[1] Proving these pair of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. For other uses, see, "↔" redirects here. Iff is used outside the field of logic as well. Another term for this logical connective is exclusive nor. Hypothesis and conclusion: Necessary and sufficient. The truth table of P However, in the preface of General Topology, Kelley suggests that it should be read differently: "In some cases where mathematical content requires 'if and only if' and euphony demands something less I use Halmos' 'iff'". ⟺ A is a proper subset of B. If X, then Y | Sufficiency and necessity. Sort by: Top Voted. "only if q then p" is just a restatement of "if q then p". A number is in B if and only if it is in C, and a number is in C if and only if it is in B. Euler diagrams show logical relationships among events, properties, and so forth. In Łukasiewicz's Polish notation, it is the prefix symbol 'E'.[12]. An Adventure in Language and Logic. Does it make a difference? Yes, because now you are left with the false statement "You get an A grade in Math 101 if you get an A- or better average on the homework." Some Uses of "if and only if" in Writing About Mathematics . [14] In logic and related fields such as mathematics and philosophy, if and only if (shortened as iff[1]) is a biconditional logical connective between statements, where either both statements are true or both are false. Speaking more generally, dropping the only from only if usually makes a significant difference to the logic … i think you and the op got things backwards. Where as -- Dave is a carpenter, only if he is a human. if P then Q), P would be a sufficient condition for Q, and Q would be a necessary condition for P. Also, given P→Q, it is true that ¬Q→¬P (where ¬ is the negation operator, i.e. If you're seeing this message, it means we're having trouble loading external resources on our website. Take an example where we discount a value by 10% dependent upon the amount a customer spends andhow many years they have been a customer. Dave is a human if he is a carpenter. The elements of X are all and only the elements of Y means: "For any z in the domain of discourse, z is in X if and only if z is in Y. 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. For example, P if and only if Q means that 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. [Read more on Conditional Diagramming: Part 1, Part 2, Part 3, Part 4, Part 5] An interesting topic came up on our forum a short time back, about the meaning of “If but only if,” and the proper way to diagram a rule that includes this phrase. The Logic of "If" vs. "Only if" This is the currently selected item. Therefore there is no way he can be a carpenter unless he is … but "only" is not a logical operator. So "A only if B" reverses the direction of the conditional from "A if B". [1] This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition). ⇔ From MathWorld--A Wolfram Web Resource. Sometimes the biconditional in the statement of the phrase “if and only if” is shortened to simply “iff.” Thus the statement “P if and only if Q” becomes “P iff Q.” {\displaystyle \Leftrightarrow } Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. Where as -- Dave is a carpenter, only if he is a human. the rwo sentences do not have the same meaning in ordinary English. ", "Iff" redirects here. This means that the relationship between P and Q, established by P→Q, can be expressed in the following, all equivalent, ways: As an example, take the first example above, which states P→Q, where P is "the fruit in question is an apple" and Q is "Madison will eat the fruit in question". they do in logic, because "only" is logically redundant. The inverse. {\displaystyle \Leftrightarrow } C is a subset but not a proper subset of B. Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. Learn how and when to remove this template message, "The Definitive Glossary of Higher Mathematical Jargon — If and Only If", "Jan Łukasiewicz > Łukasiewicz's Parenthesis-Free or Polish Notation (Stanford Encyclopedia of Philosophy)", Southern California Philosophy for philosophy graduate students: "Just in Case", https://en.wikipedia.org/w/index.php?title=If_and_only_if&oldid=990794170, Articles needing additional references from June 2013, All articles needing additional references, Creative Commons Attribution-ShareAlike License, This page was last edited on 26 November 2020, at 15:16. http://www.criticalthinkeracademy.com This video shows how to evaluate conditional statements of the form "A only if B" In logic, a set of symbols is commonly used to express logical representation. 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. Sufficiency is the converse of necessity. When one is true, you automatically know the other is true as well. When you have “only if”, the claim that precedes the “only if’ is antecedent, what follows it is the consequent. Khan Academy is a 501(c)(3) nonprofit organization. "not"). Theorems which have the form "P if and only Q" are much prized in mathematics. Now drop only from the sentence. Donate or volunteer today! "P if Q", "if Q then P", and Q→P all mean that Q is a proper or improper subset of P. "P if and only if Q" and "Q if and only if P" both mean that the sets P and Q are identical to each other. The IF function alone can only handle one condition, or comparison. The “only if” actually reverses the direction of logical dependency. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. The OR functio… The AND and OR functions are used when you want to perform more than one comparison in your formula. The authors of one discrete mathematics textbook suggest:[16] "Should you need to pronounce iff, really hang on to the 'ff' so that people hear the difference from 'if'", implying that "iff" could be pronounced as [ɪfː]. That is to say, given P→Q (i.e. A number is in A only if it is in B; a number is in B if it is in A. Iff is used outside the field of logic as well. Therefore there is no way he can be a carpenter unless he is … The Logic of "If" vs. "Only if" Our mission is to provide a free, world-class education to anyone, anywhere. ⇔ This often includes conditional statements such as “IF Bob is selected THEN Suzie is also selected” or “Suzie is selected IF Bob is selected” ‍ Dave can be a carpenter, or not as this has no bearing on him being a human. Weisstein, Eric W. The biconditional – “p iff q” or “p if and only if q” If and only if statements, which math people like to shorthand with “iff”, are very powerful as they are essentially saying that p and q are interchangeable statements. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. actually that's not true, "only if q then p" is meaningless, logically.

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