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Slide 1

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In the last chapter we began to construct derivations.

Slide 2

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So far we've studied and applied the introduction and elimination rules for the three binary connectives, conjunction, disjunction, and the conditional. These rules correspond to patterns of argumentation.

Slide 3

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Let’Äôs briefly recall the rules for disjunction.

Slide 4

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it is quite obvious that we can infer φ v ψ if we are given a proof of either φ or ψ; these patterns of reasoning give us the introduction rules.

Slide 5

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The corresponding elimination rule reflects the standard "argument by cases". We are given a proof of φ v ψ.

Slide 6

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If the assumption of φ allows us to prove ρ...

Slide 7

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and the assumption of ψ allows us to prove ρ...

Slide 8

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then we can infer ρ (without either of these assumptions).

Slide 9

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Now it's time to take a look at the rules of inference for our unary connective, negation. Negation plays obviously a crucial role in the classical indirect arguments we examined in the Introduction.

Slide 10

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We assume that the square root of 2 is rational, derive a contradiction and can infer logically the irrationality of the square root of 2.

Slide 11

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In this chapter, we look not only at the basic introduction and elimination rules for negation, but some useful derived rules as well.

Slide 12

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We'll also examine the notion of an indirect proof, and explain why the negation introduction and elimination rules are considered to be indirect.

Slide 13

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Finally, we'll present a strategy for finding contradictions to be used in applying the indirect rules.