In current practice, the single 'word' "iff" is almost always read as the four words "if and only if". It is somewhat unclear how "iff" was meant to be pronounced. Its invention is often credited to Paul Halmos, who wrote "I invented 'iff,' for 'if and only if'-but I could never believe I was really its first inventor." Usage of the abbreviation "iff" first appeared in print in John L. An alternative is to prove the disjunction "(P and Q) or (not-P and not-Q)", which itself can be inferred directly from either of its disjuncts-that is, because "iff" is truth-functional, "P iff Q" follows if P and Q have been shown to be both true, or both false. ![]() Proving these pairs of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. In most logical systems, one proves a statement of the form "P iff Q" by proving either "if P, then Q" and "if Q, then P", or "if P, then Q" and "if not-P, then not-Q". In logical formulae, logical symbols, such as ↔ via command \iff or \Longleftrightarrow. Some authors regard "iff" as unsuitable in formal writing others consider it a "borderline case" and tolerate its use. In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. For example, P if and only if Q means that P is true whenever Q is true, and the only case in which P is true is if Q is also true, whereas in the case of P if Q, there could be other scenarios where P is true and Q is false. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"-with its pre-existing meaning. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. then") combined with its reverse ("if") hence the name. ![]() The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if. Therefore, “p or q” is true in this case.Ĭonsidering the meaning of or, if both statements are false, then it is not true that “p or q”, thus we list false for this statement.In logic and related fields such as mathematics and philosophy, " if and only if" (shortened as " iff") is a biconditional logical connective between statements, where either both statements are true or both are false. ![]() We are saying “one or both of the statements is true”. Since we are working with the inclusive or, the statement “p or q” will be true in this case. This shows in the first row of the truth table, which we will now analyze: When we want to work with the exclusive or, we are specific and use different notation (you can read about this here: the exclusive or). In math, the “or” that we work with is the inclusive or, denoted \(p \vee q\). There is the inclusive or where we allow for the fact that both statements might be true, and there is the exclusive or, where we are strict that only one statement or the other is true. You may not realize it, but there are two types of “or”s. The order of the rows doesn’t matter – as long as we are systematic in a way so that we do not miss any possible combinations of truth values for the two original statements p, q. If both statements are false, then “p and q” is false. Row 4: the two statements could both be false.If this is the case, then by the same argument in row 2, “p and q” is false. Row 3: p could be true while q is false.Row 2: p could be false while q is true.įor “p and q” to be true, we would need BOTH statements to be true.In this case, it would make sense that “p and q” is also a true statement. Row 1: the two statements could both be true.To analyze this, we first have to think of all the combinations of truth values for both statements and then decide how those combinations influence the “and” statement. ![]() Conjunction – “and”Ĭonsider the statement “p and q”, denoted \(p \wedge q\). Notice that the truth table shows all of these possibilities. Negation is the statement “not p”, denoted \(\neg p\), and so it would have the opposite truth value of p. Any statements that are either true or false. They could be statements like “I am 25 years old” or “it is currently warmer than 70°”. Propositions are either completely true or completely false, so any truth table will want to show both of these possibilities for all the statements made.įor all these examples, we will let p and q be propositions. Truth tables are a way of analyzing how the validity of statements (called propositions) behave when you use a logical “or”, or a logical “and” to combine them.
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