Laws of Thought
LAWS OF THOUGHT
The term "laws of thought" traditionally covered the principles of identity, of contradiction, of excluded middle, and occasionally the principle of sufficient reason. Whereas these principles were frequently discussed from the time of the Greeks until the beginning of the twentieth century, the term has become obsolete, for at least two good reasons. One is the great and confusing variety of meanings with which it has been used, the other is the now generally acknowledged fact that no viable system of logic can be constructed in which the principles of identity, contradiction, and excluded middle would be the only axioms. Typical discussions of these principles are to be found, for example, in Friedrich Ueberweg's System der Logik and in H. W. B. Joseph's Introduction to Logic. In the following discussion the principle of sufficient reason, which, unlike the others, cannot be interpreted as a principle of formal logic, will not be dealt with.
The three laws of thought have in the main been conceived of as descriptive, prescriptive, or formal. As descriptive laws, they have been regarded as descriptive (a ) of the nature of "being as such," (b ) of the subject matter common to all sciences, or (c ) of the activity of thinking or reasoning. As prescriptive laws, they have been conceived of as expressing absolute or conventional standards of correct thinking or reasoning. As formal laws, they have been held to be propositions which are true in virtue of their form and independently of their content, true in all possible worlds, or true of any objects whatsoever, whether these objects exist or not. Distinctions between these conceptions are often blurred, since they depend on implicit and often unclear assumptions about the relations between factual, normative, and metaphysical propositions: It is, for example, rarely investigated either to what extent various kinds of rules depend for their satisfiability on what is the case or to what extent logic is or can be free from metaphysical presuppositions or implications.
All these very different conceptions of the laws of thought are compatible with their traditional formulations, which lack the precision now achievable by means of the axiomatization and formalization of theories. Examples of typical, traditional formulations are: For the law of identity, A is A ; everything is what it is; every subject is its own predicate. For the law of contradiction, A is not not-A ; judgments contradictorily opposed to each other cannot both be true. For the law of excluded middle, everything is either A or not-A ; judgments opposed as contradictories cannot both be false, nor can they admit the truth of a third or middle judgment, but one or the other must be true, and the truth of the one follows from the falsehood of the other. An obvious ambiguity concerning the law of identity is connected with the question whether is is to be taken as expressing equality or as the copula between subject and predicate, and, in the latter case, whether or not it implies the existence of the subject. Again, the term not admits of different interpretations according to different metaphysical and logical assumptions about negation.
For Aristotle, who discussed the laws of thought in his logical and metaphysical works, they are primarily descriptive of being as such and only secondarily standards of correct thinking. It is thus a metaphysical or ontological impossibility that "the same can and cannot belong to the same in the same reference" (Metaphysics III, 2, 2), from which it follows as a rule of correct thought and speech that it is incorrect to assert that "the same is and is not" (Metaphysics IV, 6, 12). Aristotle produced seven "proofs" to demonstrate the indispensability of the law of contradiction. With a similar intention, formal logicians are nowadays wont to show that its negation implies any proposition whatever (and thus also the law of contradiction itself) by some such reasoning as the following: (1) To assume that the law of contradiction is false is to assume for some proposition p that p and not-p are both true. (2) From the truth of p it follows that "p or x " is also true, where x is an arbitrary proposition and "or" is used in the nonexclusive sense of "and/or." (3) From the truth of "p or x " and the truth of not-p the truth of x follows. But x is an arbitrary proposition for which, for example, the law of contradiction may be chosen.
Aristotle's defense of the law of contradiction as descriptive of "being as such" includes implicitly a defense of the metaphysical principle of identity against Heraclitus, who held it possible for the same thing to be and not to be and who explained the concept of becoming as implying the falsehood of the principle that everything is what it is. Before Aristotle this metaphysical principle had been defended by Parmenides.
Aristotle's arguments for the truth of the principle of excluded middle are again metaphysical. They are connected with his rejection of the Platonic doctrine that attempts to mediate between Heraclitus and Parmenides. The changing sensible and material objects, which in Plato's phrase "tumble about between being and nonbeing," are placed by Plato between the eternal Forms, which fully and truly exist, and that which does not exist at all, that is, they are "a third" between being and nonbeing. The metaphysical principle of excluded middle, as understood by Aristotle, excludes any such third. This principle has sometimes been taken to imply fatalism: Since of any two contradictory statements one must be true, of any two contradictory statements about the future one must be true, so that, it is argued, the future is wholly determined. In a famous passage about "the sea fight tomorrow" Aristotle refutes this argument: It is, he points out, necessary that the sea fight will or will not take place tomorrow. But it is not true that it will necessarily take place tomorrow or necessarily not take place tomorrow. Indeed the logical necessity of a disjunction "p or not-p " does not imply that either p or not-p is a necessary proposition.
Heraclitus, Parmenides, Plato, and Aristotle conceived of the laws of thought as controversial metaphysical principles, and just as Aristotle attempted their justification on metaphysical grounds, so did G. W. F. Hegel, Karl Marx, and Friedrich Engels attempt their refutation on metaphysical grounds. Hegel's attack was based on his distinction between abstract understanding, which petrifies and thus misdescribes the ever-changing "dialectical" process that is reality, and reason, which apprehends its true nature. Hegel objected to the principle that A is A or, what for him amounts to the same thing, that A cannot at the same time be A and not-A because "no mind thinks or forms conceptions or speaks in accordance with this law, and … no existence of any kind whatever conforms to it" (Die Encyclopädie der philosophischen Wissenschaften ). For Hegel contradiction is not a relation that holds merely between propositions but one which is also exemplified in the real world, for example, in such phenomena as the polarity of magnetism, the antithesis between organic and inorganic matter, and even the complementarity of complementary colors. With such an interpretation it becomes possible for him to assert that "contradiction is the very moving principle of the world" and that "it is ridiculous to say that contradiction is unthinkable." Aristotle's metaphysics corresponds to a logic in which the metaphysical principles of identity, contradiction, and excluded middle have their logical counterpart in corresponding laws of reasoning. The counterpart of Hegel's rejection of these metaphysical principles is not any traditional logical theory but a "dialectical" logic, or dialectics.
The Hegelian point of view was adopted by Marx and Engels with the difference that they conceived reality not as ideal but as material. Engels, unlike Hegel, did not even acknowledge the law of identity as valid for the abstractions of mathematics. His arguments, based on the alleged structure of the differential and integral calculus, seem—at least today—confused. He held, for example, that under certain circumstances straight lines and curves are literally identical.
From the conception of the laws of thought as descriptive of "being as such," whatever this may mean precisely, we must distinguish the conception of them as empirical generalizations of very high order. This view was most clearly expressed by John Stuart Mill in his System of Logic (London, 1843). Thus, he regarded the principle of contradiction as one "of our first empirical generalizations from experience" and as "originally founded on our distinction between belief and disbelief as two different mutually exclusive states" (System of Logic, Book II, Ch. 7). He similarly argued that the empirical character of the law of excluded middle follows from, among other things, the fact that it requires for its truth a large qualification, namely "that the predicate in any affirmative categorical proposition must be capable of being meaningfully attributed to the subject, since between the true and the false there is the third possibility of the meaningless" (Book II, ch. 7). Mill's view must not be taken to imply that the laws of thought are psychological laws, describing the processes of thought—a view which rests on a confusion between thinking and correct thinking.
Another interpretation of the laws of thought regards them as in some sense prescriptive—based on some absolute authority, by analogy with moral laws, or based on conventions admitting of possible alternatives, by analogy with municipal laws. Traces of the former view are, for example, still found in J. N. Keynes's Formal Logic, one of the last valuable treatises on traditional formal logic. According to the preface of this work, logic deals with the laws regulating the processes of formal reasoning purely as "regulative and authoritative" and as affording criteria for the discrimination between valid and invalid reasoning.
Versions of the conception that all logical principles are based on conventions have rarely been worked out with sufficient care. According to A. J. Ayer's Language, Truth and Logic (1936; 2nd ed., 1946) every logical principle is based on conventions. Thus "not (p and not-p )" is logically necessary because the use of "and" and "not" is governed by certain linguistic conventions, which are neither true nor false. Yet given these conventions the proposition "not (p and not-p )," that is, the law of contradiction, is necessarily true. Ayer and those who have held similar views never consider the question whether, and to what extent, linguistic conventions depend on some nonconventional framework which restricts one's freedom to formulate, accept, or reject them. Can one, for example, by adopting suitable conventions for the use of "or" and "not" really think or speak in contravention of the principle that under the usual conventions is expressed by "not (p and not-p )"?
Conventionalism is most plausible when it explains the necessity of alternative systems of definitions and of alternative systems of logic as being based on conventions, in the sense of rules whose acceptance is not obligatory. In the case of the law of contradiction no alternative is conceivable, so that the "convention" on which it is based would have to be obligatory in a sense in which the other conventions are not. However, an admission of "conventions obligatory for all thinkers" would bring conventionalism much nearer to views of logic which, at least prima facie, it seems to reject.
leibniz and kant
According to Gottfried Wilhelm Leibniz there are two kinds of truths, truths of fact and truths of reason, truths of reason being true in all possible worlds and therefore descriptive of facts in such a way that not even God can change them. Leibniz regarded as a necessary and sufficient condition for a truth's being a truth of reason, and thus logically necessary, that its analysis should reveal it to depend wholly on propositions whose negation involves a contradiction, that is, on identical propositions (see, for example, Monadology, Secs. 31–35). He even held, in the second letter to Samuel Clarke, that the law of contradiction is "by itself sufficient" for the demonstration of "the whole of arithmetic and geometry." Although the thesis that all logical, as well as all mathematical, truths are demonstrable by means of the law of contradiction alone is, from the point of view of contemporary knowledge, mistaken or at least obscurely expressed, the characterization of logical truths as true in all possible worlds is still the root of the Bolzano-Tarski definition of logical validity.
Although Immanuel Kant opposed the Leibnizian doctrine that the truths of mathematics are logical truths, he adhered to the principle of contradiction as the supreme principle of all logical truths or, more precisely, as the "general and wholly sufficient principle of all analytical knowledge." Since the truth of such knowledge in no way depends on whether or not the objects which are referred to exist, the principle of contradiction is a necessary but not a sufficient condition of factual knowledge. What is true of possible objects must be true of all actual ones—what is true in all possible worlds must be true of the actual one. But since the converse statement is false, the principles of formal logic cannot be an "organon" of any particular science, that is, a means for attaining knowledge of its subject matter. (See Kritik der reinen Vernunft, 2nd ed., introduction to Part II of "Transcendentale Elementarlehre.")
In contemporary logical theory the conception of "true in every possible world" or "true of any objects whatever" has been sharpened into the conception of valid statement forms and valid statements which are well formed in accordance with the precisely formulated syntactical rules of elementary logic—propositional calculus, quantification theory, and theory of identity. A distinction is made between the logical particles, or constants, on the one hand and nonlogical constants and variables on the other. The logical constants are (1) "¬," "∨," "∧," and other connectives, whose intended interpretations are, respectively, "not," "or," "and," and so on, conceived as connecting true or false propositions so as to form other true or false propositions in such a way that the truth or falsehood of any compound statement depends only on the truth or falsehood of the component statements; (2) the quantifiers "∀" and "∃," the intended interpretation of which is such that "∀x Px " and "∃x Px " mean, respectively, that for a well-demarcated domain of individuals, which may be finite or infinite, every element x has the predicate P, and that there exists an individual x which possesses P (in addition to such monadic predicates as Px, such dyadic predicates as Pxy and polyadic predicates are also admitted, so that, for example, ∀x ∃y Pxy, ∀x ∀y Pxy, and so on, are also admitted); and (3) the sign "=" with the intended interpretation as identity of individuals.
The nonlogical constants are (a ) names of specific individuals, such as "Socrates" or, indeterminately, "x 0," (b ) names of specific predicates, such as "green" or, indeterminately, "P 0" where two predicate names which are truly asserted of the same individuals of the given domain are regarded as naming the same predicate, (c ) names of specific statements, such as "Socrates is mortal" or, indeterminately, "p 0." The variables are individual variables such as "x," predicate variables such as "P," and statement variables such as "p." Variables are either free (or, more precisely, free for substitution by names of corresponding constants) or bound by a quantifier so that, for example, "Px " contains a free individual variable and a free predicate variables, whereas ∀x Px contains only a free predicate variable.
A well-formed formula of elementary logic that contains free variables is a statement form. A statement form is valid if—with the intended interpretation of the logical constants—every substitution instance of it is valid in every nonempty domain, provided that every individual, predicate, and statement variable is replaced by the same individual, predicate, and statement constant wherever it occurs in the statement form. Clearly the laws of thought are valid statement forms in, for example, the following formulations: Principle of identity: x = x. Principle of contradiction: ¬(p ∧ ¬ p ), ∀x ¬ (Px ∧ ¬ Px ). Principle of excluded middle: p ∨ ¬ p, (∀x Px ) ∨ (∃x ¬ Px ). It is equally clear that many other well-formed formulas such as ¬¬ p ∨ ¬ p are valid. Valid statement forms that contain only statement variables have been called tautologies by Wittgenstein.
The great precision and clarity given to the conception of the laws of thought as principles of formal logic has, however, not lifted them out of the range of philosophical controversy. Thus, intuitionist philosophers of mathematics argue that the principle of excluded middle is valid only for finite domains and that the extension of its validity to the nonfinite domain of arithmetic is based on the mistaken notion of an actually infinite domain of natural numbers, a notion that unjustifiably assimilates the number sequence to a finite class of objects. Similarly, they deny the validity of other classically valid statement forms, such as ¬ ¬ p → ¬ p.
The results of modern mathematical logic have deprived the laws of thought of their privileged status as the supreme principles of all logical truths. But since these results do not imply that there is only one true logic, the choice between classical elementary logic, intuitionist logic, and perhaps some other logical theories still depends, at least at the present time, on extralogical, philosophical arguments.
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