Ling 680 10-4-00 The Terms "Necessary Condition" and "Sufficient Condition". Many logic textbooks simply relate these these to the material conditional by these two statements (and no more). 1. "p is a sufficient condition for q" is the same as (p --> q) 2. "p is a necessary condition for q" is the same as (q --> p) But maybe it will help to understand this relationship (and to remember which term goes with which formula) to think about the ordinary meanings of "necessary" and "sufficient" in relationship to the truth table for the conditional: First, here is the standard way of writing the truth table for "-->" p q | (p --> q) ------------------ i. 1 1 | 1 ii. 1 0 | 0 iii. 0 1 | 1 iv. 0 0 | 1 If we want to compare the table for (q --> p) with that for (p --> q), side by side (keeping the values in the "p" and "q" columns the same in both), then we can do this as follows: p q | (p --> q) | (q --> p) ------------------ ------------ i. 1 1 | 1 | 1 ii. 1 0 | 0 | 1 iii. 0 1 | 1 | 0 iv. 0 0 | 1 | 1 (Notice the columns of truth values for the two conditionals are different, which is because we have switched the antecedent variable and consquent variables around in the second conditional.) The expression "p is a sufficient condition for q" can be paraphrased this way: "If you know that p is true, that suffices for you to know that q is true." In what sense does this describe truth table for (p --> q)? Look closely at that truth table. The four lines (i. - iv.) correspond to the four possible combinations of truth and falsity that p and q can have. According to the table, the conditional (p --> q) is true in lines i., iii., and iv, but not in ii. What would we know if we knew that the conditional was true but did not yet know whether p and q individually are true are false? We would know only that line ii. is ruled out: p q | (p --> q) <== known to be true ------------------ i. 1 1 | 1 ii. 1 0 | 0 X (ruled out). iii. 0 1 | 1 iv. 0 0 | 1 By reading the truth table "from right to left", as it were, you can see that we merely know that the actual facts are either as in situation i, iii or iv, but we don't know which. We can't figure out the truth values of p and q individually yet, because i. ii. and iii together include situations where each of p and q are sometimes true, sometimes false. Suppose we now find out, in addition, that p is true; how does that change things? p is true only in lines i. and ii., so we can now also rule out lines iii. and iv.: p q | (p --> q) (cond is true) (p is true) ------------------ i. 1 1 | 1 ii. 1 0 | 0 X iii. 0 1 | 1 X iv. 0 0 | 1 X And now the only remaing possibly is line i., and q is true in line i. Thus, we can describe the conditional (p --> q) as follows: Because of the relationship that exists between p and q, finding out that p is true will SUFFICE for us to know that q is true. On the other hand, suppose we know that this same conditional, (p --> q), is true and now find out that q is true. Visually: p q | (p --> q) (cond is true) (q is true) ------------------ i. 1 1 | 1 ii. 1 0 | 0 X X iii. 0 1 | 1 iv. 0 0 | 1 X THat is, the new information here only tells us that iv. is not the actual state of affairs, but it still leaves both i. and iii. open as possibilies. But, p is true in i. but false in iii., so we still don't have enough information here to know whether p is true. (So with this conditional: finding out that q is true does NOT suffice for us to know that p is true.) More on this kind of situation below. Next, we turn to (q --> p) and "q is a necessary condition for p". We look now at the truth table for (q --> p) (taken from the double table above): p q | (q --> p) ------------------ i. 1 1 | 1 ii. 1 0 | 1 iii. 0 1 | 0 iv. 0 0 | 1 This time, If we know only that the conditional (q --> p) is true, we rule out line iii. as a possibility: p q | (q --> p) ------------------ i. 1 1 | 1 ii. 1 0 | 1 iii. 0 1 | 0 X iv. 0 0 | 1 This leaves only lines i., ii., and iv. If we find out in addition that q is true, the additional information rules out lines iii. and iv.: p q | (q --> p) (cond is true) (q is true) ------------------ i. 1 1 | 1 ii. 1 0 | 1 iii. 0 1 | 0 X X iv. 0 0 | 1 X still leaving i. and ii. We still do not know whether p is true or false (this is somewhat like an earlier example). However, suppose instead that we find out instead that p is FALSE --- still assuming the conditional (q --> p)? If p is false, then line i. and ii. cannot possibly represent the facts, and given that the conditional here rules out iii, p q | (q --> p) (cond is true) (p is false) ------------------ i. 1 1 | 1 X ii. 1 0 | 1 X iii. 0 1 | 0 X iv. 0 0 | 1 that leaves iv. as the only possibility. And in iv., p is not true. This, then, is why we can describe (q --> p) as saying "p is a necessary condition for q": if p fails to be true, then q cannot be possibly be true, so p is NECESSARY in order for q to be true. (But knowing that p is true is still NOT sufficient for knowing q to be true with this conditional: cf. lines i. and ii. in this last table, remembering that "p fails to be false" is the same as "p is true".) We have just shown that p can be a necessary condition for q WITHOUT being a sufficient condition for q. It's also now easy to show that p can be a sufficient condition for q WITHOUT being a necessary condition for q, but I leave the details for you. QUESTION: How is the difference between `necessary' and `sufficient' related to the fact that p, p-->q |- q is a rule of inference in statement logic but q, p-->q |- p is not? That ~q, p-->q |- p is a rule but ~p, p-->q |- q is not? Finally, it is also possible for the two conditions to hold at the same time: To say that p is a NECESSARY AND SUFFICIENT condition for q is (as you would expect) to say that both (p --> q) and (q --> p) are true conditionals. And as we know from our textbook (or will shortly), the situations where both these conditionals hold are the same as the situations where the biconditional holds: p q | (p <--> q) ------------------ i. 1 1 | 1 ii. 1 0 | 0 iii. 0 1 | 0 iv. 0 0 | 1