The Principle of Sufficient Reason

The thoughts that I’ve presented over the last few posts – about possibility and necessity, and their grounding in the causal powers of the things that exist – lead nicely into another discussion: one about what I believe to be a fundamental metaphysical principle, second only to the laws of logic in its significance for the nature of reality. It has usually been referred to, since the time of polymath Gottfried Leibniz, as the Principle of Sufficient Reason.

Roughly speaking, the principle says that everything has an explanation. But different forms of it have had an influence on philosophical thought since ancient times. We can see an echo of it in this saying (which is Latin, but it is attributed to an ancient Greek philosopher):

“Ex nihilo nihil fit.” – Parmenides

That is, out of nothing, nothing comes. Everything has to have a cause.

To be more precise, I believe the Principle of Sufficient Reason (the PSR, for short) is best stated this way: every contingent true proposition has an explanation for its truth. Or, we could say: every contingent state of affairs that occurs in reality has an explanation for its occurrence.

The PSR is restricted to contingent truths – that is, propositions that are true, but could be false – because it isn’t really clear how necessary truths can be explained in many cases, beyond explaining their truth by their necessity. For example, how could we explain why 1 + 1 = 2 is true? It seems like we just have to say something to the effect that it must be true.

However, when it comes to contingent truths, we are very familiar with how to explain them. And it seems to me that explanations for contingent truths are always ultimately causal explanations.

  • We can give conceptual explanations: the metal is hot because its atoms have high kinetic energy.
  • We can give scientific explanations: the atoms have high kinetic energy because energy transferred to them from the electric current passing through the metal, according to the laws of physics.
  • But ultimately, to explain this state of affairs, we need to cite a cause: the energy transferred from the electric current to the atoms because of the causal powers and liabilities of those electrons and atoms (mediated in this case by the electromagnetic field).

And this coheres well with the causal account of modality that I gave in my last post. If contingent states of affairs are made possible by the fact that something has or had the causal power to bring them about, then it is natural to believe that every contingent state of affairs has a causal explanation.

I think we have ample reason to believe that the PSR is not only true, but necessarily true. Here are three reasons why I think so.

Validity of Abductive and Inductive Reasoning

The first argument is that the validity of abductive and inductive inferences – the kind of inferences that we need in order to do science – depends on the necessary truth of the PSR. In inference to the best explanation, we assume that the best explanation for some phenomenon is the one that is most likely to be correct. Or, if we have eliminated all but one explanation, we assume the remaining explanation is correct. Does that remind you of a certain fictional detective?

“When you have eliminated the impossible, whatever remains, however improbable, must be the truth.” – Sherlock Holmes

Holmes’ famous saying, and inference to the best explanation in general, would not work without the Principle of Sufficient Reason. If it is possible that the PSR is false, then when we compare and evaluate different hypotheses for some fact, we have to include the possibility that the fact just has no explanation at all. And when we do that, we lose the ability to say that the best explanation is most likely to be true.

If we try to compare the no-explanation hypothesis (the NEH) to other hypotheses, we are forced to a halt:

  • The NEH postulates nothing at all, so it is simpler than any other hypothesis, and therefore cannot be ruled out by Occam’s razor.
  • The NEH completely lacks explanatory power, of course, but we can only rule it out for that reason if we presume that things must have explanations, which is now in question.
  • The NEH does not conflict with what we already believe unless we already believe that the PSR is true.
  • In fact, without the PSR, we can’t even say that the no-explanation hypothesis is unlikely, so neither can we rule it out for being less probable than other hypotheses.

If some phenomenon has an objective probability of occurring, it is because the scientific laws governing that phenomenon assign some probability to it, or more generally, its probability comes from the properties of things that could cause that phenomenon in an indeterministic way. For example, an electron in a Stern-Gerlach experiment has a certain probability of being deflected up, and a certain probability of being deflected down, because of the laws of physics.

But if a phenomenon has no explanation at all, it cannot be said to be governed by any laws, or caused by anything – otherwise it would have an explanation. But that means that a phenomenon without any explanation has no probability – its probability is not 0, 1, or anything in between. We cannot say that the NEH is probable or improbable, and no amount of evidence for other hypotheses is able to change this fact – that evidence could just have no explanation as well.

So, the possibility of the no-explanation hypothesis ruins any attempted inference to the best explanation. No matter what we do, we cannot get rid of it once it is an available alternative.

Things are similar with inductive inferences (which I think are just a subset of abductive inferences, anyways). The validity of inductive reasoning depends, essentially, on the fact that reality operates in regular, predictable ways. But if the PSR is false, or even just possibly false, we have no reason to expect reality to operate in regular, predictable ways – all kinds of things could be happening for no explanation, so any apparent regularities could just be unexplainable facts, with no guarantee that they will continue to hold.

All this means that our ability to understand reality is heavily dependent on the Principle of Sufficient Reason. In fact, if the PSR is possibly false, it is arguable that we have no knowledge at all – our experiences could all be happening with absolutely no explanation and no connection to reality, and we can’t even say this scenario would be improbable. But since we can know and understand reality through these non-deductive forms of reasoning, we should believe that the PSR is necessarily true.

No Observed Violations

The second argument is an inference to the best explanation (rather than being an argument about inference to the best explanation). The necessary truth of the Principle of Sufficient Reason is the best explanation of the fact that we do not see violations of it all the time – random phenomenon happening with no explanation or cause.

We never observe bicycles coming into existence from thin air, or planets inexplicably altering their courses. Why don’t we see things like this? (Someone might object that we do see events with no explanation, such as in the spontaneous decay of an unstable particles – but I would maintain that those events do have explanations, in the quantum state of the particle and its indeterministic causal powers.)

Can we get away with explaining the absence of these events with some kind of local causal principle that only applies to the ordinary states of affairs that we observe around us, without using the full PSR? It doesn’t look like it. There are problems with local causal principles that suggest that the full PSR is required to explain why we do not observe states of affairs that have no explanation:

  • The laws of physics seem to be valid across the whole physical universe, so if the causal principle is spatially localized, it does not seem like it could rule out the laws of physics changing everywhere for no reason at all.
  • If it is temporally localized, it does not rule out the possibility that the laws of physics have always had, for no reason at all, the disposition to suddenly change at a random point in time.
  • If it is restricted to finite sets of events, it does not seem to rule out infinite chains of events that ultimately have no explanation: for example, event1 that occurred a half second ago, which was caused by event2 a quarter second earlier than that, caused by event3 an eighth second earlier, and so on, but with no explanation for why the whole chain began just shy of one second ago.
  • If it is restricted to states of affairs within the physical universe, it does not rule out the possibility of supernatural immaterial beings coming into existence for no reason, and then proceeding to wreak havoc with the laws of physics.

A quick note on that last point: acts of interference in nature from the supernatural – in other words, miracles – are logically possible as long as it is possible that something supernatural exists. I’ll probably belabour this point again in a later post, but I’ll go over it quickly here.

Scientific laws do not rule out the possibility of miracles. They tell us what happens when only natural causes are operating, and are silent on what might happen when a supernatural cause is operating. I don’t see any good justification for claiming that scientific laws have the built-in implication that nature is impervious to every possible supernatural influence.

Further to that, I don’t see how you can rule out the possibility of the existence of a supernatural cause unless you suppose that no supernatural beings exist, and that the PSR is true to ensure it stays that way. So the last bullet above is indeed a valid problem for a restricted causal principle.

Adding all this up, a merely local causal principle is insufficient to accord with the fact that we do not see inexplicable events happening. We need something with universal applicability.

The philosopher Alexander Pruss has a more in-depth argument along the same lines. He contends that the intuitions supporting a local causal principle (so that we could rule out the possibility of bricks coming into being without a cause, for example) are able to justify a non-local causal principle just as well. Then he goes further, showing that a plausible non-local causal principle actually entails the PSR. (See Section 3 of his paper here. Be warned, it gets pretty technical.)

So, I believe that the Principle of Sufficient Reason is the best, if not only, explanation for why we never observe things like bricks or bicycles popping into existence, or planets randomly changing their orbits: without a cause, such events simply cannot occur, and the entities that do exist are not able or not disposed to cause them.

Wait, you might say, can I really use inference to the best explanation to argue for the PSR now, when my first argument was that inference to the best explanation depends on the PSR? The answer is yes, actually, I can.

The first argument reasons that if the PSR is false, than abductive reasoning is invalid. Then it directly invokes our rational intuitions that say abductive reasoning is valid, therefore it can’t be the case that the PSR is false. We are justifying the PSR by our rational intuitions, not the other way around.

The second argument, on the other hand, indirectly invokes our rational intuitions about abductive reasoning, the same way we do whenever we use inference to the best explanation. Then it reasons to the truth of the PSR from the fact that we never see any apparent violations of it. So, both arguments rely on our rational intuitions, but they use them in different ways, making them neither circular nor redundant.

Impossibility of Exceptions

Rather than being a separate argument, this is more of an extension of the first and second arguments to drive the point home. (So I am not counting it, but it’s an important enough point to have its own section.) There is good reason to believe that there simply cannot be any exceptions to the PSR – either all contingent states of affairs have explanations, or none of them do.

As I stated earlier, it seems to me, very strongly, that all contingent states of affairs are ultimately explained by causes, if they are explained at all. If you haven’t given a cause, you haven’t really finished explaining the phenomenon.

Now, the range of possible effects of some cause are constrained by that cause’s properties, or the situation that the cause is in. For example, I am able to walk from my house to the park, but I cannot fly there. A forest fire can burn trees, but it cannot assemble them into log cabins. The effects that a given cause can produce are constrained by the cause’s nature.

Suppose, contrary to the PSR, that some contingent event or state occurred that did not have any cause or explanation. What, exactly, determined that it was that event in particular that would happen? The answer, of course, is nothing. Because this particular effect was without a cause, there was not anything constraining what this effect could be – no restriction on what things it could affect, how it could affect them, or when or where it could occur. Which means that if there is any exception to the PSR, then no state of affairs is really bound by it.

In a manner of speaking, an effect that has no cause is caused by nothing. If only some kinds of effects can be caused by nothing, then, to quote William Lane Craig, what is it that makes nothing so discriminatory? Nothing has no properties to allow us to make any distinctions in what it can cause; it is literally not anything at all. So either nothing can cause nothing – ex nihilo nihil fit – or there is nothing that nothing cannot cause.

The best rebuttal to this that I can come up with is the proposal that, while it is possible for events to happen with no cause, things that exist only have the capacity to be affected in certain ways, and this is what constrains causeless events. Another way we could say this is that things can only be affected in accordance with their liabilities. (The proposal would go on to say that, in fact, most things do not have liabilities to causeless effects, which is why we do not observe the PSR being violated.)

Liabilities, however, are capacities to be affected by corresponding powers. When an event occurs without any explanation, there is no power in operation, so I don’t really see a good reason to think that a causeless event needs a liability to make it possible. And if it did – would it really be causeless? A liability to be affected by a causeless event looks an awful lot like a passive, indeterministic causal power to affect one’s self.

And do things also need liabilities to causelessly come into existence? If they do, then nothing can come into existence without a cause: existing things are not liable to be brought further into existence, and non-existing things have no liabilities at all. A thing must exist in order to constrain reality in some way. At this point, the rebuttal nearly entails the PSR all by itself.

I suspect the intuition behind this idea, that things can only be affected in accordance with their liabilities, is actually nothing more than a subconscious belief that the only possible effects are ones that are produced by causes. And that intuition, of course, is perfectly suited to the PSR.

The Causal Account of Modality

Finally, the third argument is that the necessary truth of the PSR is logically entailed by the causal account of modality, with one additional assumption: that the accessibility relation between possible worlds is symmetric. I explained in my last post why I think the causal account is correct, and in the post here why the symmetry axiom makes sense. With those assumptions, the proof (which also comes from Pruss’s paper that I linked above) is fairly straightforward:

  1. Assume that there is a contingent true proposition, P, that does not have an explanation in terms of a cause. Let P* be the following conjunction: P and (P has no causal explanation). Then P* is true by assumption.
  2. Since P is contingent and therefore possibly false, P* is possibly false: that is, it is false in some possible world accessible from the actual world.
  3. Since P* is true, it is necessarily possible: that is, it is possible in all possible worlds accessible from the actual world. This is due to the symmetry of the accessibility relation.
  4. Now we can move to a possible world where P* is false (there is at least one of them, as shown in step 2). In such a world, P* is false, but it is still possible (as shown in step 3).
  5. According to the causal account of modality, if P* is false but possible, it is because there is something that has the power to originate a chain of causes that would result in P* being made true.
  6. But that means there is something that could cause P to be true, giving P a causal explanation, and also cause P to not have a causal explanation; a contradiction.
  7. Moving back to the actual world, this shows that the contradictory state of affairs demonstrated in step 6 is possible.
  8. But such contradictions are not possible. Therefore, our original assumption, P*, is false. Either P is not contingent, or it is not true, or it has a causal explanation.
  9. Nothing in steps 1 to 8 depends on the contents of P or on any contingent facts about the actual world: the proof could have been done for any proposition in any possible world. Therefore, necessarily, all contingent true propositions have a causal explanation. As they say, QED.

Since I believe the causal account and the symmetry axiom are both well-justified by our modal intuitions, I believe this is a good argument for the necessary truth of the PSR. (And we can turn this around as well: if the PSR is necessarily true, and explanations must ultimately be causal, it implies that the causal account of modality is correct.)

A Worldview Taking Shape

The Principle of Sufficient Reason ties the concepts of possibility, causation, and explanation into a coherent view of the nature of each, creating a picture of reality with cause-and-effect in its very foundation. In doing so, it explains the validity of abductive and inductive reasoning, giving us further confidence in our ability to understand reality.

Because of its significance for the non-deductive forms of reasoning, the PSR is highly relevant to the project of science, especially if one holds to a realist view of science. If the PSR is false, our rational intuitions are on very shaky ground, and science has no hope of explaining or discovering truth about the natural world.

The catch is that the PSR also has powerful implications for what lies outside the domain of science, particularly when we use it to start asking why the universe exists at all. I will come to that question a little ways down the road. It would appear, though, that in order to have a rational, scientific view of nature, you might have to accept that nature is not all there is.

Because of the powerful implications of the PSR, it would be good to consider the objections to it to make sure we haven’t gone wrong somewhere. I will do that in the next post.

Metaphysical Modality

In this post I will turn to the question I raised two posts ago: what makes possibilities possible? What is it that grounds the possibility or necessity of certain propositions? It seems intuitive, for example, that my own existence is contingent: a reality where I was never born seems possible. On the other hand, it seems intuitively necessary that 2 + 2 = 4. What features of reality make such statements of possibility and necessity true?

The question I am asking is subtly different from seeking an explanation of why these statements are true, though that question is obviously related. Rather, I am looking for what in reality makes these statements true. Truth is correspondence to reality, so (it seems) there has to be something about reality that grounds possibility and necessity.

Philosophers have offered a few different answers to this question. In this post I will look at the common ones and present the one that I think makes the most sense.

Logic and Nothing Else

The narrowly logical account of modality is that all that constrains what is possible in reality is the laws of logic. Something is possible if and only if no contradiction can be derived from it by the laws of logic (and something is necessary if and only if a contradiction can be derived from its negation). Then necessity is equivalent to provability.

One difficulty with this is that there are metaphysically necessary truths that can’t be proven merely from the laws of logic, unless non-logical axioms are added. For example, it is plausibly a necessary truth that nothing can be the cause of its own existence. But without any axioms related to the behaviour of causation, such a truth could not be proven. But the laws of logic cannot determine which non-logical axioms are the appropriate ones to use.

(Maybe self-causation is possible if time travel is possible. In that case, assuming S5 modal logic is correct, it would be a necessary truth that self-causation is possible, and that truth would now be one that needs non-logical axioms to be proven.)

The narrowly logical account also runs into a problem when you consider Gödel’s incompleteness theorem, which effectively says there will always be true arithmetical statements that we cannot prove. If the narrowly logical account is correct, this means that there are contingent truths about arithmetic: truths about which numbers add or multiply to which that could have been false. This seems wrong.

To escape the incompleteness theorem, you could suggest that the correct logic of reality is one that has an infinite number of axioms that can’t be generated by any finite algorithm, or that allows infinitely long statements or infinitely long proofs, or all of the above.

But even that move does not seem to be enough. Again, what can be proven from our logic depends on the rules of inference and the axioms. But then the question we are trying to answer just shifts up one level: what feature of reality makes those rules and axioms the correct ones? We use the rules that we do because they seem to embody fundamental necessary inferences, and we use the axioms that we do because they seem to be fundamental necessary truths. So the narrowly logical account of modality ends up needing modality to ground it, which gets us nowhere.

Existence of Possible Worlds

In S5 modal logic, we can describe what possibility and necessity mean with the conceptual device of possible worlds: something is possible if it is true in some possible world, and something is necessary if it is true in all possible worlds. So one way to ground modality is to say that these possible worlds exist in reality.

One way of doing this, which may be called extreme modal realism, is to propose that all of these possible worlds exist just as our actual world does: as entirely separate domains of reality with their own real, physical space-time universes and everything else. On this view, “actual” is just a location-relative word like “here” or “now.”

Extreme modal realism wreaks havoc on our notions of probability, as well as on any notion of morality or ethics that gives any weight to the consequences of our actions. But even aside from those difficulties, I think we can dismiss it out of hand, because the existence of other separate domains of reality has nothing to do with possibility or necessity. “Actual” is not merely a location-relative word; it applies to everything in reality. If we found out that reality was composed of sub-realities, that would just mean there was more to the actual world than we thought.

Further to this, extreme modal realism usually takes the stance that there is some sort of principle that guarantees a diversity of its so-called possible worlds, sufficient to account for all of our intuitions about what is possible. But then the necessity of this principle is left ungrounded.

So another way of grounding possibility in the existence of possible worlds is to make the possible worlds into abstract objects, which may be called platonic modal realism. On this view, possible worlds are maximal consistent sets of propositions, just as I described when I was explaining them as conceptual devices, with the twist that now we are considering these abstract objects to really exist.

The first objection to this approach is that it postulates that abstract objects exist, when (I have argued) we do not otherwise have good reason to think so. The second objection is that, since the existence of the abstract possible worlds is unexplained, this approach amounts to saying that possibility and necessity are fundamentally irreducible, when (as I will argue next) at least a partial reduction is available. This makes platonic modal realism redundant, and leaves an unexplained coincidence between the way that possibility is grounded in the abstract realm, and the way that it appears we can ground it in the concrete realm.

Causal Powers of Reality

In many of our modal intuitions, what seems to make a state of affairs possible is that there is something that has the causal power to make that state of affairs actual. For example, I could have been a philosopher because I had the power to choose to study philosophy in a professional capacity, and the ability (I hope, given the amount of philosophy I am doing on this blog) to carry that choice out. And conversely, it seems reasonable to ground the necessity of a state of affairs in the fact that (roughly speaking) there is nothing with the causal power to prevent it from being actual.

This is the causal account of modality. On this view, a state of affairs, S, is possible either if S is actual or if something exists, or existed, or will exist, that can originate a chain of causes which would lead to the occurrence of S. And S is necessary if it is actual and there is nothing that exists, or existed, or will exist, that can originate a chain of causes that would lead to the non-occurrence of S.

This is only a partial reduction of modality, since the causal powers that things have are still modal in character, but it is better than what either the narrowly logical account or platonic modal realism achieves. Moreover, our intuitions about causation are clearer and better grounded in our experience than our intuitions about mere possibility and necessity.

I have argued that we already have good reason for thinking that causation is primitive, so reducing metaphysical possibility and necessity to causation means that we only have one irreducible concept where we would otherwise have had two. And, usefully, the causal account lets us easily handle different varieties of possibility and necessity. For example, we can define:

  • Physical possibility: by restricting the chain of causes to include only physical ones.
  • Temporal possibility: by restricting the chain of causes to include only present or future ones.

Finally, this account makes it very plausible that possibility behaves according to S5 modal logic, with a reflexive, transitive, symmetric accessibility relation between possible worlds. Consider:

  • Reflexivity: it seems very reasonable to think that the actual state of affairs came about by a chain of causes, or at least, that this is true about the part of the actual state of affairs that is wholly contingent. (Since I’m not sure it makes sense to say that necessary states of affairs can be caused.)
  • Transitivity: if X can start a causal chain leading to Y and Y can start a causal chain leading to Z, then X can start a causal chain leading to Z.
  • Symmetry: if some other state of affairs than the actual one had occurred, it would have been brought about by a causal chain that branched from the actual world at some point. But if the actual state of affairs did come about by a causal chain, as our intuition about reflexivity suggests, then the possible reality contains something that could have brought about what actually happened.

So I think that the causal account of modality best explains our intuitions about possibility and necessity, and that it most successfully grounds truths about possibility and necessity in reality. And for these reasons, I believe it is correct. Possibility and necessity are grounded in causation.

Extension and Intension

Last post I started exploring the topic of possibility and necessity. I’m getting into more complicated subject matter here, but I want to briefly (some of you may feel “briefly” should be in scare quotes here) touch on a topic relevant to the concepts of possibility and necessity, and logical systems that can handle them.

The core of our modern-day formalizations of logic is propositional logic. Propositional logic uses simple terms that represent basic propositions, and complex terms that represent compound propositions built up from others, using the logical connectives of negation (“not”), conjunction (“and”), disjunction (“or”), and material implication (“if, then”).

For example, if you have a variable A representing the proposition that it is sunny, and another variable B representing that it is raining, propositional logic allows you to form the expression A OR B, which then represents the proposition that it is sunny or it is raining (or both). These connectives produce propositions whose truth values depend only on the truth values, not the meanings, of the propositions that they join.

There are two major ways to build upon propositional logic, and combining them together has proven difficult. The first is predicate logic. Predicate logic introduces new ways to form propositions, by applying predicates to terms representing objects of some kind. Essentially, propositional logic only lets you act on sentences to produce compound sentences, while predicate logic lets you build sentences out of nouns and verbs.

For instance, you could have a predicate N meaning “is a natural number,” and you could apply it to a variable x to produce the proposition N(x) meaning “x is a natural number.” Predicate logic also introduces quantifiers, allowing you to express statements such as “for all natural numbers x, there is a natural number y that is 1 greater than x.” These kind of statements can be fully represented in a formal way within predicate logic.

The second way of building on propositional logic is modal logic, designed to handle sentences expressing possibility and necessity by adding modal operators that act on propositions. Unlike the logical connectives in propositional logic, the modal operators produce propositions whose truth value is not determined solely by the truth value of propositions they are created from. Whether P is possible depends on more than whether or not P is true: it depends, in some sense, on the meaning of P.

Unifying Predicate and Modal Logic

In my readings on this topic, figuring out how to combine predicate logic and modal logic together has sometimes seemed like the logician’s version of trying to unify quantum mechanics and general relativity – a problem that has vexed physicists for the last half century. The main difficulty is in figuring out what it means for things in different possible worlds to be the “same.” For example, in the actual world I have an engineering degree, but in another possible world I have a philosophy degree. What does it mean for me to be me across different possible worlds?

There is also a difficulty with combining the usual rules for S5 modal logic with the usual rules for quantifiers in predicate logic: doing so means that the following somewhat unintuitive inference is valid (known as the Barcan formula):

  • If it is possible for there to be something that has property F, then there is something that possibly has property F.

For example, if it is possible for there to be a unicorn (in some possible world, there is something which is a unicorn), then there is something that could be a unicorn (in the actual world, there is something that, in some possible world, is a unicorn).

The result formally implies that the quantifiers (“there is an x such that…” and “for all x, it is true that…”) have the same domain (the values available to be assigned to x) in all possible worlds. Naively, this could be interpreted as saying that everything exists necessarily, which certainly isn’t right. After all, our own existence is highly contingent: there are an untold multitude of things that could have happened to prevent the existence of every single one of us.

Fortunately, I think that the problem of combining modal and predicate logic has effectively been solved. First, the fact that the quantifiers have the same domain in every possible world is easily handled by re-interpreting the quantifiers as expressing possible existence, and introducing a specific predicate to express actual existence. Then “there is something that could be a unicorn” above just means that there is something that could exist and could be a unicorn, not that anything actually exists that could be a unicorn.

Some philosophers object to this by saying that quantifiers should express actual existence, and that it is somehow wrong to use a predicate to express existence. I find their arguments for these assertions to be pretty weak. One of the main arguments seems to be that “existence is not a property;” that is, you don’t add anything to the concept of something by saying that it exists. That’s fine, but it is certainly still informative to say that something exists, and we use predicates in natural language to express this all the time. So I don’t see any problem with doing that in a formal language. (Neither do proponents of free logic.)

Intensional Logic

The way to handle the more serious difficulty of tracing individuals across possible worlds (or possible cases, more generally) seems to be to use intensional logic. In intensional logic, every expression in the logic has an extension in any given world, and an intension. The extension captures the reference of the expression; roughly, it is what the expression refers to in a given world. The intension captures the sense or meaning of the expression in a formal way as the pattern of its extensions across all worlds. By doing this, and by allowing predicates to differentiate expressions by their intensions and not just their extensions, the logic is made highly expressive, easily able to handle subtle modal concepts.

I’ll try to give an example to clarify the distinction between extension and intension. Imagine that Plato is standing on the street corner at noon, giving a lecture. Later in the afternoon, he goes somewhere else, and Aristotle takes his place. The expression “the philosopher on the corner” represents a certain intension: it picks out whichever philosopher is standing on the corner in the present case. At noon, “the philosopher on the corner” has the same extension as the expression “Plato.” Later, it no longer shares its extension with “Plato,” but it does share it with “Aristotle.” “Aristotle” also represents an intension with a pattern of extensions across different times: it picks out a young man in 365 BC, and an older man around 325 BC.

Intensional logic (specifically, the so-called case-intensional logic developed in the paper I linked above) allows different variables to be assigned to different intensions like Plato, Aristotle, and the philosopher on the corner. It can distinguish between when variables merely share the same extension in the present case (as “Plato” and “the philosopher on the corner” do at noon, but not at other times) and when they represent the same intension, thereby sharing the same extension in every case (as perhaps do “Socrates” and “the famous ancient Greek philosopher executed by hemlock”).

By distinguishing between extensional identity and intensional identity, it seems to me that intensional logic can clarify a number of philosophical conundrums, such as the apparent problems of contingent identity or material constitution. Importantly, though, it allows a logically rigorous way of tracing individuals across different times or different possible worlds, by assigning some intensions with special properties (called absolute properties in case-intensional logic) that mark them as individuals of a certain kind.

Let’s say we want to trace the individual referred to by “the philosopher on the corner” across different times. Since this expression picks out different people at different times (Plato at noon, and later, Aristotle), it does not have the absolute property corresponding to being a single human individual. But at each of these times there is an intension with the same extension as “the philosopher on the corner” that has the right absolute property, namely, “Plato” at noon, and later, “Aristotle.” And we use those intensions to trace Plato and Aristotle between times (or cases, or worlds).

So if we have an intension x referring to an individual or object that we want to trace between possible cases, we find the intension y that (i) shares an extension with x in the present case and (ii) has the right absolute property. Then, y correctly represents that individual across cases. This solves the problem of tracing individuals across possible worlds, the main difficulty in unifying predicate and modal logic.

For any aspiring logicians reading this, there is plenty of work to be done developing case-intensional logic, so that is my advice if you’re looking for a research direction. If you go in that direction, you could introduce yourself by saying that what you’re working on is like quantum gravity, but for logic instead of physics. (You’ll be so cool.)

Further reading: check out my more detailed introduction to case-intensional logic.

Possibility and Necessity

The concept of possibility is another important notion when it comes to thinking about the fundamental nature of reality. There are some propositions that we think are logically possible even if they are not true. It is false that I majored in philosophy in undergrad – but certainly I could have. What is it that makes possibilities possible?

There are also some propositions that we think are logically necessary: propositions which could not possibly have been false. For example, not only are there infinitely many prime numbers, it could not possibly have failed to be true that it is so. A proposition that is true, but could have been false, is called contingent. So it is necessarily true that there are infinitely many prime numbers, but only contingently true that I have a degree in engineering instead of philosophy.

Broadly speaking, there are two distinct concepts that we can talk about when we talk about possibility: epistemic possibility and metaphysical possibility.

  • Epistemic possibility: what could be true given our knowledge and rational capabilities; in other words, what could be true for all we know.
  • Metaphysical possibility: what could be true given the nature of reality itself, independent of our knowledge about it.

As an example of the difference between these: truths about numbers are usually thought to be metaphysically necessary, so that whatever is true about numbers, must be true. But there are things about numbers that we don’t know. It is unknown whether, for instance, the Riemann hypothesis is true. So it is epistemically possible that the Riemann hypothesis is true, and epistemically possible that it is false: it could go either way, given what we know. But metaphysically speaking, whichever way it turns out to be, it could not possibly have been anything else: it was always true that it had to turn out that way.

In considering the fundamental nature of reality, I am concerned with what is absolutely metaphysically possible, not merely what is epistemically possible. I am interested in how reality itself is constrained to be, not just what our current knowledge permits or prohibits.

Possible Worlds

Philosophers have a conceptual device for thinking about possibility and necessity: possible worlds. We can think of possible worlds as consistent “world-stories,” or maximal descriptions of the different ways that reality could be. Each possible world is an infinite set of propositions that, taken together, could all be true. They are maximal descriptions in this sense: if P is a proposition and W is a possible world, W either contains P, or it contains the negation of P (and it does not contain both). So, every proposition is either true or false (and not both) in every possible world, as the laws of logic require.

One of the possible worlds contains all and only those propositions which are actually true, so we call it the actual world. So, using the abstract object language of possible worlds, the actual world is the infinite list of all true statements:

{… “2+2=4”, “The sky is blue”, “Unicorns do not exist”, “I have an engineering degree”, …}

And there are other possible worlds, some of which may be, for example:

{… “2+2=4”, “The sky is blue”, “Unicorns do not exist”, “I have a philosophy degree instead of an engineering degree”, …}

{… “2+2=4”, “The sky is blue”, “Unicorns existed in the past but are now extinct”, “I was never born because someone who would have been my ancestor was gored by a unicorn”, …}

Modal logic is the study of logical deduction that can handle the possibility and necessity of propositions. In it, possible worlds can be made more general, and they are sometimes just called cases. Depending on what the logic is trying to capture, cases do not have to be complete descriptions of all of reality; instead they can be complete descriptions of individual moments in time, or complete descriptions of possible outcomes of an experiment or an event.

Once we have our various possible worlds, we consider an accessibility relation between the different worlds or cases. (Roughly, world X is accessible from world Y if X is possible from the standpoint of Y.) Different accessibility relations correspond to different conceptions of possibility. Then, we say that a proposition P is possible in a world W if P is true in some world accessible from W, and P is necessary in W if it is true in all worlds accessible from W.

Axioms of Possibility

What should the accessibility relation look like in order to properly characterize what is absolutely metaphysically possible? I think we can impose a few different conditions on it:

First, the accessibility relation should be reflexive: every world should be accessible from itself. This amounts to saying that if something is true, then it is possible, which makes perfect sense. It also amounts to saying that if something is necessary, then it is true, which also makes perfect sense.

Second, the accessibility relation should be transitive: this essentially means that worlds can be accessed through other worlds, so that if W1 can access W2, and W2 can access W3, then W1 can also access W3. This amounts to saying that if something is possibly possible, then it is possible, which makes sense for an absolute concept of possibility. By contraposition, it also amounts to saying that if something is necessary, then it is necessarily necessary.

Third, I think it makes sense for the accessibility relation to be symmetric: accessibility goes both ways, so that if W1 can access W2, then W2 can access W1. What this amounts to saying is that if something is true, then it is necessarily possible. (Or, if something is true, it is impossible that it is impossible.) In other words, however else things could have happened, it would still be true that they could have happened the way they actually did.

Symmetry is a slightly more contested feature of the accessibility relation, because it also implies that if something is possibly necessary, then it is true. If this result seems somewhat weird and surprising, another way to phrase it is that unless something is true, it cannot possibly be necessary. I think, once we recall that the possibility being thought of here is metaphysical and not epistemic (what is actually possible in reality, and not just what is possible for all we know), that the initial weirdness of this fact goes away.

When philosophers and logicians formalize these three properties of the accessibility relation, the result is a system known as S5 modal logic. Reflexivity, transitivity, and symmetry together imply a further property, which says that if something is possible, then it is necessarily possible. In other words, what is possible (and also what is necessary) does not change from one possible world to another. All possible worlds can access all other possible worlds. S5 is nice, since it lets us forget about the accessibility relation altogether: we just say that something is possible if it is true in some possible world, and something is necessary if it is true in all possible worlds.

S5 also implies that if something is possibly necessary, then it is necessary. Like the closely associated result of the symmetry property, this seems a little surprising at first, but it makes sense as a feature of an absolute concept of possibility, where possibility and necessity do not change based on what the actual world turns out to be.

(S5 also has the nifty feature that if you have a long chain of possibles and necessaries, you can throw away all but the last one: if something is necessarily possibly possibly … necessarily possible, then we can just say that it is possible.)

So that is a little bit of modal logic for you. I haven’t yet answered the question I asked at the beginning of this post: what makes possibilities possible? I will take up that question in – well, not the next post, since that will instead discuss a somewhat technical aspect of modal logic – but the post after that.