THOUGHT, LAWS OF

traditionally, the three fundamental laws of logic: (1) the law of contradiction, (2) the law of excluded middle (or third), and (3) the principle of identity. That is, (1) for all propositions p, it is impossible for both p and not p to be true, or symbolically, ~(p ~p), in which ~ means not and means and; (2) either p or ~p must be true, there being no third or middle true proposition between them, or symbolically p ~p, in which means or; and (3) if a propositional function F is true of an individual variable x, then F is indeed true of x, or symbolically F(x) F(x), in which means formally implies. Another formulation of the principle of identity asserts that a thing is identical with itself, or (x) (x = x), in which means for every; or simply that x is x. Aristotle cited the laws of contradiction and of excluded middle as examples of axioms. He partly exempted future contingents, or statements about unsure future events, from the law of excluded middle, holding that it is not (now) either true or false that there will be a naval battle tomorrow, but that the complex proposition that either there will be a naval battle tomorrow or that there will not is (now) true. In the epochal Principia Mathematica (191013) of A.N. Whitehead and Bertrand Russell, this law occurs as a theorem rather than as an axiom. That the laws of thought are a sufficient foundation for the whole of logic, or that all other principles of logic are mere elaborations of them, was a doctrine common among traditional logicians. It has been shown, however, that these laws do not even comprise a sufficient set of axioms for the most elementary branch of logic, the propositional calculus, nor for the traditional theory of the categorical syllogism or the logic of terms. The law of excluded middle and certain related laws have been rejected by L.E.J. Brouwer, a Dutch mathematical intuitionist, and his school, who do not admit their use in mathematical proofs in which all members of an infinite class are involved. Brouwer would not accept, for example, the disjunction that either there occur ten successive 7's somewhere in the decimal expansion of p or else not, since no proof is known of either alternative; but he would accept it if applied, for instance, to the first 10100 digits of the decimal, since these could in principle actually be computed. In 1920 Jan Lukasiewicz, a leading member of the Polish school of logic, formulated a propositional calculus that had a third truth-value, neither truth nor falsity, for Aristotle's future contingents, a calculus in which the laws of contradiction and of excluded middle both failed. Other systems have gone beyond three-valued to many-valued logics e.g., certain probability logics having various degrees of truth-value between truth and falsity. Types of thinking The spectrum or range of thinking reflects the relative intensity of intrinsic and extrinsic influences. When intrinsic processes operate strongly and are relatively free of environmental constraints, a person thinks expressively: he imagines, fantasizes, dreams, hallucinates, or has delusions. As his thinking becomes dominated by external stimuli, he tends to become more logical, directed, disciplined; the process then is identified by such terms as judging, conceptualizing, and problem solving. Sigmund Freud recognized this distinction between expressive and disciplined function in contrasting what he called primary and secondary process thinking. Freud held that one's impulses and wishes arise from unconscious sources and determine primary process thinking, while the pursuit of exterior objects and goals determines secondary process thinking, which he associated with planning, rational control, and continuous organization. These two aspects of thinking also can be called, respectively, autistic (determined by subjective emotional-motivational activities) and realistic (oriented toward the external environment). The terms are not mutually exclusive but rather correspond to relative degrees of influence of different conditions that enter into thinking. In a broad sense, then, activities called thinking are internally adaptive responses to intrinsic and extrinsic stimuli; not only do they express inner impulses but they also serve to generate environmentally effective, goal-seeking behaviour. Realistic thinking Convergent thought processes It has been proposed that certain forms of thinking call on one's abilities to assemble and organize information. The result of such thinking satisfies a defined goal in the achievement of an effective solution to a problem. These forms are called convergent thinking and become apparent when situations arise in which one's ability to cope with a task demands resources beyond the explicit stimuli presented; i.e., converges the components of one's past and present experience in organizing or directing one's response.