If A → B and B → C, what can we conclude about A → C?

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Multiple Choice

If A → B and B → C, what can we conclude about A → C?

If A leads to B and B leads to C, then A leads to C. This is the idea of transitivity for implications: when A occurs, it forces B to occur; and when B occurs, it forces C to occur. So whenever A is true, C must be true as well. The implication A → C is thus always true in this chain, and even if A is false, A → C remains true by the standard truth-functional definition of implication. The other possibilities don’t fit this chain of reasoning: reversing the implication would not be supported by the given statements, saying A and C are independent ignores the forced chain, and saying A implies not C contradicts the fact that A forces B and B forces C, which makes C true whenever A is true.

If A leads to B and B leads to C, then A leads to C. This is the idea of transitivity for implications: when A occurs, it forces B to occur; and when B occurs, it forces C to occur. So whenever A is true, C must be true as well. The implication A → C is thus always true in this chain, and even if A is false, A → C remains true by the standard truth-functional definition of implication.

The other possibilities don’t fit this chain of reasoning: reversing the implication would not be supported by the given statements, saying A and C are independent ignores the forced chain, and saying A implies not C contradicts the fact that A forces B and B forces C, which makes C true whenever A is true.

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