A first book in logic by Henry Bradford Smith

By Henry Bradford Smith

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N , then the height of µ is max{height(ν1 ), . . 19) The height of a tree is defined to be the height of its root (cf. 15)). (d) A path from node ν to node µ is a set of nodes {ν0 , . . , νk } where ν0 is ν, νk is µ, and for each i < k, νi is the mother of νi+1 . A path from the root to a leaf µ is called a branch (to µ). 20) has one leaf and no non-leaves, and its height is 0. The centre tree has two leaves, two nodes of arity 1 and one node of arity 2; the root has height 3, so the height of the tree is also 3.

So the second symbol determines whether we are in case (b) or (c). In case (b) the occurrence of the head is uniquely determined as in the lemma. So φ is everything to the left of this occurrence, except for the first ‘(’ of χ; and ψ is everything to the right of the occurrence, except for the last ‘)’ of χ. Similarly in case (c), φ is the whole of χ except for the first two symbols and the last symbol. The theorem allows us to find the parsing tree of any formula of LP, starting at the top and working downwards.

As in case (β), the fact that ν1 and ν2 satisfy (a) implies that the depth is at least 2. (ζ) (ν1 ∧ ν2 The depth is 1 + 0 + 0 + 0 = 1. (η) µ itself The depth is 1 + 0 + 0 + 0 − 1 = 0 as required. This proves that µ satisfies (a). To prove that it satisfies (b), we note from (δ) that the head symbol has depth 1. If t is any other occurrence of a functor in µ, then t must be inside either ν1 or ν2 , and it’s not the last symbol since the last symbol of a complex formula is always ‘)’. Hence t is the end of an initial segment as in case (β) or (ε), and in both these cases the depth of t is at least 2.

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