Normal forms in Software Integration 3 of 9 barcode in Software Normal forms

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1.5 Normal forms use software code 39 extended writer toincoporate code 3/9 with software Codabar the left-han barcode 3 of 9 for None d sides of these four sequents. This is the place where we rely on the law of the excluded middle which states r r, for any r. We use LEM for all propositional atoms (here p and q) and then we separately assume all the four cases, by using e.

That way we can invoke all four proofs of the sequents above and use the rule e repeatedly until we have got rid of all our premises. We spell out the combination of these four phases schematically:. 1 2 3 4 5 6 7 8 9 p q p p q p p q p p q p e p q p p q p p q p e e p p p q q q .. .

. ..

ass q .. .

. ..

ass p q .. .

. ..

ass q .. .

. ..

LEM q q ass LEM ass LEM ass. As soon as y USS Code 39 for None ou understand how this particular example works, you will also realise that it will work for an arbitrary tautology with n distinct atoms. Of course, it seems ridiculous to prove p q p using a proof that is this long. But remember that this illustrates a uniform method that constructs a proof for every tautology , no matter how complicated it is.

Step 3: Finally, we need to nd a proof for 1 , 2 , . . .

, n . Take the proof for 1 ( 2 ( 3 (. .

. ( n ) . .

. ))) given by step 2 and augment its proof by introducing 1 , 2 , . .

. , n as premises. Then apply e n times on each of these premises (starting with 1 , continuing with 2 etc.

). Thus, we arrive at the conclusion which gives us a proof for the sequent 1 , 2 , . .

. , n . Corollary 1.

39 (Soundness and Completeness) Let 1 , 2 , . . .

, n , be formulas of propositional logic. Then 1 , 2 , . .

. , n is holds i the sequent 1 , 2 , . .

. , n is valid..

1.5 Normal forms In the last Software barcode 39 section, we showed that our proof system for propositional logic is sound and complete for the truth-table semantics of formulas in Figure 1.6..

1 Propositional logic Soundness me USS Code 39 for None ans that whatever we prove is going to be a true fact, based on the truth-table semantics. In the exercises, we apply this to show that a sequent does not have a proof: simply show that 1 , 2 , . .

. , 2 does not semantically entail ; then soundness implies that the sequent 1 , 2 , . .

. , 2 does not have a proof. Completeness comprised a much more powerful statement: no matter what (semantically) valid sequents there are, they all have syntactic proofs in the proof system of natural deduction.

This tight correspondence allows us to freely switch between working with the notion of proofs ( ) and that of semantic entailment ( ). Using natural deduction to decide the validity of instances of is only one of many possibilities. In Exercise 1.

2.6 we sketch a non-linear, tree-like, notion of proofs for sequents. Likewise, checking an instance of by applying De nition 1.

34 literally is only one of many ways of deciding whether 1 , 2 , . . .

, n holds. We now investigate various alternatives for deciding 1 , 2 , . .

. , n which are based on transforming these formulas syntactically into equivalent ones upon which we can then settle the matter by purely syntactic or algorithmic means. This requires that we rst clarify what exactly we mean by equivalent formulas.

. 1.5.1 Semant ic equivalence, satisfiability and validity Two formulas and are said to be equivalent if they have the same meaning.

This suggestion is vague and needs to be re ned. For example, p q and p q have the same truth table; all four combinations of T and F for p and q return the same result. Coincidence of truth tables is not good enough for what we have in mind, for what about the formulas p q p and r r At rst glance, they have little in common, having di erent atomic formulas and di erent connectives.

Moreover, the truth table for p q p is four lines long, whereas the one for r r consists of only two lines. However, both formulas are always true. This suggests that we de ne the equivalence of formulas and via : if semantically entails and vice versa, then these formulas should be the same as far as our truth-table semantics is concerned.

. De nition 1. barcode 3/9 for None 40 Let and be formulas of propositional logic. We say that and are semantically equivalent i and hold.

In that case we write . Further, we call valid if holds. Note that we could also have de ned to mean that ( ) ( ) holds; it amounts to the same concept.

Indeed, because of soundness and completeness, semantic equivalence is identical to provable equivalence.
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