By Joseph Melia
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That is all that is new about our language. In particular, it does not contain any new logical operators, such as ~ and . As this is nothing more than a first-order language, the usual rules of well-formedness apply to this language and we can use what we know from the predicate calculus to put strings of symbols together to form grammatical sentences. 16 Numerical quantification revisited Sentences such as There are three ways in which Joe could win his chess match count the possible ways in which a certain event could occur.
The predicate letters F, G, H, . . the two-place predicate = the names a, b, c, . . the brackets ( and ). These items are to be thought of as the letters of the language. The predicate calculus also contains rules for which strings of letters are well formed. We are familiar with sensible strings of symbols, such as Fa & Gb and ∀x∃yRxy, but any old string of symbols does not MODAL LANGUAGE AND MODAL LOGIC 23 count as a well-formed formula of the language. For example, aF→x∃ is ungrammatical: this particular string of symbols is not well formed.
Note that nothing is F at w2. The rightmost world again contains two elements, this time b and c, both of which are F, but only b is G. For non-modal sentences, that is, sentences that do not contain ~ or , determining which sentences are true or false at a world is just the same as determining which sentences are true or false in a model in the first-order predicate calculus. To work out whether or not a modal sentence is true or false in the model, we just have to see which non-modal sentences are true at the various worlds.
Modality by Joseph Melia