06-prednd

06-prednd - Natural deduction for predicate logic Readings:...

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Natural deduction for predicate logic Readings: Section 2.3. In this module, we will extend our previous system of natural deduction for propositional logic, to be able to deal with predicate logic. The main things we have to deal with are equality, and the two quantifiers (existential and universal). All of the rules from propositional logic carry over to predicate logic, and there are six new rules (introduction and elimination for each of the new features). We will also introduce one derived rule not in the textbook, for convenience. 1
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Equality The rule for introducing equality is fairly simple: it simply says that any term t is equal to itself, and that no premises are needed to conclude this. = i t = t The “elimination” rule for equality is more interesting; it describes how substitution may be used in formulas. 2
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If t 1 and t 2 are free for z in φ , then we may substitute the term t 2 for t 1 in φ . But our definition of substitution substitutes a term for a variable. Hence the following phrasing: t 1 = t 2 φ [ t 1 /z ] = e φ [ t 2 /z ] Of course, any variable can be used in this rule, not just z . In using this rule, there are four important components: the equality t 1 = t 2 , the formula φ [ t 1 /z ] , the variable z involved in the substitution, and the formula φ . We extend the textbook’s notation to put all four of these as justification for any use of the rule. 3
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In Math 135, you learned that any notion of equality should be reflexive, symmetric, and transitive. Let’s prove these properties using our new rules. Reflexivity follows immediately from the = i rule. The sequent schema expressing symmetry is t 1 = t 2 t 2 = t 1 , where t 1 and t 2 represent specific terms. 1 t 1 = t 2 premise 2 t 1 = t 1 =i 3 t 2 = t 1 =e1 , 2 , z, z = t 1 Note that the justification for line 3 cites t 1 = t 2 as the equality to be eliminated, z as the variable to be substituted for in the formula z = t 1 , t 1 = t 1 as the result of substituting t 1 for z , and t 2 = t 1 as the result of substituting t 2 for z . 4
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The sequent expressing transitivity is t 1 = t 2 ,t 2 = t 3 t 1 = t 3 . 1 t 1 = t 2 premise 2 t 2 = t 3 premise 3 t 1 = t 3 =e2 , 1 , z, t 1 = z The justification for line 3 cites t 2 = t 3 as the equality to be eliminated, z as the variable to be substituted for in the formula t 1 = z , t 1 = t 2 as the result of substituting t 2 for z , and t 1 = t 3 as the result of substituting t 3 for z . 5
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The = e rule allows us to substitute the right-hand side of an equality for the left-hand side. It does not say anything about substituting the left-hand side of an equality for the right-hand side. In order to do this, we must switch around the equality using the three lines on the previous slide and then apply = e. This is annoying enough that we will introduce a derived rule not in
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This note was uploaded on 12/08/2011 for the course CS 246 taught by Professor Wormer during the Spring '08 term at Waterloo.

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06-prednd - Natural deduction for predicate logic Readings:...

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