it would seem that the only plausible answers to questions of the above sort would invoke haecceities. (Formally, a haecceity can be defined as the property expressed by a lambda term whose matrix is an equation where one and only one side is a rigid designator. Example: λx.x = r. Informally, a haecceity is the property of, for a certain individual, being that individual: being Socrates, being Plato. Etc.) For it would seem that love cannot depend solely on generic, repeatable qualities like beauty, wit, kindness etc. At any rate that seems to be the thesis of much of our best poetry: “love is not love which alters when it alteration finds”. When one thinks why the lover loves the beloved, it seems the lover would love the beloved no matter what the beloved were like, as long as the beloved were the beloved indeed, and not someone else. Or so it would seem if we are perhaps in love ourselves, and inclined to romanticizing the circumstances.

So let us assemble these considerations into an argument which, if not persuasive, should at least be amusing to some of you; and maybe that’s just as worthy of a purpose for a piece of philosophy.

1) there are true explanations concerning questions of love

2) these explanations carry an ontological commitment to haecceities

3) there are haecceities! QED