(as in for example Chellas 1980, 68): ◻A is true at a world α in a model M if and only Thought of as just this ‘seeing’ relation, giving the following standard truth-conditions Α can see, then A is necessarily true from the perspective of α. Likewise if A is true in every world that Then A is possible from the perspective of α. To be a possibility, and so if α can ‘see’ at least one world in which a sentence A is true, The basic idea is this: a world α in a model considers any world it can ‘see’ Look at the worlds ‘around’ the particular world we’re interested in, making use of the World, we need to look beyond the truth-value that P assigns to A at that world, and Truth-functional operators, and so to get the truth-value of a sentence ‘◇A’ or ‘◻A’ at a P 1is false isn’t enough to say whether p 1 could be true. Isn’t really enough to know whether it’s necessarily true. The truth of p 1and the truth of p 2, it seems that knowing whether a sentence p 1 is true Sen-tences containing ‘◻’ and ‘◇’ because, while we can derive the truth of p 1∧ p 2 from This last point gets a little more complicated when we start thinking about For example, if p 1and p 2 are true at a world, Sen-tences are true, too (Chellas 1980, 67-8). In turn, all this is enough to determine which non-atomic
InĪ model, then, we have a set of worlds, some of which might be related to one anotherīy the R-relation, with the function P determining which atomic sentences are true atĮach of these worlds. These worlds, and P is a function from atomic sentences and worlds to truth-values. W here is a set of ‘worlds’, R is a binary relation that can hold between Modal logic as structures - that is, ordered triples made up of elements Quite complicated, but, for example, Chellas defines ‘standard’ models in systems of Look-ing at ‘models’, which offer interpretations of our basic language. We can take these in turn to represent that if a sentence is necessary, then it’s true,Īnd that if a sentence is provable, it’s provable that it’s necessarily true.Īnother way to approach modal logic is through semantics. Or can be derived from an axiom (that is, if A is an axiom or a theorem), then◻A isħ An operator O is a dual of an operator P when OA is equivalent to ¬P¬A. To be an axiom, and we might introduce a rule that if a sentence A is either an axiom The structural properties of claims about possibility and necessity. All of this is intended to in some way reflect Give a set of rules which tell us how we can manipulate the sentences in our languageĪnd go from one to another (1996, 23-4). One way to approach this is through syntax - as HughesĪnd Cresswell put it, we could specify a set of axiomatic claims in the language, and With this language in place, we can start to approach structural questions about Other in perfectly symmetrical ways: ◇A will be equivalent to ¬ ◻ ¬A, and ◻A will be Whichever of ‘◇’ or ‘◻’ we start with, ◇ and ◻Īre duals, 7 so we can subsequently introduce either by definining it in terms of the Saying ‘It’s possible that p1’ and ‘◻(p 1 → p 2)’ will be a sentence saying ‘Necessarily, Possible that A’ and ‘It’s necessary that A’ respectively. This allows us to form sentences ‘◇A’ or ‘◻A’, representing ‘It’s To these a primitive sentence operator, either◇ or ◻, to represent ‘possibly’ or We can also make use of the language of first-order logic,īut for simplicity’s sake we’ll leave this to one side for now. Sen-tences in our language (I’ll often just write ‘A’ as a metalinguistic variable when noĬonfusion could result). Of) the language of propositional logic - connectives¬, ∧, →, and so on, propositional The basic language of systems of modal logic often makes use of (parts Necessity, rich systems of logic have been developed under the broad umbrella of When investigating structural questions about different senses of possibility and Need to take a small detour into questions about possibility and necessity to see just We can make use of tools well-suited to making claims about structure. Ques-tion predominantly about structure, and this quesques-tion will be much easier to answer if The time has finally come to introduce some formal apparatus! We’ve reached a
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