midterm - Midterm PHIL 151 Feb 5 2010 Apart from the...

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Midterm PHIL 151 Feb. 5, 2010 Apart from the natural deduction proofs you are asked to provide in problem 6, you may in your proofs use any lemma or theorem from the van Dalen text. Problem 1 . Suppose we have a set of symbols that includes: a propositional symbol p i for each natural number i , symbols and ; and parentheses ( and ). For this problem take PROP to be the smallest set such that p i is in PROP for each natural number i . • ⊥ is in PROP . If ϕ and ψ are in PROP , then so is ( ϕ ψ ). Part a) (4points) Define the function length from PROP to the natural num- bers recursively such that length ( ϕ ) is the number of symbols in ϕ (counting parentheses as well as the other symbols). Part b) (4 points) Suppose ψ is a proposition in PROP . Define the function sub ψ,p 0 : PROP PROP recursively, so that sub ψ,p 0 ( ϕ ) is the result of substituting ψ for p 0 in ϕ . Part c)
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midterm - Midterm PHIL 151 Feb 5 2010 Apart from the...

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