phi_270_text_3_2

The law alternatives via assumptions tells us that, when sentences φ and φ^⋈ are contradictory, having one as a premise comes to the same thing as having the other as a conclusion—that is,

Γ ⊨ φ, Σ if and only if Γ, φ^⋈ ⊨ Σ

If we apply this to the contradictories φ and ¬ φ, we get a pair of principles

Γ ⊨ ¬ φ, Σ if and only if Γ, φ ⊨ Σ
Γ, ¬ φ ⊨ Σ if and only if Γ ⊨ φ, Σ

where we get the second by reversing the contradictory pair. The two together tell us that having a negation as either a premise or alternative comes to the same thing as having the unnegated sentence in the opposite role (where the opposition in question is the duality mentioned in 1.4.7).

We do not study conditional exhaustiveness directly, and we use of the basic law for conditional exhaustiveness mainly to exchange alternatives for premises so that a claim of conditional exhaustiveness may be converted into a claim of entailment. But suppose we apply it to entailment instead; that is, suppose we begin with only a single alternative (so the set Σ is empty). In this case, when φ and φ^⋈ are contradictory, we can say that

Γ ⊨ φ if and only if Γ, φ^⋈ ⊨

where the right-hand side says that φ^⋈ is inconsistent with (or is excluded by) Γ. When we express that inconsistency as the validity of a reductio argument, we get the following principle:

if φ and φ^⋈ are contradictory, then Γ ⊨ φ if and only if Γ, φ^⋈ ⊨ ⊥

We get our basic principles for negation by applying this principle to the case of negation by choosing the contradictory pair as a sentence and its negation, both in that order and its reverse. Turning the second if and only if principle around so that clause concerning negation comes first, the two principles are these:

Law for negation as a conclusion. Γ ⊨ ¬ φ if and only if Γ, φ ⊨ ⊥.

Law for negation as a premise. Γ, ¬ φ ⊨ ⊥ if and only if Γ ⊨ φ.

Although these principles are dual in something like the way that the earlier pair for conditional exhaustiveness were, each has a rather different significance. The first captures the core properties of negation while the second is closely tied to the equivalence of ¬ ¬ φ with φ (which, as was noted in 3.1.3, is about as controversial as anything gets in logic). Also, while the first will provide us with straightforward ways of planning for negative goals and carrying out these plans, the second gives an account of the role of negative premises only in the context of reductio arguments and, for this reason, has a less straightforward implementation as a derivation rule. We will go on to explore the implementation of the first now and postpone a discussion of the second until 3.3.