Download Algebra: An Elementary Text-Book for the Higher Classes of by George Chrystal PDF

By George Chrystal

This Elibron Classics ebook is a facsimile reprint of a 1904 version through Adam and Charles Black, London.

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Extra resources for Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1

Example text

8. , a wff, that is provable in one but not in the other system. ] As a last remark concerning certain peculiarities of LK , let us note a discrepancy in the formulation of the two-premise rules. The ( ∧) and (∨ ) rules assume that the two premises are identical except A and B . On the other hand, the (⊃ ) rule does not prescribe that Γ and Θ , or Δ and Λ are the same. The rule could have been formulated instead as follows. Γ Δ, A B, Γ Γ, A ⊃ B Δ Δ ⊃ e We switched A ⊃ B to the other edge in the antecedent (in view of our previous complaint related to typing), but the main difference is that now A and B must be proved in the same sequents, though on different sides of the turnstile.

Conversely, prove that if the left introduction rule for ⊃ is (⊃ e ) , then (⊃ ) is derivable (when the rest of LK is kept unchanged). We listed some features of the original formulation of LK that later turned out to be puzzling, undesirable or suboptimal. To further motivate our formulation of the cut rule, let us consider the sequent calculus again as a system to reason about inferences. The right premise of the cut rule says that Λ can be derived from Θ1 , C , Θ2 . If from Γ the formula C is derivable within Δ1 and Δ2 (as given by the left premise), then placing Γ in the spot where C is (that is, starting with Θ1 , Γ , Θ2 ), should suffice for the derivation of Λ within Δ1 and Δ2 (that is, Δ1 , Λ, Δ2 ).

Classical first-order logic 37 the principal formula p . Accordingly, there are four subcases. ) (a) (id), (id) . The proof tree is of the form given on the left below. We can transform the proof by retaining one of the premises, as shown on the right. p p p p cut p p p p (b) (id) , ( K ) . The proof given and the proof after the transformation are as follows. The justification is exactly the same as for the proof tree rooted in the right premise of the cut. . Γ Δ . K Γ Δ p p p, Γ Δ cut K p, Γ Δ p, Γ Δ (c) ( K ), (id) .

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