Semantics

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By Joseph Melia

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Perhaps they think that Harry is not a dog. Perhaps they are sceptical of all knowledge of the external world. Whatever. The argument is still valid even if its premises are not true, for whatever the truth of the premises, it is still not possible for the premises to be true and the conclusion false. We can recognize the validity of an argument even if we do not know the truth of the premises. Indeed, we can see that the fourth argument is valid even though we may not know what the words in the argument mean, for we know that, as long as the words are of the right grammatical category (as long as “boojum” and “beejoom” are predicates and “Tharrpx” is a name), if the premises are true then the conclusion must be true also.

If a model assigns to the predicate R, we can think of the model as saying that a does bear R to b. If is not assigned to the predicate R by the model, we can think of the model as saying that a does not bear R to b. Similarly, when it comes to modelling a three-place predicate H, the model needs to tell us which triples of objects belong to the predicate H. First-order languages contain names as well as predicates. Just as our model has to tell us which predicates of our formal language are satisfied by which n-tuples, so it has to tell us which names of our language refer to which objects.

13 As usual, our language (let us call it PWL for possible worlds language) will have the familiar existential and universal quantifiers plus the truth-functional connectives. It will also contain the usual supply of names and variables. Naturally, it will also contain predicates, but whereas in QML one-place predicates of English, such as “x is red”, are treated as one-place predicates Fx, PWL treats them as two-place predicates Fxw. This is because, in the possible worlds framework, one object can have different properties at different worlds.

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