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Single Idea 15934

[filed under theme 5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle ]

Full Idea

The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction.

Gist of Idea

Mathematical proof by contradiction needs the law of excluded middle

Source

Shaughan Lavine (Understanding the Infinite [1994], VI.1)

Book Ref

Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.155

A Reaction

This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku.

The 28 ideas with the same theme [propositions must be either true or false]:

1312 If everything is and isn't then everything is true, and a midway between true and false makes everything false [Aristotle on Heraclitus]
1701 A prayer is a sentence which is neither true nor false [Aristotle]
12363 Everything is either asserted or denied truly [Aristotle]
21668 Epicurus rejected excluded middle, because accepting it for events is fatalistic [Epicurus, by Cicero]
6023 Every proposition is either true or false [Chrysippus, by Cicero]
2669 Dialectic assumes that all statements are either true or false, but self-referential paradoxes are a big problem [Cicero]
21595 Excluded middle is the maxim of definite understanding, but just produces contradictions [Hegel]
18844 You would cripple mathematics if you denied Excluded Middle [Hilbert]
2947 Questions wouldn't lead anywhere without the law of excluded middle [Russell]
21539 Excluded middle can be stated psychologically, as denial of p implies assertion of not-p [Russell]
18944 Russell's theories aim to preserve excluded middle (saying all sentences are T or F) [Sawyer on Russell]
15941 For intuitionists excluded middle is an outdated historical convention [Brouwer]
9357 Excluded middle is just our preference for a simplified dichotomy in experience [Lewis,CI]
19192 The truth definition proves semantic contradiction and excluded middle laws (not the logic laws) [Tarski]
9024 Excluded middle has three different definitions [Quine]
9195 Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett]
8179 The law of excluded middle is the logical reflection of the principle of bivalence [Dummett]
7334 Anti-realism needs an intuitionist logic with no law of excluded middle [Dummett, by Miller,A]
17587 The 'Law' of Excluded Middle needs all propositions to be definitely true or definitely false [Inwagen]
9775 Excluded Middle, and classical logic, may fail for vague predicates [Fine,K]
10251 The law of excluded middle might be seen as a principle of omniscience [Shapiro]
8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
9605 If a proposition is false, then its negation is true [Brown,JR]
21605 Excluded Middle is 'A or not A' in the object language [Williamson]
10111 Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman]
15934 Mathematical proof by contradiction needs the law of excluded middle [Lavine]
8709 The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend]
17924 Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan]