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Single Idea 8709

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

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.

Gist of Idea

The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false

Source

Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2)

Book Ref

Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.108

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?

Related Ideas

Idea 9024 Excluded middle has three different definitions [Quine]

Idea 17924 Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan]

The 41 ideas from 'Introducing the Philosophy of Mathematics'

8713 In classical/realist logic the connectives are defined by truth-tables [Friend]
8663 Raising omega to successive powers of omega reveal an infinity of infinities [Friend]
8662 The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
8661 The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]
8664 Cardinal numbers answer 'how many?', with the order being irrelevant [Friend]
8671 The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend]
8669 Between any two rational numbers there is an infinite number of rational numbers [Friend]
8665 A 'proper subset' of A contains only members of A, but not all of them [Friend]
8672 A 'powerset' is all the subsets of a set [Friend]
8666 Infinite sets correspond one-to-one with a subset [Friend]
8667 The 'integers' are the positive and negative natural numbers, plus zero [Friend]
8668 The 'rational' numbers are those representable as fractions [Friend]
8670 A number is 'irrational' if it cannot be represented as a fraction [Friend]
8675 Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend]
8674 The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend]
8677 Set theory makes a minimum ontological claim, that the empty set exists [Friend]
3678 Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend]
8676 Is mathematics based on sets, types, categories, models or topology? [Friend]
8678 Most mathematical theories can be translated into the language of set theory [Friend]
8680 Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend]
8681 The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend]
8682 Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend]
8685 Studying biology presumes the laws of chemistry, and it could never contradict them [Friend]
8688 Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend]
8694 Free logic was developed for fictional or non-existent objects [Friend]
8696 Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend]
8695 Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend]
8701 The number 8 in isolation from the other numbers is of no interest [Friend]
8702 In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend]
8699 Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend]
8700 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]
8704 Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend]
8706 Constructivism rejects too much mathematics [Friend]
8705 Anti-realists see truth as our servant, and epistemically contrained [Friend]
8708 Double negation elimination is not valid in intuitionist logic [Friend]
8709 The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend]
8707 Intuitionists typically retain bivalence but reject the law of excluded middle [Friend]
8711 Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend]
8712 Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend]
8716 Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]
8721 An 'impredicative' definition seems circular, because it uses the term being defined [Friend]