by Corry Shores

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[Logic and Semantics, entry directory]

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[Priest,* Introduction to Non-Classical Logic*, entry directory]

[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]

Summary of

Graham Priest

*An Introduction to Non-Classical Logic: From If to Is *

Part I

Propositional Logic

2.

Basic Modal Logic

2.7

Modal Actualism

Brief summary:

(2.7.1) Under modal actualism, possible worlds are understood not as physically real entities, like in modal realism, but rather as abstract entities, like numbers. (2.7.2) One version of modal actualism understands a possible world as a set of propositions or other language-like entities and as being “individuated by the set of things true at it, which is just the set of propositions it contains” (29). (2.7.3) One problem with the propositional understanding of possible worlds is that there are many sorts of sets of propositions, but not all constitute worlds. For example, “a set that contains two propositions but not their conjunction could not be a possible world” (29). (2.7.4) A big problem with the propositional understanding of possible worlds is that in order for propositions to form a world, we need to know which inferences follow validly from others. Then, after knowing that, we can apply the mathematical machinery to explain which inferences are valid. But as you can see, the mathematical machinery, which is was we are trying to substantiate with this propositional account, is made useless, as it is what is supposed to determine validity, not take validity ready-made and redundantly confirm it. (2.7.5) To avoid this problem of validity, there is another sort of modal actualism called *combinatorialism*. Here a possible world is understood as “the set of things in this world, rearranged in a different way. **So in this world, my house is in Australia, and not China; but rearrange things, and it could be in China, and not Australia**” (30). (2.7.6) Because arrangements are abstract objects, combinatorialism is a sort of modal actualism. And because combinations can be explained without the notion of validity, combinatorialism avoids the problems of validity that the propositional understanding suffered from. (2.7.7) One big problem with combinatorialism is that it is unable to generate all possible worlds. For, there could be objects in other possible worlds not found in our world or in any other possible world obtained by rearranging the objects in our world.

[Modal Actualism: Possible Worlds as Abstract Entities]

[Version 1: Possible Worlds as Sets of Propositions]

[Problem of Defining Possible Worlds in Terms of Proposition Sets]

[The Problem of Assuming Validity in the Propositional Understanding]

[Version 2: Combinatorialism]

[Combinatorialism as a Modal Actualism and as Free from Problems of Validity]

[Combinatorialism’s Limitation]

Summary

[Modal Actualism: Possible Worlds as Abstract Entities]

[Under modal actualism, possible worlds are understood not as physically real entities, like in modal realism, but rather as abstract entities, like numbers.]

[Let me quote from our review for the prior section 2.6, found at the beginning of section 2.6.1:

Recall from section 2.5 some of the following ideas. Modal logic uses the intuitive notion of a possible world, but as we saw, it is formulated using mathematical machinery where it is not obvious what any of it has to do with the metaphysics of possible worlds. The assumption is that the mathematics somehow represents “something or other which underlies the correctness of the notion of validity” (p.28, section 2.5). For example:

no one supposes that truth is simply the number 1. But that number, and the way that it behaves in truth-functional semantics, are able to represent truth, because the structure of their machinations corresponds to the structure of truth’s own machinations. This explains why truth-functional validity works (when it does).

(Priest p.28, section 2.5)

So Priest ended by asking, “what exactly, in reality, does the mathematical machinery of possible worlds represent? Possible worlds, of course (what else?). But what are they?” (p.28, section 2.5).

Then in section 2.6 we gave as one potential answer that the mathematical entities for possible worlds are simply other real physical worlds that exist in different times or places. This is called ‘modal realism’. Now we consider another view, called ‘modal actualism’. It regards possible worlds as existing, but not physically so. Rather, they are abstract entities, like numbers.]

Another possibility (frequently termed ‘modal actualism’) is that, though possible worlds exist, they are not the physical entities that the modal realist takes them to be. They are entities of a different kind: specifically, abstract entities (like numbers, assuming there to be such things).

(29)

[Version 1: Possible Worlds as Sets of Propositions]

[One version of modal actualism understands a possible world as a set of propositions or other language-like entities and as being “individuated by the set of things true at it, which is just the set of propositions it contains” (29).]

[Priest notes there are different ways to construe possible worlds as abstract entities. The first that he considers is to think of them as sets of propositions or “other language-like entities.” As such, we could understand a possible world as being “individuated by the set of things true at it, which is just the set of propositions it contains” (29).]

What kind of abstract entities? There are several possible candidates here. A natural one is to take them to be sets of propositions, or other language-like entities. Crudely, a possible world is individuated by the set of things true at it, which is just the set of propositions it contains.

(29)

[Problem of Defining Possible Worlds in Terms of Proposition Sets]

[One problem with the propositional understanding of possible worlds is that there are many sorts of sets of propositions, but not all constitute worlds. For example, “a set that contains two propositions but not their conjunction could not be a possible world” (29).]

[Priest next notes a problem with this understanding. It is not clear which sets will qualify as a world, because surely there are many sets that are not possible worlds; “For example, a set that contains two propositions but not their conjunction could not be a possible world” (29).]

But a problem arises with this suggestion when one asks which sets are worlds? Clearly not all sets are possible worlds. For example, a set that contains two propositions but not their conjunction could not be a possible world.

(29)

[The Problem of Assuming Validity in the Propositional Understanding]

[A big problem with the propositional understanding of possible worlds is that in order for propositions to form a world, we need to know which inferences follow validly from others. Then, after knowing that, we can apply the mathematical machinery to explain which inferences are valid. But as you can see, the mathematical machinery, which is was we are trying to substantiate with this propositional account, is made useless, as it is what is supposed to determine validity, not take validity ready-made and redundantly confirm it.]

[Priest notes now a big problem with this conception. The mathematical machinery of possible worlds semantics is supposed to explain why certain inferences are valid and why certain others are invalid. (But for this propositional understanding of possible worlds to work, that means whenever true propositions in a world entail another, the other must be true too ((that is to say, the world must be closed under valid inference.)) So there is a problem here. I probably will express it incorrectly, so see the quotation below. Let me first put together what Priest is saying. The mathematical machinery is supposed to explain which propositions are valid. This also means that the notion of a world is needed to explain validity. But under this propositional notion of possible worlds, we need already to have a notion of validity to determine which propositions form a world. This means that the notion of validity is required to explain the notion of a world. And thus, it is not the mathematical machinery that explains why certain inferences are valid and others not. But then, what is the point of the mathematical machinery if it is not what is determining validity? See the quote, as I probably have this wrong.)]

For a set of propositions to form a world, it must at least be closed under valid inference. (If a proposition is true at a world, and it entails | another, then so is that.) But there’s the rub. The machinery of worlds was meant to explain why certain inferences, and not others, are valid. But it now seems that the notion of validity is required to explain the notion of world – not the other way around.

(29-30)

[Version 2: Combinatorialism]

[To avoid this problem of validity, there is another sort of modal actualism called *combinatorialism*. Here a possible world is understood as “the set of things in this world, rearranged in a different way. **So in this world, my house is in Australia, and not China; but rearrange things, and it could be in China, and not Australia**” (30).]

[Priest next mentions another sort of modal actualism called combinatorialism, which can avoid this problem mentioned above in section 2.7.4. Here we think of a possible world simply as a variation on our world where the things in it are arranged in a different way. For example, “So in this world, my house is in Australia, and not China; but rearrange things, and it could be in China, and not Australia.” (**Note the possible relevance of this to Leibnizian compossible worlds. God calculates all combinations of predicates for each individual substance, along with all combinations of individual substances, along with all combinations of laws for the worlds. In order for the combination to form a singular world, the predicates of one individual substance cannot preclude those of another substance, like your parents meeting only after you are supposed to be born. ****Deleuze notes that this incompossibility is not a matter of the logical contradiction of the predicates but rather that the overall combination of all individual substances with their predicates cannot form one world where they all allow for one another’s combined existence.**** Rather than all individual substances converging into one world, they rather diverge into incompossible worlds. And so since this divergence is not determined by contradiction, Deleuze says that convergence and divergence are matters of “alogical” compatibilities and incompatibilities. In a vaguely similar way, we are here understanding combinations of worldly facts as not inherently involving validity. And since logic is the science of valid inference, then perhaps this combinational understanding of possible worlds can be seen as being vaguely similar to what Deleuze calls the alogical notion of compatibility and incompatibility.**)]

A variation of actualism which avoids this problem is known as ‘combinatorialism’. A possible world is merely the set of things in this world, rearranged in a different way.

So in this world, my house is in Australia, and not China; but rearrange things, and it could be in China, and not Australia.(30)

[Combinatorialism as a Modal Actualism and as Free from Problems of Validity]

[Because arrangements are abstract objects, combinatorialism is a sort of modal actualism. And because combinations can be explained without the notion of validity, combinatorialism avoids the problems of validity that the propositional understanding suffered from.]

[The reason why combinatorialism is a sort of modal actualism is because the arrangement itself is an abstract object, even if the arranged things are not. Also, it avoids the objection regarding validity that we saw in section 2.7.4, because we can “explain what combinations there are without invoking the notion of validity” (30).]

Combinatorialism is still a version of actualism, because an arrangement is, in fact, an abstract object. It is a set of objects with a certain structure. But it avoids the previous objection, since one may explain what combinations there are without invoking the notion of validity.

(30)

[Combinatorialism’s Limitation]

[One big problem with combinatorialism is that it is unable to generate all possible worlds. For, there could be objects in other possible worlds not found in our world or in any other possible world obtained by rearranging the objects in our world.]

[But there is a problem also with combinatorialism. Possible worlds are different arrangements of the things in our world. But there could be an object in another world that neither exists in our world nor in any other world obtained by rearranging the objects in our world. “Hence, there are possible worlds which cannot be delivered by combinatorialism” (30).]

But combinatorialism has its own problems. For example, it would seem to be entirely possible that there is an object such that neither it nor any of its parts exist in this world. It is clear, though, that such an object could not exist in any world obtained simply by rearranging the objects in this world. Hence, there are possible worlds which cannot be delivered by combinatorialism.

(30)

From:

Priest, Graham. 2008 [2001]. *An Introduction to Non-Classical Logic: From If to Is*, 2nd edn. Cambridge: Cambridge University.

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