Conditionals are sentences of the forms “If φ, [then] ψ” and “ψ if φ,” such as

1. a. If the village is flooded, then the dam must have broken.

2. b. If Henry had come to the party, Sue would have come too.

3. c. Paul would have bought the house if it hadn't been so expensive.

One may also classify as conditionals sentences that can be naturally put in the above forms, such as

1. a. They will leave in an hour, unless John changes his mind.

2. b. No guts, no glory.

3. c. Give Louis a toy and he'll ruin it.

which can be rephrased as, respectively, (1.3a), (1.3b), and (1.3c):

1. a. If John does not change his mind, they will leave in an hour.

2. b. If a person lacks courage, there will be no glory for him or her.

3. c. If one gives Louis a toy, he ruins it.

In “If φ, then ψ,” φ is called the “antecedent” and ψ the “consequent.”

Conditionals are special. They are special for a number of reasons, but probably most conspicuously for the heated controversy that they have generated, and continue to generate. Not that controversy is anything out of the ordinary in philosophy. But even in philosophy, controversies commonly take place against a shared background of basic assumptions. For instance, while there is ongoing controversy about the concept of knowledge, there is at the same time broad (if not universal) agreement on many core issues surrounding that concept. Few dispute that knowledge is factive; that it requires belief as well as justification; that justified true belief is not sufficient for knowledge, however; that coherence amongst one's beliefs is not enough to elevate these to the status of knowledge; that we can gain knowledge from testimony; and so on and so forth. Not so in the case of conditionals. For almost any claim about conditionals that is not downright trivial, it will be exceedingly hard to find a majority, or even a sizable minority, of philosophers who adhere to it.