He seems to means 'humanist logic', however, since he ignores the rich 'scholastic' tradition of the 15 th and early 16 th centuries. It is of much less interest to look at Valla and Vives, as Parreiah does, since they were not interested in the theory of syllogisms at all. For them, it was the symbol of the corruption of philosophy and learning in the universities.

Perreiah is right, however, in his assessment that their criticisms makes no sense, if they are taken as serious logical arguments against the theory of syllogisms. Instead, they should be seen as ideological and political arguments that seek to replace a theoretical pursuit with a practical one. Stephen Gaukroger contributes the next, although very short, article, which contrasts the theory of the syllogisms with Descartes' idea of a logic of discovery.

## Leibniz: Logic

It contains ideas similar to those in his earlier work. A similar confusion, although perhaps not as serious, can be witnessed in Douglas Jesseph's article on Hobbes. Hobbes seems to have known more about logic than Descartes, but his knowledge was limited to the very rudimentary and sometimes inaccurate summaries published in his time. An interesting note, made by the author but not developed, is that Hobbes seems to suggest a reading of the syllogism that eliminates Aristotle's notion of existential import.

I am not sure that this is a systematic suggestion by Hobbes, since in other places, for example, in De corpore I. Hobbes is considerably more interesting when he turns to demonstration and develops his so-called scientific method, which is founded on Aristotle's account of demonstration. This part of Jesseph's article is very interesting.

Although they follow the ideas of Descartes, Antoine Arnauld and Pierre Nicole have a much better grasp of logic than Descartes had. Russell Wahl presents an illuminating take on the Port-Royal logic and on syllogisms.

They push an interpretation of the categorical proposition that relies on an idea that the predicate or attribute, as they call it is in the extension of the subject. It is not entirely clear what this comes to and several interpretations have been proposed, which Wahl discusses. What is clear is that they push the same kind of reading in their interpretation of the syllogisms. They manage to get a consistent interpretation and show how the second and third figure moods can be reduced to the first, but their view has clear limitations, as Wahl brings out.

It is an illuminating article, one of the better articles on the Port-Royal logic that I have read. Locke had very little interest in syllogisms, as Davide Poggi shows in his article. The perspective on syllogistics as forms of reasoning that he seems to emphasize is very limited.

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Wolfgang Lenzen's contribution on Leibniz, on the other hand, displays a more interesting account of logic and syllogisms. Leibniz clearly builds on the Port-Royal logic when he develops what Lenzen calls, an 'operator of conceptual containment', that is, 'A is B' is read as 'A contains B'. Log In Sign Up. Paul Redding. Thus, Leibniz would boast in a letter to his future employer Duke Johann Friedrich of Hannover, October , that: In philosophy, I have found a means of accomplishing in all sciences what Descartes and others have done in Arithmetic and Geometry by Algebra and Analysis, by the Ars Combinatoria … By this all composite notions in the whole world are reduced to a few simple ones as their alphabet; and by the combination of such an alphabet a way is made of finding, in time by an ordered method, all things with their theorems and whatever is possible to investigate concerning them.

Algebra had been an area of mathematics that remained relatively undeveloped by the Greeks but that had flourished under the Arabs. First, geometers required proofs that were linked to their labelled diagrams, so as to prove theorems from axioms. More specifically, Aristotle seemed to draw on geometry for his logic in a variety of ways. Aristotle had used Greek letters alpha, beta, gamma like those used to label diagrams in geometry, but here as schematic letters for the representation of subject and predicate terms of judgments,3 and it has been suggested that in his teaching these symbolic composites were similarly related to labelled diagrams in quasi-geometric ways Netz ,15; Kneale and Kneale , 71— We have no direct evidence for this, but even if it turned out that Aristotle had not used diagrams in this way, the spatial imagery he employed in relation to concepts, suggesting some concepts as contained within others, makes the use of such diagrams seem natural, and diagrams are known to have been used for this purpose in the Middle Ages and after.

But if proper names could have no real role in syllogisms, they would thereby be unable to be used in arguments about individuals. Such letters could be raised to higher degrees. In Aristotle, schematic letters stand in for sentences, and are thus replaceable by sentences. Variables are what are bound by quantifiers.

Dutilh Novaes , 71 and 71 n. Particular judgments talk about some As—for example, some men— but such a group could now be thought to be narrowed down to one: some man, for example, Socrates. However, the use of numbers might also be thought to be in line with the modern impulse to allow reasoning to be applied to individuals per se—that is, to Socrates and Alcibiades, say, just as numerals can be naturally thought of as the proper names of the particular numbers they represent Auerbach It is here that the issue of just how diagrams are interpreted becomes important.

Viewing an Euler diagram that employs three concentric circles A, B and C, such that circle C is contained in B, and B in A, it will be immediately apparent that that every point in a circle C is thereby in the circle B that contains it, and is thus also in the circle A that contains it. For example, if Socrates is represented by a specific point in the circle C, representing the class of all humans, with B representing the class of animals and A the class of mortals, it is immediately obvious that Socrates is also a mortal thing. Rather, Aristotle thought of points in terms of lines and lines in terms of planes—a point being a place where two lines intersected, and a line representing the intersection of two planes—and this was in keeping with the primacy of geometry in Greek mathematics.

In short, had Aristotle used diagrams to show the relations of inclusion among line-segments 5 This logical representation of individuals by universal terms seems to have been accompanied by an ontological consideration of individuals as universals, in the sense of conceiving of individuals as having their own individual essences Tarlazzi Classically, only kinds had essences, not their individual instances.

Rather, they would more have been meant to be read in what would later be described as an intensional way, as representing relations among concepts rather than classes of things. This equivalence of intensional and extensional readings would then seem to be implicit in his technique of translating between the two ways of representing individuals in syllogisms. Either way the representation connects with the same thing, but it is the same thing known differently.

From the perspective of analytic geometry, these too could be dissolved into the atomic entities constituting them, sets of points. A logic that modelled itself on algebra and the arithmetic to which it gave expression would thus seem to be an appropriate logic for a nominalistically conceived world in which natural kinds, say, were to be dissolved into classes of entities that simply shared certain properties.

Leibniz had acknowledged that we finite beings cannot in fact ascend all the way, but Kant had insisted on a more principled restriction on human knowledge. As such it signalled the limits of any Cartesian arithmetization of geometry and it marked a reassertion of the Greek notion of geometry as resisting reduction to arithmetic, and a reassertion of the place for Greek kinds in the modern world of individuals.

At the end of the nineteenth century, the French mathematician Henri Poincare employed the term for the discipline that would become known as algebraic topology, and understood broadly as the application of algebra to the geometry of continuous multi-dimensional surfaces. The roots of projective geometry can be found in the ancient world in the interest in ratios that remain invariant under changes to diagrams Shenitzer , but in its modern form it had a distinctive pre-history in Renaissance attempts to theorise the perspectival aspects of pictorial representation Andersen Thus, lines that were parallel in the represented horizontal plane train tracks disappearing into the distance, in the standard contemporary example would, in the representing plane, be represented as converging towards an horizon, or a circular feature on the represented plane would be represented as an ellipse on the representing plane.

Projective geometry would therefore have offered a new determinate conception of infinity in relation to which the differences between all monadic perspectives might be envisaged as able to be resolved. With the idea of parallel lines meeting at infinity, projective geometry had removed a restriction found in Euclidean geometry to the idea that any two lines can be conceived as intersecting, as even parallel lines were now thought to meet—that is, meet at infinity.

## The Algebra of Logic Tradition (Stanford Encyclopedia of Philosophy)

In retrospect, however, as has been argued by Bassler , one might take the type of reciprocity of the extensional and intensional interpretations of conceptual containment found in Leibniz as itself as the earliest manifestation of this underlying phenomenon. Thus, the line is conceived as bounded by two opposing poles which are the origins of opposing directionalities and which expressed opposing points of view.

From one point of view we grasp the world and ourselves in it naturalistically, as constituted by determinate happenings subject to law-like regularities. From another perspective, however, we consider ourselves as free agents, able to introduce effects into the world, effects that would not have occurred without the exercise of our wills. This is the dimension that is seized upon and elevated to the ideal of moral subjectivity urged upon us by Fichte.

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The single line able to be read as going in contrary directions is meant to give expression to this type of duality. Beside its Spinozist and Leibnizian resonances, the constructed line is meant to expresses definite Kantian features as well. Given the fact that each of those one-sided limited perspectives necessarily 14 Kant had linked directionality to intentional action in an essay otherwise directed to explaining the meaning of negative numbers Kant a.

For someone on a sailing ship travelling east to Brazil, the 3 miles travelled on being blown back in a westerly direction would be counted as —3 in relation to the trip as a whole. Of course, in an absolute sense, the distance travelled would be the same whatever direction the ship was moving ibid.

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These mathematically grounded dualism represent, I suggest, a largely unexplored position within the modern metaphysics of mind. Much of the fate of subsequent German idealism would, from this point, be fought out over how this situation was to be understood, with Schelling and Hegel traditionally understood as interpreting it in opposed ways. Indeed, the imagery of opposed perspectives would seem to be a quite natural one for thought about hermeneutics—the theory of interpretation—and the thesis of the reciprocities of speech and thought and expression and understanding, that one finds there.

For example, he describes the acts of linguistic production and understanding in terms of such an Umkehrung Schleiermacher , 7. Therein lies the duality of the talent for language. The comparative grasping of languages in their difference, the extensive talent for language, is different from the penetration into the interior of language in relation to thought, the intensive talent for language.

See Stalnaker , chs 1 and 3. Bibliography Andersen, Kirsti. New York: Springer. Arnauld, Antoine and Nicole, Pierre Logic or the Art of Thinking. Translated and Edited by Jill Vance Buroker. Cambridge: Cambridge University Press. Auerbach, David. Here mathematics shall mean mostly algebra, i. Hailperin Algebraic logic can be already found in the work of Leibniz, Jacob Bernoulli and other modern thinkers, and it undoubtedly constitutes an important antecedent of Boole's approach.

In a broader perspective, both are part of the tradition of symbolic knowledge in the formal sciences, as first conceived by Leibniz see Esquisabel This idea of algebraic logic was continued to some extent in the French Enlightenment in the work of Condillac and Condorcet see Grattan-Guinness 14 ff. Later, in his Pure Logic from , Jevons changed Boole's partial operation of union of disjoint sets to the modern unrestricted union and eliminated Boole's questionable use of uninterpretable terms see Jevons In addition, he extended the algebra of logic for classes to the algebra of logic for binary relations and introduced general sums and products to handle quantification.

The contributions of Gottlob Frege — to logic from the period —, based on an axiomatic approach to logic, had very little influence at the time and the same can be said of the diagrammatic systems of C. Peirce developed at the turn of the century. Whitehead and Russell rejected the algebra of logic approach, with its predominantly equational formulations and algebraic symbolism, in favor of an approach strongly inspired by the axiomatic system of Frege, and using the notation developed by Giuseppe Peano, namely to use logical connectives, relation symbols and quantifiers.

In particular, elimination theorems in the algebra of logic influenced decision procedures for fragments of first-order and second-order logic see Mancosu, Zach, Badesa After WWI David Hilbert — , who had at first adopted the algebraic approach, picked up on the approach of Principia , and the algebra of logic fell out of favor. However, in , Tarski treated relation algebras as an equationally defined class.

Such a class has many models besides the collection of all binary relations on a given universe that was considered in the s, just as there are many Boolean algebras besides the power set Boolean algebras studied in the s. In the years — Tarski, along with his students Chin and Thompson, created cylindric algebras as an algebraic logic companion to first-order logic, and in Paul Halmos — introduced polyadic algebras for the same purpose.

## The Algebra of Logic Tradition

As Halmos b, c and d noted, these new algebraic logics tended to focus on studying the extent to which they captured first-order logic and on their universal algebraic aspects such as axiomatizations and structure theorems, but offered little insight into the nature of the first-order logic which inspired their creation. Unfortunately, he was never able to incorporate his best ideas into a significant system. His omission of a symbol for equality made it impossible to develop an equational algebra of logic.

It seems that synthesis was not De Morgan's strong suit. Boole approached logic from a completely different perspective, namely how to cast Aristotelian logic in the garb of symbolical algebra. Using symbolical algebra was a theme with which he was well-acquainted from his work in differential equations, and from the various papers of his young friend and mentor Duncan Farquharson Gregory — , who made attempts to cast other subjects such as geometry into the language of symbolical algebra.

Since the application of symbolical algebra to differential equations had proceeded through the introduction of differential operators, it must have been natural for Boole to look for operators that applied in the area of Aristotelian logic. In his book, Boole realized that it was simpler to omit selection operators and work directly with classes. However he kept the selection operators to justify his claim that his laws of logic were not ultimately based on observations concerning the use of language, but were actually deeply rooted in the processes of the human mind. From now on in this article, when discussing Boole's book, the selection operators have been replaced with the simpler direct formulation using classes.

Since symbolical algebra was just the syntactic side of ordinary algebra, Boole needed ways to interpret the usual operations and constants of algebra to create his algebra of logic for classes. Addition was defined as union, provided one was dealing with disjoint classes; and subtraction as class difference, provided one was subtracting a subclass from a class. In other cases, the addition and subtraction operations were simply undefined, or as Boole wrote, uninterpretable.

To eliminate the middle term in a syllogism Boole borrowed an elimination theorem from ordinary algebra, but it was too weak for his algebra of logic. This would be remedied in his book. Boole found that he could not always derive the desired conclusions with the above translation of particular propositions i. The symbolic algebra of the s included much more than just the algebra of polynomials, and Boole experimented to see which results and tools might apply to the algebra of logic. For example, he proved one of his results by using an infinite series expansion.

His successors, especially Jevons, would soon narrow the operations on classes to the ones that we use today, namely union, intersection and complement. As mentioned earlier, three-quarters of the way through his brief book of , after finishing derivations of the traditional Aristotelian syllogisms in his system, Boole announced that his algebra of logic was capable of far more general applications.

Then he proceeded to add general theorems on developing expanding terms, providing interpretations of equations, and using long division to express one class in an equation in terms of the other classes with side conditions added. Boole's theorems, completed and perfected in , gave algorithms for analyzing infinitely many argument forms. This opened a new and fruitful perspective, deviating from the traditional approach to logic, where for centuries scholars had struggled to come up with clever mnemonics to memorize a very small catalog of valid conversions and syllogisms and their various interrelations.

De Morgan's Formal Logic did not gain significant recognition, primarily because it was a large collection of small facts without a significant synthesis. Boole's The Mathematical Analysis of Logic had powerful methods that caught the attention of a few scholars such as De Morgan and Arthur Cayley — ; but immediately there were serious questions about the workings of Boole's algebra of logic: Just how closely was it tied to ordinary algebra?

How could Boole justify the procedures of his algebra of logic? In retrospect it seems quite certain that Boole did not know why his system worked. Nonetheless it is also likely that he had checked his results in a sufficient number of cases to give substance to his belief that his system was correct.

In his second book, The Laws of Thought , Boole not only applied algebraic methods to traditional logic but also attempted some reforms to logic. He started by augmenting the laws of his algebra of logic without explicitly saying that his previous list of three axioms was inadequate , and made some comments on the rule of inference performing the same operation on both sides of an equation. But then he casually stated that the foundation of his system actually rested on a single new principle, namely it sufficed to check an argument by seeing if it was correct when the class symbols took on only the values 0 and 1, and the operations were the usual arithmetical operations.

Let us call this Boole's Rule of 0 and 1. No meaningful justification was given for Boole's adoption of this new foundation, it was not given a special name, and the scant references to it in the rest of the book were usually rather clumsily stated. The development of the algebra of logic in the Laws of Thought proceeded much as in his book, with minor changes to his translation scheme, and with the selection operators replaced by classes.

This theorem also played an important role in Boole's interpretation of Aristotle's syllogistic. From an algebra of logic point of view, the treatment at times seems less elegant than that in the book, but it gives a much richer insight into how Boole thinks about the foundations for his algebra of logic.

The final chapter on logic, Chapter XV, was an attempt to give a uniform proof of the Aristotelian conversions and syllogisms. It is curious that prior to Chapter XV Boole did not present any examples of arguments involving particular propositions. The details of Chapter XV are quite involved, mainly because of the increase in size of expressions when the Elimination and Development Theorems are applied. Boole simply left most of the work to the reader.

Later commentators would gloss over this chapter, and no one seems to have worked through its details. Aside from the Rule of 0 and 1 and the Elimination Theorem, the presentation is mainly interesting for Boole's attempts to justify his algebra of logic. He argued that in symbolical algebra it was quite acceptable to carry out equational deductions with partial operations, just as one would when the operations were total, as long as the terms in the premises and the conclusion were interpretable.

The geometric interpretation of complex numbers was recognized early on by Wessel, Argand, and Gauss, but it was only with the publications of Gauss and Hamilton in the s that doubts about the acceptability of complex numbers in the larger mathematical community were overcome.

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Jevons, who had studied with De Morgan, was the first to offer an alternative to Boole's system. Boole completely rejected this suggestion it would have destroyed his system based on ordinary algebra and broke off the correspondence. Jevons published his system in his book, Pure Logic reprinted in Jevons But he kept Boole's use of equations as the fundamental form of statements in his algebra of logic.

However, he refined his system of axioms and rules of inference until the result was essentially the modern system of Boolean algebra for ground terms , that is, terms where the class symbols are to be thought of as constants, not as variables. It must be noticed that modern equational logic deals with universally quantified equations which would have been called laws in the s. Terms that only have constants no variables are called ground terms.

By carrying out this analysis in the special setting of an algebra of predicates or equivalently, in an algebra of classes Jevons played an important role in the development of modern equational logic. As mentioned earlier, Boole gave inadequate sets of equational axioms for his system, originally starting with the two laws due to Gregory plus his idempotent law; these were accompanied by De Morgan's inference rule that one could carry out the same operation Boole's fundamental operations in his algebra of logic were addition, subtraction and multiplication on equals and obtain equals.

Boole then switched to the simple and powerful but unexplained Rule of 0 and 1. Having replaced Boole's fundamental operations with total operations, Jevons proceeded, over a period of many years, to work on the axioms and rules for his system. Some elements of equational logic that we now take for granted required a considerable number of years for Jevons to resolve:.

In Jevons listed this as a postulate , p.