Symbolizing Non-categorical Propositions

As demonstrated on the previous two pages, categorical propositions are surprisingly flexible and can be used to represent a wide variety of English sentences. Nevertheless, many sentences cannot be paraphrased as categorical propositions and thus cannot be symbolized in that way. For example, the sentence Socrates is a philosopher is not a categorical proposition, since it does not relate two categories of things. (Socrates is not a category; he is a named individual.) The sentence Socrates is a philosopher is symbolized simply ‘Ps’ as we saw at the beginning of this chapter. Many compound propositions involving truth-functional connectives can’t be represented as categorical propositions either.

When symbolizing sentences that cannot be paraphrased as categorical propositions, look for truth-functional connectives and try to identify the overarching structure of the sentence as a whole. A good strategy is to identify the main connective first and deal with each component of the sentence separately, just as we did in propositional logic. For example, consider the sentence:

If all philosophy professors teach logic, then some students learn to reason well.

This sentence as a whole cannot be paraphrased as a categorical proposition, though it does contain categorical propositions as components. The sentence as a whole is a conditional, so we can symbolize its antecedent and consequent separately, then join them using ‘⊃’.

The antecedent of the conditional is All philosophy professors teach logic, which is a categorical proposition. It can be symbolized ‘(∀x)(Px ⊃ Tx).’ The consequent of the conditional is Some students learn to reason well. That, too, is a categorical proposition, symbolized ‘(∃x)(Sx • Rx).’ Joining the antecedent and consequent together with a ‘⊃’, the sentence as a whole can be symbolized as follows:

((∀x)(Px ⊃ Tx) ⊃ (∃x)(Sx • Rx))

This complete formula is not a categorical proposition, but it accurately represents the meaning and logical form of the original English sentence.