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When you have a propositional sentence of the form PQ  — which we might read as "if P, then Q" — how can you tell when it is true, or false, based on the truth-values of P and Q in classical logic? When is this different from QP? And what is the connection to 'necessary' conditions, or sufficient conditions?

Also: does PQ mean that P causes Q?

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How to think about "P ⊃ Q" in plain English

In propositional logic, PQ is what is called a material implication. It doesn't mean that P and Q mean the same thing (they might not have the same truth value); all that it is, is a claim that if P is true, then Q is also true — without making any more claims than this.

An alternative way of considering PQ is as a "constraint" that someone claims holds for the state of affairs. That constraint is either satisfied (in which case P ⊃ Q is true) or it is violated (in which case P ⊃ Q is false). This picture is somewhat different from the usual way that we think of "if-then" statements, which is more like causation than constraint. For this reason, it might be helpful to describe P ⊃ Q as either

  • "P only if Q" (if P holds, then Q had better hold for the constraint to be satisfied); or
  • "either ¬P or Q" (if ¬P is false, then Q has to be true if the constraint is to be satisfied).

In this respect, we can think of "satisfying the constraint P ⊃ Q" as being something like a theory about how the world is. If it is true, then it is never violated; but if you can find one counterexample, then it must be false.

Necessary conditions and sufficient conditions

In the case that P ⊃ Q actually does hold, the two different phrasings "if P then Q" and "P only if Q" allow us to easily describe the relationship between P and Q in terms of either necessary conditions or sufficient conditions.

  • "If P then Q" intuitively means that P is enough to ensure that Q holds. In this respect, P is called a sufficient condition for Q.

  • Conversely, "P only if Q" intuitively means that Q is a precondition of P holding true; even though P implies Q, P also cannot hold without Q holding. In this respect, Q is called a necessary condition for P.

The notions of a "sufficient condition" and a "necessary condition" are complementary to one another; if one condition is sufficient for another, then that second condition is necessary for the first condition.

The truth-table for P ⊃ Q

Thinking about P ⊃ Q in terms of constraints allows us to explain the truth-table for conditionals:

 P  Q  |  P ⊃ Q
-------|--------
 F  F  |    T 
 F  T  |    T
 T  F  |    F
 T  T  |    T

We can describe these simply in terms of the conditions in which P is "allowed" to be true.

  • If P is false — no matter what the value of Q is — then we haven't violated the constraint P ⊃ Q, because it only imposes constraints on when P can be true.

    For instance, suppose P is "you succeed in climbing Mt. Everest", and Q is "you try to climb Mt. Everest". Because climbing Mt. Everest is difficult enough that we would presume that it has to be a deliberate undertaking, it is reasonable to say that you can only succeed in climbing Mt. Everest if you try: that is, P only if Q. But it is possible to fail to climb Mt. Everest in two different ways:

    1. either by not trying, or
    2. by trying but having problems (weather conditions, a serious accident, etc.) which prevent you from succeeding.

    In the case where P is false, we say that P ⊃ Q is vacuously true — it is true only because the constraint that it imposes on P being true is not put to the test.

  • In the case that P is true, the constraint P ⊃ Q is put to the test, and that constraint is satisfied only if Q is true. If P and Q are both true, then the constraint is satisfied, and so P ⊃ Q is true. However, if P is true and Q is false, then it cannot be that P only if Q; the constraint is violated, so that P ⊃ Q is false.

Is Q ⊃ P equivalent to P ⊃ Q, or does it allow you to infer P ⊃ Q ?

The short answer is that they are not equivalent, and neither one allows you to infer the other. There are two easy ways to see this.

  • P ⊃ Q is a constraint on when P can be true, while Q ⊃ P is a constraint on when Q can be true. In general, these are not comparable constraints; neither one allows you to infer the other.

  • We can make reference to the truth-tables for each, using the table we've already computed for P ⊃ Q to find out the values for each row in Q ⊃ P:

     P  Q  |  P ⊃ Q  |  Q ⊃ P
    -------|---------|---------
     F  F  |    T    |    T
     F  T  |    T    |    F
     T  F  |    F    |    T
     T  T  |    T    |    T
    

    We see that when only one of P or Q is true, then one of P ⊃ Q or Q ⊃ P is true — but not the other. Because P ⊃ Q and Q ⊃ P are not both satisfied under all of the same conditions as each other, we see that they are not logically equivalent.

We can see from the above that both P ⊃ Q and Q ⊃ P are true, if P and Q are either both true or both false — if P and Q are themselves equivalent. This is just the observation that P ≡ Q ("P is equivalent to Q") is logically equivalent to (P ⊃ Q) & (Q ⊃ P). But precisely because P ⊃ Q and Q ⊃ P are not logically equivalent, P ≡ Q is a stronger logical statement — a tougher constraint to satisfy — than either conditional alone.

Note that because P ≡ Q is equivalent to (P ⊃ Q) & (Q ⊃ P), it is possible to summarize P ≡ Q as saying that "P is a necessary and sufficient condition for Q", where the 'necessity' comes from the conditional Q ⊃ P, and the 'sufficiency' comes from P ⊃ Q.

Contrapositives

The conditional statement P ⊃ Q does have another equivalent conditional form, known as the contrapositive: ¬Q ⊃ ¬P. This is a way of rephrasing the constraints on when P can be true, as a constraint on when Q can be false.

  • "P only if Q" means that if Q fails, then P had better also fail; that is, "if ¬Q then ¬P".
  • Similarly, "either ¬P or Q" means that if Q fails, then ¬P must hold; again, "if ¬Q then ¬P".

We can show that ¬Q ⊃ ¬P also implies P ⊃ Q using a symmetric argument, and double negation (¬¬P ≡ P). And if you compute the truth-table for both formulae, you will find that they have the same values:

 P  Q  | ¬P  ¬Q  |  P ⊃ Q  |  ¬Q ⊃ ¬P
-------|---------|---------|---------
 F  F  |  T   T  |    T    |     T  
 F  T  |  T   F  |    T    |     T
 T  F  |  F   T  |    F    |     F
 T  T  |  F   F  |    T    |     T

So we can see just by calculation that they are logically equivalent propositions, or "constraints".

Does P ⊃ Q mean that there's a cause-and-effect relationship?

Something that trips up a lot of people is thinking about logical conditional statements as though they were cause-and-effect relationships. While cause-and-effect is one kind of conditional statement, it is not the only kind; so you should be careful not to assume that a logical "if-then" statement is saying anything about cause and effect.

For instance, you might worry about conditional statements such as

"If I eat an apple, then I will die"

— which might be true for someone severely allergic to apples, but is also true for absolutely anyone who is mortal. (Though I'm not personally allergic to apples, I do expect that I will die some day. So it is true that I will die whether or not I eat any more apples; so if I eat an apple, I will die.) You might complain that the condition Pif I eat an apple — has little to do with the conclusion QI will die. This doesn't seem to get much better if we present it as a constraint of the form "P only if Q" (I will only eat an apple if I will die) — again, what does dying have to do with whether I could eat an apple? What makes the proposition hold is the formulation "either I won't eat an apple, or I will die" is true. Indeed, in the chain of reasoning I applied to myself, I asserted the proposition QI will die — and then reasoned that the logical status of P (my eating an apple) had no bearing on that truth. My arriving at the conclusion "if I eat an apple, then I will die" is in this sense true, but somewhat perverse; it omits the fact that if I don't eat an apple, I will also die then.

We can make similar comments for vacuously true statements, such as

"If I were elected the King of France, I would make it rain ice cream every day".

With our current understanding of weather — or at least the cost of ice cream production and jet fuel — it seems unlikely that I could cause ice cream to rain down every day under any circumstances, regardless of whether or not I were the King of France. If my claim seems outlandish to you, it is because you're not evaluating P ⊃ Q (where P = "I am elected the King of France" and Q = "I would make it rain ice cream every day"), but only Q on its own (you can't see any reasonable circumstance in which I could make Q true). Of course, the conditional P ⊃ Q is a weaker assertion than Q: it is possible for Q to be false and P ⊃ Q to be true; it's equivalent to "either I will not be elected the King of France, or I will cause it to rain ice cream from the sky every day" (which now sounds like a surreal ultimatum rather than a promise). Given that France doesn't have any kings at all at present, let alone elected kings, this proposed constraint on the way the world works will likely be satisfied by virtue of the fact that I will not be elected the King of France.

In both cases, the condition had little to do with the consequence; eating apples is unlikely to cause me to die, and becoming King of France is unlikely to give me ice-cream-rain-god powers. But both of the conditional are true, because the constraints that they propose for the nature of the world are satisfied in either case.

When we do want to think of conditionals P ⊃ Q in terms of cause and effect, we usually want to work in models of the world in which it is possible for P to be true, and where it is also possible for Q to be false. The reason why the examples above might seem strange is exactly because in one case Q is unavoidable (I can't avoid dying), and in the other P is unreasonable (I can't be elected the King of France) — at least not with the current governmental systems and levels of medical technology.

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Excellent. Thank you. –  iphigenie Oct 30 '12 at 18:40
3  
Can we "sticky" this answer somewhere and link everyone to it who asks this question? It seems that every second question on this site is some form of this one, and this was the most perfect answer yet! –  Ryno Oct 31 '12 at 11:27
    
@NieldeBeaudrap Nice answer but wouldnt this be easier if there was a plain explanation about what tautology is? Am I missing something? :o –  Geo Papas Nov 26 '13 at 10:49
    
@GeoPapas: it would be slightly more comprehensive, but one must stop explaining somewhere, and start relying on background knowledge. A tautology is any sentence which is true solely by its logical form. See e.g. the wikipedia article on logical tautologies for more details. –  Niel de Beaudrap Nov 26 '13 at 22:05
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