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5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle

[propositions must be either true or false]

28 ideas
If everything is and isn't then everything is true, and a midway between true and false makes everything false [Aristotle on Heraclitus]
     Full Idea: The remark of Heraclitus that all things are and are not effectively renders all assertions true, and that of Anaxagoras that there is an intermediary between assertion and negation makes all assertions false.
     From: comment on Heraclitus (fragments/reports [c.500 BCE]) by Aristotle - Metaphysics 1012a
     A reaction: Compare Idea 416. Heraclitus is discussing truth-value 'gluts', as in paraconsistent logic, and Anaxagoras is discussing truth-value 'gaps', as in three-valued Kleene logic.
A prayer is a sentence which is neither true nor false [Aristotle]
     Full Idea: A prayer is a sentence which is neither true nor false.
     From: Aristotle (On Interpretation [c.330 BCE], 17a01)
Everything is either asserted or denied truly [Aristotle]
     Full Idea: Of the fact that everything is either asserted or denied truly, we must believe that it is the case.
     From: Aristotle (Posterior Analytics [c.327 BCE], 71a14)
     A reaction: Presumably this means that every assertion which could possibly be asserted must come out as either true or false. This will have to include any assertions with vague objects or predicates, and any universal assertions, and negative assertions.
Epicurus rejected excluded middle, because accepting it for events is fatalistic [Epicurus, by Cicero]
     Full Idea: Epicurus said that not every proposition is either true or false. ...Epicurus was afraid that if he admits that every proposition is true or false he will also have to admit that all events are caused by fate (if they are so from all eternity).
     From: report of Epicurus (fragments/reports [c.289 BCE]) by M. Tullius Cicero - On Fate ('De fato') 10.21
     A reaction: Epicurus proposed his 'swerve' in the movements of atoms to avoid this fatalism. Epicurus is agreeing with Aristotle, who did not accept excluded middle for a future contingent sea-fight.
Every proposition is either true or false [Chrysippus, by Cicero]
     Full Idea: We hold fast to the position, defended by Chrysippus, that every proposition is either true or false.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by M. Tullius Cicero - On Fate ('De fato') 38
     A reaction: I am intrigued to know exactly how you defend this claim. It may depend what you mean by a proposition. A badly expressed proposition may have indeterminate truth, quite apart from the vague, the undecidable etc.
Dialectic assumes that all statements are either true or false, but self-referential paradoxes are a big problem [Cicero]
     Full Idea: It is a fundamental principle of dialectic that every statement is either true or false. So is this a true proposition or a false one: "If you say that you are lying and say it truly, you lie"?
     From: M. Tullius Cicero (Academica [c.45 BCE], II.xxix.95)
Excluded middle is the maxim of definite understanding, but just produces contradictions [Hegel]
     Full Idea: The law of excluded middle is ...the maxim of the definite understanding, which would fain avoid contradiction, but in doing so falls into it.
     From: Georg W.F.Hegel (Logic (Encyclopedia I) [1817], p.172), quoted by Timothy Williamson - Vagueness 1.5
     A reaction: Not sure how this works, but he would say this, wouldn't he?
You would cripple mathematics if you denied Excluded Middle [Hilbert]
     Full Idea: Taking the principle of Excluded Middle away from the mathematician would be the same, say, as prohibiting the astronomer from using the telescope or the boxer from using his fists.
     From: David Hilbert (The Foundations of Mathematics [1927], p.476), quoted by Ian Rumfitt - The Boundary Stones of Thought 9.4
     A reaction: [p.476 in Van Heijenoort]
Questions wouldn't lead anywhere without the law of excluded middle [Russell]
     Full Idea: Without the law of excluded middle, we could not ask the questions that give rise to discoveries.
     From: Bertrand Russell (An Inquiry into Meaning and Truth [1940], c.p.88)
Excluded middle can be stated psychologically, as denial of p implies assertion of not-p [Russell]
     Full Idea: The law of excluded middle may be stated in the form: If p is denied, not-p must be asserted; this form is too psychological to be ultimate, but the point is that it is significant and not a mere tautology.
     From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.41)
     A reaction: 'Psychology' is, of course, taboo, post-Frege, though I think it is interesting. Stated in this form the law looks more false than usual. I can be quite clear than p is unacceptable, but unclear about its contrary.
Russell's theories aim to preserve excluded middle (saying all sentences are T or F) [Sawyer on Russell]
     Full Idea: Russell's account of names and definite descriptions was concerned to preserve the law of excluded middle, according to which every sentence is either true or false (but it is not obvious that the law ought to be preserved).
     From: comment on Bertrand Russell (On Denoting [1905]) by Sarah Sawyer - Empty Names 3
     A reaction: That is the strongest form of excluded middle, but things work better if every sentence is either 'true' or 'not true', leaving it open whether 'not true' actually means 'false'.
For intuitionists excluded middle is an outdated historical convention [Brouwer]
     Full Idea: From the intuitionist standpoint the dogma of the universal validity of the principle of excluded third in mathematics can only be considered as a phenomenon of history of civilization, like the rationality of pi or rotation of the sky about the earth.
     From: Luitzen E.J. Brouwer (works [1930]), quoted by Shaughan Lavine - Understanding the Infinite VI.2
     A reaction: [Brouwer 1952:510-11]
Excluded middle is just our preference for a simplified dichotomy in experience [Lewis,CI]
     Full Idea: The law of excluded middle formulates our decision that whatever is not designated by a certain term shall be designated by its negative. It declares our purpose to make a complete dichotomy of experience, ..which is only our penchant for simplicity.
     From: C.I. Lewis (A Pragmatic Conception of the A Priori [1923], p.365)
     A reaction: I find this view quite appealing. 'Look, it's either F or it isn't!' is a dogmatic attitude which irritates a lot of people, and appears to be dispensible. Intuitionists in mathematics dispense with the principle, and vagueness threatens it.
The truth definition proves semantic contradiction and excluded middle laws (not the logic laws) [Tarski]
     Full Idea: With our definition of truth we can prove the laws of contradiction and excluded middle. These semantic laws should not be identified with the related logical laws, which belong to the sentential calculus, and do not involve 'true' at all.
     From: Alfred Tarski (The Semantic Conception of Truth [1944], 12)
     A reaction: Very illuminating. I wish modern thinkers could be so clear about this matter. The logic contains 'P or not-P'. The semantics contains 'P is either true or false'. Critics say Tarski has presupposed 'classical' logic.
Excluded middle has three different definitions [Quine]
     Full Idea: The law of excluded middle, or 'tertium non datur', may be pictured variously as 1) Every closed sentence is true or false; or 2) Every closed sentence or its negation is true; or 3) Every closed sentence is true or not true.
     From: Willard Quine (Philosophy of Logic [1970], Ch.6)
     A reaction: Unlike many top philosophers, Quine thinks clearly about such things. 1) is the classical bivalent reading of excluded middle; 2) is the purely syntactic version; 3) leaves open how we interpret the 'not-true' option.
Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett]
     Full Idea: It must not be concluded from the rejection of excluded middle that intuitionistic logic operates with three values: true, false, and neither true nor false. It does not make use of true and false, but only with a construction being a proof.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 8.1)
     A reaction: This just sounds like verificationism to me, with all its problems. It seems to make speculative statements meaningless, which can't be right. Realism has lots of propositions which are assumed to be true or false, but also unknowable.
The law of excluded middle is the logical reflection of the principle of bivalence [Dummett]
     Full Idea: The law of excluded middle is the reflection, within logic, of the principle of bivalence. It states that 'For any statement A, the statement 'A or not-A' is true'.
     From: Michael Dummett (Thought and Reality [1997], 5)
     A reaction: True-or-not-true is an easier condition to fulfil than true-or-false. The second says that 'false' is the only alternative, but the first allows other alternatives to 'true' (such as 'undecidable'). It is hard to challenge excluded middle. Somewhat true?
Anti-realism needs an intuitionist logic with no law of excluded middle [Dummett, by Miller,A]
     Full Idea: Dummett argues that antirealism implies that classical logic must be given up in favour of some form of intuitionistic logic that does not have the law of excluded middle as a theorem.
     From: report of Michael Dummett (works [1970]) by Alexander Miller - Philosophy of Language 9.4
     A reaction: Only realists can think every proposition is either true or false, even if it is beyond the bounds of our possible knowledge (e.g. tiny details from remote history). Personally I think "Plato had brown eyes" is either true or false.
The 'Law' of Excluded Middle needs all propositions to be definitely true or definitely false [Inwagen]
     Full Idea: I think the validity of the 'Law' of Excluded Middle depends on the assumption that every proposition is definitely true or definitely false.
     From: Peter van Inwagen (Material Beings [1990], 18)
     A reaction: I think this is confused. He cites vagueness as the problem, but that is a problem for Bivalence. If excluded middle is read as 'true or not-true', that leaves the meaning of 'not-true' open, and never mentions the bivalent 'false'.
Excluded Middle, and classical logic, may fail for vague predicates [Fine,K]
     Full Idea: Maybe classical logic fails for vagueness in Excluded Middle. If 'H bald ∨ ¬(H bald)' is true, then one disjunct is true. But if the second is true the first is false, and the sentence is either true or false, contrary to the borderline assumption.
     From: Kit Fine (Vagueness, Truth and Logic [1975], 4)
     A reaction: Fine goes on to argue against the implication that we need a special logic for vague predicates.
The law of excluded middle might be seen as a principle of omniscience [Shapiro]
     Full Idea: The law of excluded middle might be seen as a principle of omniscience.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3)
     A reaction: [E.Bishop 1967 is cited] Put that way, you can see why a lot of people (such as intuitionists in mathematics) might begin to doubt it.
Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
     Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements.
     From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2)
     A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle.
If a proposition is false, then its negation is true [Brown,JR]
     Full Idea: The law of excluded middle says if a proposition is false, then its negation is true
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1)
     A reaction: Surely that is the best statement of the law? How do you write that down? ¬(P)→¬P? No, because it is a semantic claim, not a syntactic claim, so a truth table captures it. Semantic claims are bigger than syntactic claims.
Excluded Middle is 'A or not A' in the object language [Williamson]
     Full Idea: The logical law of excluded middle (now the standard one) is the schema 'A or not A' in the object-language.
     From: Timothy Williamson (Vagueness [1994], 5.2)
     A reaction: [He cites Henryk Mehlberg 1958] See Idea 21606. The only sensible way to keep Excluded Middle and Bivalence distinct. I would say: (meta-) only T and F are available, and (object) each proposition must have one of them. Are they both normative?
Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman]
     Full Idea: A hallmark of our realist stance towards the natural world is that we are prepared to assert the Law of Excluded Middle for all statements about it. For all statements S, either S is true, or not-S is true.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4)
     A reaction: Personally I firmly subscribe to realism, so I suppose I must subscribe to Excluded Middle. ...Provided the statement is properly formulated. Or does liking excluded middle lead me to realism?
Mathematical proof by contradiction needs the law of excluded middle [Lavine]
     Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction.
     From: Shaughan Lavine (Understanding the Infinite [1994], VI.1)
     A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku.
The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend]
     Full Idea: The law of excluded middle is purely syntactic: it says for any well-formed formula A, either A or not-A. It is not a semantic law; it does not say that either A is true or A is false. The semantic version (true or false) is the law of bivalence.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2)
     A reaction: No wonder these two are confusing, sufficiently so for a lot of professional philosophers to blur the distinction. Presumably the 'or' is exclusive. So A-and-not-A is a contradiction; but how do you explain a contradiction without mentioning truth?
Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan]
     Full Idea: The law of excluded middle (for every proposition P, either P or not-P) must be carefully distinguished from its semantic counterpart bivalence, that every proposition is either true or false.
     From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3)
     A reaction: So excluded middle makes no reference to the actual truth or falsity of P. It merely says P excludes not-P, and vice versa.