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Unformatted text preview: Completeness in Propositional vs Predicate Logic Branden Fitelson 04/03/07 In Section 2 of Hunter, we examined a Henkin-type proof of the completeness of the formal system PS of propositional logic. In section 3, completeness of the system QS of predicate logic is also proved in the style of Henkin. Although the basic strategy of the two proofs is the same, there are several significant differences, which is what this handout is about. The following table summarizes the heart of the proof for PS — 32.13: Every p–consistent set Γ of wffs of P has a model. Proof Theory Model Theory An arbitrary p-consistent set Γ of P Γ has a model. ∩ (is a subset of : 32.12) ⇑ A maximal p-consistent set Γ of PS 32.13 = ⇒ Γ has a model There is a great deal more complexity in the language Q than in the language P, the semantics for Q versus the semantics for P, and in the formal systems QS vs PS. This complexity leads to some different notions in the completeness proof, the heart of which is — 45.14: Every consistent, negation-complete, closed first order theory (fot) K has a denumerable model. Proof Theory Model Theory An arbitrary consistent fot K K has a denumerable model. û (is extended by : 45.13) ⇑ A consistent, negation-complete, closed fot K 45.14 = ⇒ K has a denumerable model A trivial difference is that we speak of “consistent” sets of formulas of Q (or Q + ) rather than “p-consistent” sets of formulas of P. The reason for this is that there is no notion in the semantics for Q corresponding to that of m-consistency for P. In the semantics for P, if a set has no model, then for all interpretations, there is at least one formula in the set that is false . But in the semantics for Q, some sets have no models because some of the formulas in the set are simply not true in the sense that they lack truth values (truth and falsehood are defined in terms of being satisfied by all or no I-sequences). We might have formulas in the set which contain free variables, and which are satisfied by some I-sequences but not others. We do not want to say that such sets are inconsistent , since they have some “positive semantic value” (so to speak). Specifically, they are simultaneously satisfiable . The correlate of an m-inconsistent set would be a set that is not simultenously satisfiable ....
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- Spring '07
- Logic, Order theory, Mathematical logic, Model theory, First-order logic, p-consistent set