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150 Hardegree, Symbolic Logic 1. INTRODUCTION In an earlier chapter, we studied a method of deciding whether an argument form of sentential logic is valid or invalid – the method of truth-tables. Although this method is infallible (when applied correctly), in many instances it can be tedi-ous.
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intuitively clear; the most intuitively clear system is not the most parsimonious. Our particular choice will accordingly be a compromise solution. We have to select from the infinitely-many valid argument forms of sentential. logic a handful of very fertile ones, ones that will generate the rest.
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154 Hardegree, Symbolic Logic Note carefully: it is understood here that if a formula replaces a given letter in one place, then the formula replaces the letter in every place.
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Chapter 5: Derivations in Sentential Logic 151 2. THE BASIC IDEA Underlying the method of formal derivations is the following fundamental idea. Granting the validity of a few selected argument forms, we can demonstrate the validity of other argument forms. A simple illustration of this procedure might be useful. In an earlier chapter,
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SI MPLE INFERENCE RULES. In the present section, we lay down the ground work for constructing our sys-. tem of formal derivation, which we will call system SL (short for ‘sentential logic’). At the heart of any derivation system is a set of inference rules .
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be discussed later.] (5) In each of the rules, d and e are arbitrary formulas of sentential logic. Each rule is short for infinitely many substitution instances. (6) In each of the rules, the order of the premises is completely irrelevant.
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