2 \u03b1 1 \u0393 3\u03b1 2 \u03931 \u0393 2 \u0393 3\u0393 1 \u03b11 \u03b1 2 \u0393 2 \u0393 3 Left \u03b1 1 \u0393 1 \u03b1 2 \u0393 2 \u0393 3 \u0393 1 \u0393 2 \u03b1 1

2 α 1 γ 3α 2 γ1 γ 2 γ 3γ 1 α1 α 2 γ 2 γ 3

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2 α 1 , Γ 3 α 2 , Γ 1 , Γ 2 Γ 3 Γ 1 , ( α 1 α 2 ), Γ 2 Γ 3 Left- α 1 , Γ 1 α 2 , Γ 2 , Γ 3 Γ 1 Γ 2 , ( α 1 α 2 ), Γ 3 Right- Γ 1 , Γ 2 α , Γ 3 Γ 1 , ¬α , Γ 2 Γ 3 Left- ¬ α , Γ 1 Γ 2 , Γ 3 Γ 1 Γ 2 , ¬α , Γ 3 Right- ¬ Γ 1 , ( α 1 α 2 ), ( α 2 α 1 ), Γ 2 Γ 3 Γ 1 , ( α 1 α 2 ), Γ 2 Γ 3 Left- Γ 1 Γ 2 , ( α 1 α 2 ), Γ 3 Γ 1 Γ 2 , ( α 2 α 1 ), Γ 3 Γ 1 Γ 2 , ( α 1 α 2 ), Γ 3 Right-
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Propseq1.doc:1998/04/14:page 9 of 31 At this point, it might appear that we are no better off with G than we are with the Hilbert system. We have an infinite number of axioms, and so a infinite number of proofs. Q: To begin a proof, must we “divine” the correct set of starting axioms? A: No! With G , we will construct the proof backwards. That is, we will start with the statement which we wish to prove, and work backwards towards the axioms. There is an algorithm to construct such proofs. Before developing formally notions of soundness and completeness, we will work out a variety of examples.
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Propseq1.doc:1998/04/14:page 10 of 31 Examples: First of all, we will provide a proof, within G , that the wff ((A 1 A 2 ) ( ¬ A 2 ¬ A 1 )) is a tautology. As with resolution and semantic tableaux, the proof is best represented as a directed graph. A solution is shown below on the next slide, in both notations. Notice the following: In constructing this proof, we work backward from the conclusion, to reach valid atomic sequents. The formula ϕ to be validated is represented as the sequent ϕ . As we work backwards, each step up the tree produces a “simpler” formula. (This means that the tree cannot grow forever.) A path ends whenever an atomic sequent is encountered. Such a sequent cannot be reduced further. The original formula is valid iff each path ends with a valid atomic sequent.
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Propseq1.doc:1998/04/14:page 11 of 31 »… » ((A 1 A 2 ) ( ¬ A 2 ¬ A 1 )) »… » A 1 , ( ¬ A 2 ¬ A 1 ) » ¬ A 2 » ¬ A 1 , A 1 » (A 1 A 2 ) » ( ¬ A 2 ¬ A 1 ) » A 2 » ( ¬ A 2 ¬ A 1 ) » ¬ A 2 , A 2 » ¬ A 1 Right- Right- Left- Right- Valid Valid » A 2 » A 2 , ¬ A 1 » A 1 , ¬ A 2 » A 1 » A 1 » A 1 , ¬ A 2 Right- ¬ Left- ¬ Left- ¬ » A 1 , A 2 » A 2 ((A 1 A 2 ) ( ¬ A 2 ¬ A 1 )) A 1 , ( ¬ A 2 ¬ A 1 ) ¬ A 2 ¬ A 1 , A 1 (A 1 A 2 ) ( ¬ A 2 ¬ A 1 ) A 2 ( ¬ A 2 ¬ A 1 ) ¬ A 2 , A 2 ¬ A 1 Right- Right- Left- Right- Valid Valid A 2 A 2 , ¬ A 1 Right- ¬ A 1 , ¬ A 2 A 1 A 1 , A 2 A 2 Left- ¬ Right- ¬ A 1 A 2 , A 1
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Propseq1.doc:1998/04/14:page 12 of 31 In establishing validity it is not necessary to expand a node until it becomes an atomic sequent. For validity, it suffices that there be a common proposition name (as a wff; not as a component of one) in each sequence. Thus, the expansions in the previous example could actually have been halted without expanding the boxes which are outlined with dashes. It is in fact possible to halt the expansion of a node when there is a common wff in each sequence. However, testing for such common formulas at each step involves substantial overhead, and so the decision to do this should be weighed carefully. In these notes, only checking for matching proposition names will be performed.
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