In logic, the hypothetical syllogism says That given three statements P, Q. and

Question

In logic, the hypothetical syllogism says That given three statements P, Q. and R, the hypotheses P rightarrow Q and Q rightarrow R logically imply P rightarrow R. (Symbolically this is written (P rightarrow Q) ^ (Q rightarrow R) (P rightarrow R)). Use induction to show that, given n statements A1, A2, ..., An that (A1 rightarrow A2) ^ (A2 rightarrow A3) ^ ... ^ (An-1 rightarrow An) (A1 rightarrow An) for n 2.

Explanation / Answer

Let n=3 (A1->A2)^(A2->A3) => (A1->A3), from the given hypothesis. Let this be true for n=p, where p>3 So, (A1->A2)^(A2->A3)....^(Ap-1^Ap)=>(A1->Ap) For n=p+1, (A1->A2)^(A2->A3)....^(Ap-1^Ap)^(Ap->Ap+1) =>(A1->Ap)^(Ap->Ap+1) From the given hypothesis, it can be derived as (A1->Ap)^(Ap->Ap+1) => (A1->Ap+1) So, (A1->A2)^(A2->A3)....^(Ap-1^Ap)^(Ap->Ap+1) => (A1->Ap+1) Hence it holds for all 'n' By Principle of Mathematical Induction, we can say that the law is true.

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