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1 Copi, Irving and Cohen, Carl, Introduction to Logic, 10th ed. (Upper Saddle River, NJ: Prentice-Hall, 1998).
2 Groarke, Leo, Tindale, Christopher W., and Fisher, Linda, Good Reasonig Matters! 2nd ed. (Toronto: Oxford University Press, 1997).
3 A nice “user-friendly” introduction to algebraic semantics can be found in Dunn's, J. Michael chapter, “Intensional Algebras,” in Entailment: The Logic of Relevance and Necessity, edited by Anderson, A. R. and Belnap, Nuel D. (Princeton: Princeton University Press, 1975). (Thanks to a referee for reminding us of this reference.) A detailed discussion can be found in Rasiowa, Helena, An Algebraic Approach to Non-Classical Logics (Amsterdam: North Holland, 1974), or Rasiowa, Helena and Sikorski, Roman, The Mathematics of Metamathematics, 2nd ed. (Warsaw: Polish Scientific Publishers, 1968).
4 A BA is a lattice B with 0 and 1, 0≠1, such that for a, b, c ∈ B, a ∧ (b ∨ c) = (a ∨ b) ∧ (a ∨ c), and such that every element has a complement.
5 Another example which makes the same point and which might be familiar to some readers is Bas van Fraassen's notion of supervaluations. A supervaluation starts with a partial valuation, i.e., an assignment of T to some sentences, F to others, and no value to the rest. Then take all possible consistent classical extensions of the partial valuation to a total valuation, i.e., consider all consistent extensions in which every sentence which has no value under the partial valuation is assigned either T or F according to classical valuation rules. Truth (falsity) in the supervaluation over the given partial valuation, is truth (falsity) in all classical extensions of the given partial valuation. If a sentence gets different values in different classical extensions, then it has no value in the supervaluation. Suppose P has no value in a given partial valuation. Then, in all classical extensions, either P is T or ¬ P is T, and so P ∧ ¬P is T in the supervaluation. Hence, the use of classical extensions validates excluded middle, but bivalence fails because there may be sentences in the supervaluation with no value. Van Fraassen introduced supervaluations in a series ofpapers in the mid-1960s. A simplified introduction can be found in Bas van Fraassen and Karel Lambert, Derivation and Counterexample (Encino, CA: Dickenson Publishing, 1972). A more detailed introduction is included in McCawley, James D., Everything Linguists Have Always Wanted to Know about Logic (But Were Ashamed to Ask), 2nd ed. (Chicago: University of Chicago Press, 1993).
6 Bell, John L., Boolean-Valued Models and Independence Proofs in Set Theory, 2nd ed. (Oxford: Clarendon, 1985), is the standard reference for Booleanvalued models of set theory. It includes an interesting foreword by Dana Scott which discusses the history of the Boolean-valued-models approach to forcing.
7 An HA is a lattice L with 0 such that for any a, b ∈ L the element a → b = l.u.b {x: x ∧ a ≤ b} exists.
8 In intuitionistic logic, both bivalence and excluded middle fail, of course. However, it is useful to have it available for our discussion of reductio arguments in the next section.
9 Because if excluded middle holds as well, then ¬\a is a complement for a, and so every element in the lattice has a complement. But aBA is just a complemented distributive lattice.
10 A lattice is orthocomplemented if every member a of the latticecan be assigned a unique element ¬a such that a ∧ ¬a = 1, a ∧ ¬a = 0, and such a ≤ b implies ¬b ≤ ¬a. Some philosophers prefer to work with lattices which are also orthomodular, i.e., such that a ≤ b implies b = a ∨ (b ∧ ¬a), but this does not affect the present discussion. A natural deduction system for a logic of the sort of interesthere can be found in Peter Gibbins, “A User-Friendly Quantum Logic,” Logique et Analyse, 112 (1985): 353–62.
11 This condition is not a requirement that contraposition must hold for all relations which can be represented by a conditional connective if thesystem also includes a genuine negation. Many conditional connectives (e.g., W. T. Parry's “analytic implication”) are used to capture some notion much stronger than logical consequence—often thisis the explicit justification for their introduction—and, for them, contraposition may, indeed, fail. But this does not mean that the described conditionfails. Indeed, in some logics which include Parry's analytic implication, there is also a weaker conditional which does correspond to that logic's version of logical consequence (in the sense that the conditional is true if and only if the antecedent has the consequent as a logicalconsequence) for which contraposition does hold. Our thanks to a referee who pointed out the need to say something about this.
12 One can also get logics where excluded middle is valid and yet the logic is nonclassical by giving up the law of non-contradiction. We have suggested elsewhere (i.e., in “□ in Intuitionistic Modal Logic,” Australasian Journal of Philosophy, 75 [1997]: 201–13) that a “negation” where this is the case probably does not deserve the name, and adopted the name “supplementation” for the particular notion under discussion there. Advocates of paraconsistent logics would beg to differ, of course.
13 Bell, John L., Toposes and Local Set Theories: An Introduction (Oxford: Oxford University Press, 1988), pp. 141–42, 147–49. The models in question are the toposes Set^{M}, where M is a monoid which is not a group.
14 We owe this example to a suggestion by a referee.
15 The secondary literature on Tarski's work on truth is vast and prone to confusions about relations between object languages and metalanguages, and to using “language” when “theory” would be more appropriate. For a (we hope) careful treatment of relevant subtleties, see our article “Tarski and ‘Essentially Richer’ Metalanguages,” Journal of Philosophical Logic, 28 (1999): 1–28.
16 Kalish, Donald and Montague, Richard, Logic (New York: Harcourt, Brace, Jovanovich, 1964).
17 There are less familiar logics in which excluded middle is not equivalent to double negation elimination, and so this claim is context-dependent in important ways, but we will not pursue this matter here.
18 Machover, Moshé, Set Theory, Logic and Their Limitations (Cambridge: Cambridge University Press, 1996), p. 128.
19 Williamson, Timothy, Vagueness (London: Routledge, 1994). Williamson's discussion is notably sophisticated in its careful treatment of the relevant distinctions.
20 Lukasiewicz, Jan, “Philosophical Remarks on Many-Valued Systems of Prepositional Logic,” in Polish Logic 1920–1939, edited by McCall, Storrs and translated by H. Weber (Oxford: Oxford University Press, 1967), p. 52.
21 Lukasiewicz, Jan, Aristotle's Syllogistic (Oxford: Oxford University Press, 1951), p. 82.
22 Rescher, Nicholas, Many-Valued Logic (New York: McGraw-Hill, 1969), pp. 148–54, lists five different claims which have all at some time or other been called the “law of excluded middle.” We have focused on the most common.
23 Routley, Richard, “The Need for Nonsense,” Australasian Journal of Philosophy, 47 (1969): 84–367.
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