Many logic students struggle with understanding the constructive dilemma rule, also known as “CD.” However, when you consider the meaning of the symbols and form of the constructive dilemma rule, the constructive dilemma rule is actually quite intuitive.
The constructive dilemma (CD) rule for natural deduction logic proofs is as follows:
Premise 1: (p → q) & (r → s)
Premise 2: p v r
Conclusion: q v s
(Note: there are other forms of the constructive dilemma rule, but this form is the most commonly used.)
The constructive dilemma rule tells you that if you have a statement of the form (p → q) & (r → s) and a statement of the form p v r, then you can derive the statement q v s on a subsequent line of a logic proof. But what does this mean?
The meaning of constructive dilemma is actually very easy to explain in ordinary language. The first premise tells you that some statement p implies another statement q, and also that some statement r implies another statement s. The second premise tells you that either statement p or statement r is true. Well, if either p or r must be true, and if each of these statements implies some other statement, then one of the two implied statements, either q or s, must be true. This is the reason that the conclusion, q v s, follows from the two premises.
To use the constructive dilemma rule in a natural deduction logic proof, simply cite the two premises used for the constructive dilemma step by number, and cite “CD” as the justification for the new line of your proof. The constructive dilemma rule is named “constructive dilemma” because you are literally constructing a new dilemma (also called a “disjunction,” or an “either…or…” statement) from the premises cited previously.