by Corry Shores
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[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 distracting mistakes, because I have not finished proofreading.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
2. Basic Modal Logic
2.5. Possible Worlds: Representation
Brief summary:
Possible world semantics is mathematical machinery. But it represents certain real features of truth and validity. We wonder, what exactly do possible worlds and their semantics represent, philosophically speaking?
Summary
2.5.1
[We now ask philosophically what modal semantics mean.]
The remainder of chapter 2 will ask the philosophical question of what do modal semantics mean.
2.5.2
[They might mean nothing more than the mathematical machinery underlying them.]
[Recall from section 2.3 how the modal semantics were formulated. And note as well how the propositional language it was built on is formulated (section 1.2); and while we are at it, recall from section 0.1 the set theoretical notation that at times comes into play. Without any of the intuitive elaborations on what the formulations mean, we might be able to simply say that all of this has no other meaning than its bare mathematical machinery.]
One might suggest that they do not mean anything. They are simply a mathematical apparatus – interpretations comprise just bunches of objects (W) furnished with some properties and relations – to be thought of purely instrumentally as delivering an appropriate notion of validity.
(28)
2.5.3
[But the mathematical machinery generates the right kinds of answers for our philosophical concerns, and so its philosophical value must be more than its bare mathematical machinery.]
But Priest notes that the machinery produces what seems to be the right answers. The reason why it does so is perhaps because it accurately models logical aspects of reality. “There must be some relationship between how it works and reality, which explains why it gets things right” (28). Thus it is not satisfying to simply say that the philosophical meaning of modal semantics is nothing more than its mathematical machinery. It must have some greater philosophical value.
2.5.4
[Probably modal semantics represents something fundamental to validity itself.]
[Specifically the mathematics behind modal semantics models validity in the way we want it to. We might say then that the mathematics represents something about validity.]
The most obvious explanation in this context is that the mathematical structures that are employed in interpretations represent something or other which underlies the correctness of the notion of validity.
(28)
2.5.5
[One example of how our mathematicized logical machinery represents real things of philosophical importance is how we represent truth with the symbol ‘1’ (and ‘o’ for falsity). The truth-functional semantics that calculates these symbols models the way validity actually works.]
Priest then gives an example for how the mathematical structures of logic correspond to real things that they model. He notes how we use the number 1 to represent truth; the truth-functional semantics that we use to assign and calculate truth values corresponds to the properties of truth itself.
In the same way, 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).
(28).
2.5.6
[We then ask, what do the possible worlds and the machinery of possible world semantics represent, philosophically speaking?]
Priest ends by asking: “what exactly, in reality, does the mathematical machinery of possible worlds represent? Possible worlds, of course (what else?). But what are they?” (28).
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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