Mark Weinstein
Montclair State University, USA
Logic today is heir to three competing images, two of which reflect the deepest intuitions in its articulation. The first is that logic identifies the norms of good reasoning in the most general sense. The second is that logic serves as the foundation for abstract inquiry, traditionally, metaphysics and in the 20th century, mathematics. The third, condemned in the 20th century as ‘psychologism’ is the view that sees logic as the structure of reasoning minds. Each of these intuitions is reflected in the history of logic and each has been the basis of the efforts of the most important thinkers in the field.
All three are found in Plato (c.428-347 B.C.) who in The Republic saw abstract inquiry as continuous with good reasoning and reasoning as a fundamental faculty of mind. Aristotle (384-322 B.C.) distinguished the theoretic and the practical role of logic in separate treatises, the Prior Analytic and the Rhetoric, and offered an account of mind in De Anima. Aristotle’s logic and his rhetoric have served as the foundation of logical studies ever since, although contributions were made by the Stoics who built on the ideas of the Megarian school (c. 430-c. 360B.C). Aristotle was the focus of Arab scholars such as Abu ibn Sina known as Avicenna (980-1037) and Ibn Rushd, called Averroes (1126-1198) leading to medieval philosophers such as Peter Abelard (1079-1142) and William of Ockham (c. 1285-1349) who elaborated technical logical concepts, explored metaphysical assumptions and refined rhetorical analysis.
Manuals of logic replaced scholastic disputation, as in The Port Royal Logic (1662) by Antoine Arnaud (1612-1694) and Pierre Nicole (1625-1695). Among the contributors to the movement to simplify scholastic accounts of logic and to incorporate rhetorical elements within the teaching of logic are Lorenzo Valla (1407-1457) and Peter Ramus (1515-1572) a critic of Aristotle who attempted to replace scholastic terminology and disputation by more ordinary language, The period, called the interregnum, from the middle of the fifteenth century until the revival of logical theory by Gottfried Wilhelm Leibniz (1646-1716) on the continent and William Hamilton (1788-1856) in England, was focused on logic as a practical concern
Psychologism, the analysis of logical relations in psychological terms, is characteristic of the interregnum. Rene Descartes (1596-1650) assimilated logical necessity to psychological certainty; John Locke (1632-1704) took judgments to be mental operations rather than logical propositions. This set the stage for a three hundred year tradition in which logic was seen as an indication of the underlying structure of the intellect. This is best reflected in the work of Immanuel Kant (1724-1804) who used the forms of logical judgment as a clue to the categories that he took to underlie all experience.
Kant, who saw Aristotle’s logic as a “closed and completed body of doctrine,” reflected the tradition from Plato onward that logic embodies timeless and necessary truths. Aristotelian syllogism, however, does not offer a complete theory of logical inference. Construing the terms of propositions as naming natural kinds creates difficulties for arguments that refer to possibly non-existent kinds, such as hypothetical entities. Modern logic followed Augustus De Morgan (1806-1871), George Boole (1815-1864) and John Venn (1834-1923) in interpreting logical propositions as statements about sets, which may be empty. This furnished the connection between logic, elementary set theory and mathematics characteristic of logic in the 20th century. Aristotle’s logic is also inadequate to inferences based on relational predicates, which can be easily characterized in mathematical terms.
The main project of last century can be seen to grow out of the preliminary efforts of Charles Sanders Pierce (1839-1914) and Ludwig Wittgenstein (1889-1951). Extending the work of Gottlob Frege (1848-1935), Bertrand Russell (1872-1979) and Alfred North Whitehead (1861-1947) in Principia Mathematica (1910-1913) furnished the foundation for the project, identified with David Hilbert (1862-1943), to define a formal system, which contained in its set of consequences all and only true mathematical statements. The foundation of the formal system was problematic, as shown by a paradoxical construction discovered by Russell: the set whose members are non-members of itself. Kurt Godel (1906-1978) showed that if a formal system is powerful enough to contain number theory there are some statements expressible in it that are undecidable. Since these statements are, if true, unprovable, truth in a formal rendering of mathematics cannot be reduced to provability. A corollary of Godel’s Theorem is that the consistency of a system can never be proved within the system itself. This brought into question the oldest intuition in logic, Plato’s vision that the true and the provable coincide. The mathematical correlate of that intuition was the distinction between logic as a formal syntax distinct from its semantics, the objects that formed a possible interpretation of the formal system.
The study of the relation between formal languages and their interpretations is associated with Alfred Tarski (1902-1983), reflecting studies of the foundations of arithmetic, required mathematically well-defined models, and given the availability of such domains guaranteed by the Lowenheim-Skolem theorem, the natural domain for logical explorations became the numbers. This raised the crucial issue of the size of the domains that served as models for formal languages. Mathematics requires infinite sets of objects, among them infinities that transcend the size of the natural numbers, as demonstrated by George Cantor (1848-1925). Exploring the size of the domains of mathematical models in which logic is complete, that is, where truth and provability coincide, became a major concern of mathematical logic in the twentieth century, notably in the work of Thoralf Skolem (1887-1963) and Alonzo Church (1903-1995). The disjunction between provability and truth, however, brought into question the deepest principle within the second intuition: that the real and the rational coincide.
This opened the door for extensions of logic that reflect the first and especially the third intuitions. Nonmonotonic and paraconsistent logics furnish systems that reflect the success of mathematical logic in their formal rigor and metalogical transparency. But they also reflect intuitions of what reasonableness requires if logic is to make sense of epistemic practices that rely on inferences. For nonmonotonic logics, inference becomes sensitive to new information, as is the case in natural and scientific reasoning. In paraconsistent logic contradictions cease being catastrophic to logical functioning. Since new and contradictory information is all too common in inquiry of all sorts, such approaches to logic confront the practical concerns of reasoning. Paraconsistent logics are seen to accommodate belief revision and include relevant and many-valued logics within their scope.
Modern logic addressed a number of other philosophical dimensions, including the logic of possibility and necessity, modal logic, explored by C. I. Lewis (1883-1964), deontic logic, the logic of normative judgments and epistemic logic, the logic of belief.
In a very different logical frame of mind, eschewing, for the most part, the formal structures and metatheory of mathematical logic, informal logic enriches the image of logic with insights from the field of argumentation theory. This requires an engagement with dialogue and a dynamical view of argumentation that reflects in an informal way the same issues that move the nonmontonic and paraconsistent logicians. This points to an area of possible collaboration between formal and informal logicians.
The third intuition, logic as the structure of reasoning minds is evidenced in the work of experimental psychologists Jean Piaget (1896–1980) and P. N. Johnson-Laird (b. 1936). Frege, among others, considered psychologism a fundamental error in philosophical logic, confusing the ‘is’ of psychological practice with the ‘ought’ of normative inquiry. But this merely distinguishes the third intuition, logic as the structure of reasoning minds, from the first, logic as the most general norms governing good reasoning. Empirical studies by R. E. Nisbett and L. D. Ross have shown divergence between actual reasoning strategies and the norms of logic. The role this should play in our understanding of logic is an open question.
Independently, the third intuition, reconsidered as the structure of computers, drew on the work of Alan Turing (1912-1954) and the rapid development of computer languages, foundationally tied to propositional logic. Artificial intelligence, especially in expert systems that attempted to mimic human inferences based on evidence in the light of body of knowledge, called for flexibility that quickly transcended the initial successes with purely logical tasks as in the theorem proving programs of the mid-century. To what extent the new dialogical logics will add to our ability to represent a more human information processing perspective within artificial intelligence remains an area for continuing exploration.
Along with mathematics, physical science was among the areas most identified with logical constructions, in particular, the work of Rudolf Carnap (1891-1970), Karl Popper (1902-1994) and Carl Hempel (1905-1997). The failure of these accounts to offer a persuasive rational reconstruction of empirical science led to the application of model theory and other aspects of logic to the understanding of scientific inquiry as in the work of W.V.O Quine (1908-2000), Patrick Suppes (b. 1922) and B. van Fraassen (b. 1941). To what extent logicians can learn from a focus on science in place of the traditional focus on arithmetic remains a relatively unexplored area for future logical advance.
