Necessary v. Sufficient Conditions

by Jason Stotts

The Case of Morrell v. Stotts

At Philosophy Department Senior Symposium a question was raised in which Matt Morrell and I took polemical sides – the question was on the nature of necessary conditions and the nature of sufficient conditions. This article will chronicle the debate point by point with Matt and me adding to it in turn. Comments are welcome, especially if you discover a problem with either of our arguments or if we seem to be speaking at cross-purposes.

Matt #1:

My position is this: If the exhaustive set of necessary conditions is satisfied for some action X, then this is a sufficient condition for X.

Jason #1:

My problem, Matt, with your position is two-fold:

1. Is this exhaustive of the class of sufficient conditions? I would object, whole-heartedly, if you asserted that all sufficient conditions are only exhaustive sets of necessary conditions – however, I shall reserve that objection in case this is not your position.

2. What difference does it matter that this set of necessary conditions is exhaustive – no matter how many necessary conditions you fulfill, unless one of them is sufficient you will never have X. That is, of course, unless you want to argue that sufficiency is something that only comes about from having this set of exhaustive necessary conditions (see “1” above). Otherwise all I’ve done, by exhausting the set of necessary conditions, is to make it extremely easy to effect the sufficient condition – a condition which, it seems to me, is fundamentally different from a necessary condition.

Matt #2:

Your intuition is correct that (1) is not included in my position, so I will move on to address point (2). I’m a bit surprised to see that you’ve taken this argumentative tact. To say “you can not have X unless one [condition] is sufficient” mandates that, for every X, there exists a single sufficient condition. By definition, if a sufficient condition is fulfilled for X, no other conditions are necessary to establish X. Thus, your stipulation that every X must be accompanied by a sufficient condition renders senseless the notion of conditions that are necessary but not sufficient. By taking this tact, then, you destroy the distinction between necessary and sufficient conditions — something of which you accused me earlier.

The problem with your argument is that you establish a false dichotomy between these two claims:

a) A sufficient condition can not be merely an exhaustive set of necessary conditions (your position)

b) A sufficient condition must be an exhaustive set of necessary conditions (1).

In reality, there is a third possibility. This states that a sufficient condition can, but need not be, merely an exhaustive set of necessary conditions. In other words, an exhaustive set of necessary conditions forms one instance of a sufficient condition, though not all situations require that a sufficient condition be an exhaustive set of necessary conditions. Whether this sufficient condition for X is an exhaustive set of necessary conditions will depend on the particular instance of X at hand. This third option is my position.

Jason #2:

Matt, I’m not sure that your “third way” actually presents a viable option significantly different from the alternatives; furthermore, my position is that no sufficient conditions are sets of necessary conditions. This is substantially different from the position which you wish you to attribute to me. My position is the categorical claim:

No SC are {NC}.

This is, I think, a position which does not destroy the distinction between NC and SC. I can still say that some action A is a NC for X, but that A is not a SC for X – I can also still say that some action Y is a SC for X, but that Y is not a NC for X. I see no problem with this, to particularize my point: the former would be something like “having normally functioning legs is a necessary condition for jumping, but having normally developed legs is not sufficient for jumping”, while the latter would be something like “Jumping is a sufficient condition for leaving the ground, but jumping is not a necessary condition for leaving the ground”.

I firmly belief that when one talks of the “cause” of some X, that one is only looking for the sufficient condition for X – to ask about the necessary conditions goes beyond the standard explanation and requires an explicit question pertaining to it. However, just because some action Y is a SC for X does not mean that I cannot go further and say that {A,B,C} were NC. Think about it less abstractly for a minute and you’ll see how evident this is: it was necessary for me to go to preschool, go to grade school, go to middle school, and go to high school – but none of those were sufficient for me to get into college, whereas actually applying was sufficient.

Because of the foregoing, I don’t see how I’ve destroyed the distinction, nor do I see how you have substantiated your claim that it’s ever possible for a SC to be a {NC}.

Also, it seems to me that your original position relies on an equivocation between senses of “necessary”. The first sense is the logical sense where A must happen for X, but does not “cause” X. The second sense is the common language(CL) sense where A must happen for X, but this can “cause” X. For example, the CL statement “running was necessary for me to sweat” makes sense to us, but is not true. The true statement would be “running was sufficient for me to sweat”, since clearly running caused me to sweat and others things could have caused it. The CL path seems to be what you are relying on, since you say that some {NC} are SC for X – but this only seems to follow from the CL definition of necessary and not the proper logical definition of necessary

Matt #3:

Jason, it seems you are, in fact, holding the position I attribute to you. I maintain that your claim renders senseless the idea of conditions which are necessary but not sufficient. You have given me an example of a situation that has a single sufficient condition (i.e. jumping -> leaving the ground), and of a situation that has a condition that is necessary but not sufficient (i.e. having legs -> jumping). You must provide me with a single instance that has both:

a) a sufficient condition

and

b) a distinct necessary-but-not-sufficient condition (bearing in mind that this can not be included within the sufficient condition, based on your categorical claim).

Otherwise, your argument will face my earlier problem that it renders the idea of necessary-but-not-sufficient conditions to be unintelligible.

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