Necessary and sufficient conditions

A club’s rules say that only paid-up members can vote in its election, and that members vote at the annual meeting. Dana has paid her dues but will be away on the day of the meeting.

Being paid up is necessary for voting: without it, no vote. It isn’t sufficient, because Dana is paid up and still won’t vote. Being paid up and attending the meeting are, together, sufficient.

“If P, then Q” can be described in either of two ways:

• P is sufficient for Q: whenever P holds, Q does.
• Q is necessary for P: P never holds without Q.

These say the same thing. “If it’s a square, it has four sides” states that being a square is sufficient for four sides, and equally that four sides are necessary for being a square. Hurley and Watson’s textbook gives a mnemonic, SUN: in “if S, then N”, the sufficient condition goes in the “if” part and the necessary condition in the “then” part.

Everyday English puts conditions in several places. The ones that trip people up:

The rule of thumb from the same textbook: what follows “if” is the antecedent (the “if” part of the standard form), and what follows “only if” is the consequent (the “then” part). “The dishwasher runs only if the door is latched” means if the dishwasher runs, the door is latched. A latched door is necessary, not sufficient: someone still has to press start.

The two most common formal fallacies about conditionals are both confusions between these ideas.

• Treating a necessary condition as sufficient is Affirming the consequent. “If P, then Q” makes Q necessary for P. Concluding P because Q holds treats Q as if it were enough. “Every square has four sides. This shape has four sides. So it’s a square.”
• Treating a sufficient condition as necessary is Denying the antecedent. “If P, then Q” makes P sufficient for Q. Concluding that Q fails because P fails treats P as the only way to get Q. “Every square has four sides. This shape isn’t a square. So it doesn’t have four sides.”
• Reading “only if” as “if” swaps the two. “You’ll get a refund only if you have the receipt” makes the receipt necessary. Hearing it as “if you have the receipt, you’ll get a refund” turns it into a guarantee the store never offered.

The valid moves use each condition for what it is. A sufficient condition that holds gives you the result (Modus ponens). A necessary condition that fails rules the result out (Modus tollens).

The question to ask: is this condition enough, or only required?

Most things have several necessary conditions, and often several different sufficient ones. Several necessary conditions can add up to a sufficient one, as being paid up and attending did in the club example. Two consequences follow:

• Showing that one necessary condition holds, however important, doesn’t show the result will happen. The other necessary conditions still have to be met.
• Showing that one sufficient condition is absent doesn’t show the result won’t happen. Another sufficient condition may be present.

“Cause” can mean either one. Hurley and Watson note that in ordinary English “cause” is ambiguous among the three: sometimes a sufficient condition (a cause that is enough by itself, even though other things would work too), sometimes a necessary one (something the effect can’t happen without, though it isn’t enough alone), and sometimes both. A claim that X “causes” Y needs to be read with that in mind before anyone argues from it.

Causal talk uses “necessary” loosely. Everyday causes are rarely sufficient by themselves or necessary without qualification. The philosopher J. L. Mackie proposed in 1965 that a cause is typically what he called an INUS condition: an Insufficient but Necessary part of a condition that is itself Unnecessary but Sufficient for the effect. A cigarette dropped in dry grass isn’t enough to start a fire alone (it needs dry fuel, air, no rain), but in that particular set of circumstances the fire wouldn’t have started without it; and that whole set is enough, though other sets, like a lightning strike, could also start a fire. According to Brennan’s Stanford Encyclopedia entry, a schema like Mackie’s became the basis of the “sufficient cause model” of disease in epidemiology. So when someone says a cause was “necessary”, they usually mean necessary in those circumstances, not in the strict sense of the table above.

“If” doesn’t always state the same kind of condition. The tidy two-way reading, where “if P, then Q” makes Q necessary for P, is what the Stanford Encyclopedia entry calls the “standard theory”, used in many logic and critical thinking texts. It fits inferences well. But philosophers and linguists have pointed out cases where it sounds wrong. On the standard theory, “if you touch me, I’ll scream” makes my screaming a necessary condition of your touching me, which gets the dependency backwards: the scream depends on the touch, not the other way round. What survives is the inferential sense. No scream is a reason to think there was no touch, even though the scream isn’t needed to bring the touch about. So “necessary” in the table above is a claim about inference (from P you may conclude Q, and from not-Q you may conclude not-P), which isn’t always the same as “needed to make it happen”.