Topics Logic

Question Referring to propositional logic conditionals, if we say that an antecedent A is a necessary and sufficient condition for consequent B, can we say that A caused B?

Accepted:
June 6, 2007

Comments

No.

That A is a necessary condition for B means that B presupposes A, that B cannot hold without A also holding.

That A is a sufficient condition for B means that A implies B, that A cannot hold without B also holding.

That A is a necessary and sufficient condition for B thus states a symmetrical relation between A and B: Neither can hold without the other, that is, both hold or neither.

If A being a necessary and sufficient condition for B indeed implied that A caused B then, given symmetry, it would likewise imply that B caused A. A and B would have caused each other -- a rather odd way for them to come about.

An example may help. In this example, A is that you are an unmarried male human adult at some given time t, and B is that you are a bachelor and this same time t.

Your being an unmarried male human adult at t is a necessary condition for your being a bachelor at t. (For you to be a bachelor at t, you must be an unmarried male human adult at t. Your being a bachelor at t presupposes your being an unmarried male human adult at t.) Your being an unmarried male human adult at t is also a sufficient condition for your being a bachelor at t. (That you are an unmarried male human adult at t implies that you are a bachelor at t.)

So, your being an unmarried male human adult at t is a necessary and sufficient condition for your being a bachelor at t.

Yet, clearly, your being an unmarried male human adult at t was not caused by your being a bachelor at t. Nor was your being a bachelor at t caused by your being an unmarried male human adult at t. In fact, for all I know, you may be female -- in which case any claims about what caused you to be a bachelor would be meaningless. Yet, even if you are female, it would remain true that your being an unmarried male human adult at t is a necessary and sufficient condition for your being a bachelor at t. This statement does not assert or entail that you are both a bachelor and an unmarried male human adult at t. Rather, it merely asserts that, at t, you are either both or neither, that you cannot be one without being the other. As it happens, it is true at any time of any X -- including men, girls, dolphins, atoms, and vacuum cleaners -- that X's being a an unmarried male human adult at this time is a necessary and sufficient condition for X's being a bachelor at this time. At any time, anything is either both (a bachelor and an unmarried male human adult) or neither.

Question Referring to propositional logic conditionals, if we say that an antecedent A is a necessary and sufficient condition for consequent B, can we say that A caused B? Accepted:June 6, 2007

Question Referring to propositional logic conditionals, if we say that an antecedent A is a necessary and sufficient condition for consequent B, can we say that A caused B?

Accepted:
June 6, 2007