7.5. General arguments

7.5.0. Overview

We have answered questions about entailment concerning truth-functional compounds by turning them into questions about their immediate components (or sentences contradictory to them). The largest component formulas of sentences formed by quantifiers usually contain free variables, so we will look at the sentences that are the result of putting closed terms in place of these variables.

7.5.1. Conjunction and universal quantification
An unrestricted universal sentence behaves like a conjunction of sentences saying of each particular thing what the universal says of everything.

7.5.2. Instantiation
The laws of entailment for unrestricted universals treat them as conjunctions of their instances for particular things. However, a universal behaves like a conjunction with indefinitely many conjuncts: it entails each of its instances but cannot be replaced by them.

7.5.3. Generalization
The instances of a universal are all predications of the same abstract, and this makes it possible to establish a universal by way of a single typical instance.

7.5.4. Adding instances
Because a universal has indefinitely many instances, we cannot consider each in a derivation. Instead, we exploit a generalization only partially to extract those instances that are relevant to the argument we are considering.

7.5.5. General arguments in derivations
To insure that we establish an instance of a universal in a way that admits generalization, we construct it for a new term that is permitted only a limited scope in the generalization.

7.5.6. Syllogisms
The rules for the unrestricted universal enable us to establish, among other things, the validity of arguments from a special class traditionally labeled syllogisms (in a narrow sense of the term).