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I read this article Modal Metaphysics weeks ago, but one thing that struck and perplexed me was the definition of consistency offered in this article. According to this article, it seems that one issue plaguing many theories of modality is their reliance on this modal notion consistency. Here is a quotation from the article:

...consistency is indeed modal; a set of sentences is consistent if and only if it is possible for those sentences to be jointly true.

Why does consistency have to be construed in this way as a modal notion? Why isn't the following a coherent definition of consistency?

A set of sentences is consistent if and only if the sentences are jointly true.

I am not sure if consistency in modality bears any relation to consistency in mathematics, but consistency in mathematics doesn't appear to rely on any modal notions---at least from my very cursory reading on the topic.

I have tried searching the internet for some treatment of the idea of consistency in modal theories, but I couldn't find much. Perhaps I don't know which search terms I need to use, or perhaps I ought to further investigate this on my own and write a philosophy paper!

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  • The "modal-logic" tag should actually be "modality." – Eli Bashwinger Jan 24 '17 at 20:01
  • It is hard to avoid "modal" notions in the very basic concepts of logic; consider a "standard" def of Logical Consequence : "A formula A is a (semantic) consequence of a set of statements Gamma if and only if there is no model in which all members of Gamma are true and A is false." Alternatively : "if and only if it is necessary that if all of the elements of Gamma are true, then A is true." – Mauro ALLEGRANZA Jan 24 '17 at 20:15
  • This is so since Aristotle : "A deduction is speech (logos) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so. (Prior Analytics I.2, 24b18–20)" – Mauro ALLEGRANZA Jan 24 '17 at 20:16
  • "Earth is the third planet in the Solar system" is consistent with "There is no life on Earth", they are not however jointly true. Consistency is not about what actually is the case but what could possibly be the case even if in fact it isn't. This is why it is modal. – Conifold Jan 24 '17 at 20:33
  • @Conifold Ah, and according to my definition of consistency they are not jointly true, as you point out; but "deep down" we know they ARE consistent and we want to affirm this, so this provides grounds for rejecting my definition. – Eli Bashwinger Jan 24 '17 at 21:49
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Eli Bashwinger has a point, and there does indeed seem to be something wrong with the quotation.

I think the article makes an implicit distinction along these lines: a presumably modal notion of consistency applies to a set of uninterpreted sentences, whereas a non-modal notion of consistency would apply to a set of interpreted sentences; where a sentence is interpreted when it is assigned a truth value, when all its variables, if any, are bound by quantifiers, when quantifiers are assigned domains of objects, when predicates are assigned extensions, when logical constants are assigned referents, etc.

Given a distinction between interpreted and uninterpreted sentences, we may distinguish modal and non-modal consistency as follows:

Modal consistency. A set of uninterpreted sentences "is consistent if and only if it is possible for those sentences to be jointly true." Okay. But what seems to be missing from the article is the idea that the right-hand side of the biconditional may in turn be further analyzed like this: it is possible for a set of uninterpreted sentences to be jointly true if and only if there is an interpretation or model in which those sentences are jointly true. (A given possible world may provide such an interpretation.)

Non-Modal consistency. As Eli Bashwinger puts it, a set of (fully interpreted) sentences is consistent if and only if the sentences are jointly true (at a given world.)

Now that we have a tentative distinction between modal and non-modal consistency, we have to ask whether or not Kripke's and Lewis's accounts of possible worlds turn on this or any other notion of modal consistency, and whether this is a problem if either does.

I'm not sure about Lewis's account. Kripke's, however, does indeed seem to turn on some modal notion of consistency as applied to sets of uninterpreted sentences.

What is unclear is whether or not this is a problem for him. On the one hand, yes, modal consistency appeals to some notion of possibility, and so this may look like a problem. But, on the other hand, the notion of possibility to which it appeals is the notion of there being an interpretation in which a set of sentences come out jointly true. In other words, Kripke doesn't seem to appeal to a primitive notion of possibility, but to one which is already analyzed: existence of an interpretation in which sentences are jointly true. The structure of the argument is unclear to me, so I am not sure whether this is a problem for him.

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