On formulas with valid cases

Extract

Commonly, when we succeed in showing a formula of quantification theory to be consistent, we do so by producing a true interpretation. Sometimes we achieve the same effect without even exceeding the resources of quantification theory itself: we show a formula consistent by producing a valid formula from it by substitution. Example: ‘(∃x)Fx ⊃ (x)(∃y) (Gxy ▪ Fy)’ is seen consistent by noting its valid substitution case ‘(∃x)Fx ⊃ (x)(∃y)(Fx ∨ Fy ▪ Fy)’. How generally available is this latter method? I shall show that it is available if and only if the formula whose consistency is shown is satisfiable in a one-member universe.

The “only if” part is immediate. For, if ψ is a substitution case of ϕ, then ϕ is satisfiable wherever ψ is; and ψ, if valid, is satisfiable in a one-member universe.

Conversely, suppose a true interpretation of ϕ in a one-member universe. Each predicate letter of ϕ is thereby interpreted outright as true or false, in effect, since there is no varying the values of ‘x’, ‘y’, etc. Now form ψ from ϕ by substitution as follows: change each atomic formula ϕ_i(e.g. ‘Fx’, ‘Fy’, ‘Gxy’) to ⌜ϕ_i⊃ϕ_i⌝ if its predicate letter is one that was interpreted as true, or to ⌜ϕ_i ▪ − ⊃ϕ_i⌝ if its predicate letter is one that was interpreted as false. Clearly ψ under all interpretations even in large universes, will receive the truth value that ϕ received under the supposed interpretation in the one-member universe. So ψ is valid.