By Christos Evangeliou
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Extra resources for Aristotle's Categories and Porphyry
3. An expression is a wff only if it can be shown to be a wff on the basis of 1 and 2. 4 Secondly, we add a new one-place predicate, E, for existence. When talking about modality, we often wish to talk about things that might or might not have existed; although such talk can be captured in the resources we already have, it can be represented more perspicuously with this new predicate E. Ea can be thought of as saying a exists. Accordingly, the vocabulary of quantified modal logic, as well as containing clauses (a)(g) above, also contains: (h) the operators ~ and (i) the one-place predicate letter E.
A semantics for modal logic might help us make progress on these questions. In the familiar non-modal quantificational logic, we could test our system of rules of inference against the semantics to make sure that the rules are sound and complete. If the system is sound then we have a guarantee that the rules and axioms of the system are not too strong: they cannot take us from a truth to a falsehood. Conversely, if the system is complete, we know that the rules dont miss anything out: no further rules or axioms are required for any valid inference to be carried out within the system.
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.
Aristotle's Categories and Porphyry by Christos Evangeliou