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Quantification Theory

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.

Individuals, Predicates, and Quantifiers

In addition to the familiar symbols of the propositional calculus, quantification theory also employs special symbols of four special sorts:

Writing an individual symbol directly after a predicate symbol signifies that this thing has that property. Thus, using the above examples,  Fa, Gb, and Hc  would signify "Allison is free," "Bill is greedy," and "This car is heavy" respectively. Since these are complete statements, they can be combined using the statement connectives to form such compound statements as  Fa • Gb, Hg ⊃ Gb, and Hc ∨ ~Hc . Put together, these symbols can be used to represent each of the four major forms of categorical proposition: Any of the rules of replacement from the propositional calculus may be applied to these statements of quantification theory. Thus, for example, we can easily demonstrate the converse of an E proposition:
		1. (x)(Sx ⊃ ~Px)		premise
		2. (x)(~~Px ⊃ ~Sx)		1 Trans.
		3. (x)(Px ⊃ ~Sx)		2 D.N.

Quantification Rules

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:

Using these quantification rules together with the inference and replacement rules of the propositional calculus, it is possible to prove the validity of any of the valid categorical syllogisms. For EIO-1, for example:
		 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

Asyllogistic Inferences

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.

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Last modified 12 November 2011.
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