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In this final lesson on symbolic logic, we'll take a very brief look at modern methods of representing the internal structure of propositions in first-order predicate calculus (or quantification theory).
Incorporating all of the propositional calculus along with a few new symbols and rules of inference, the predicate calculus provides another way of handling the same logical forms we examined in our study of categorical logic.
In addition to the familiar symbols of the propositional calculus, quantification theory also employs special symbols of four special sorts:
|constants||A, B, C...||a, b, c...||F, G, H...|
|variables||p, q, r...||...x, y, z|
|operators|| ~, &, v, |
1. (x)(Sx ⊃ ~Px) premise 2. (x)(~~Px ⊃ ~Sx) 1 Trans. 3. (x)(Px ⊃ ~Sx) 2 D.N.
In order to prove the validity of syllogisms, however, we first need to strip the quantifiers from each statement, apply the appropriate rules of inference, and then restore quantifiers to each statement. The four quantification rules dictate the conditions under which it is permissible to delete or add a quantifier:
(x)( Øx ) _______ ØuThe "u" in this case can be any arbitrarily chosen individual constant or variable. In the context of a proof, for example, the truth of "(x)[Fx ⊃(Gx ∨ Hx)]" could be used to justify that of "Fb ⊃ (Gb ∨ Hb)." If the statement holds for all x, then it certainly must hold for b.
Øy _______ (x)( Øx )In this case, however, it is crucial that the "y" is an arbitrarily chosen individualthat is, an individual that was introduced into the proof by an application of UI. Only then can we be sure that what holds of it is not some special feature but something that would hold equally well of all " x."
(∃x)( Øx ) _______ ØuIn this case, the individual constant "u" must be one which has never been used in any earlier line of the proof; otherwise, we might mistakenly associate two things which have nothing in common. Thus, it is usually best to employ EI as soon as possible (certainly, before any application of UI) in a proof.
Øu _______ (∃x)( Øx )Here, as in UI, "u" may be any individual constant or variable.
1. (x)(Mx ⊃ ~Px) premise 2. (∃x)(Sx • Mx) premise 3. Sd • Md 2 EI 4. Md ⊃ ~Pd 1 UI 5. Sd 3 Simp. 6. Md • Sd 3 Comm. 7. Md 6 Simp. 8. ~Pd 4, 7 M.P. 9. Sd • ~Pd 5, 8 Conj. 10. (∃x)(Sx • ~Px) 9 EG
Quantification theory makes it possible to prove the validity of many arguments that could not easily be expressed in categorical logic at all.
In fact, Russell and Whitehead showed less than a hundred years ago that
a higher-order predicate calculus (one that fills in the chart above by including variables and quantifiers for predicates as well as for individuals) is sufficient to provide
logical demonstrations of simple arithmetical truths.
A few decades later, however, Gödel showed that any such powerful system must contain at least one proposition whose truth or falsity cannot be proven.