Boolean Logic, Fregean Logic, Wittgensteinian Logic and the Processing of Natural Language

Gottlob Frege (1848-1925) transformed the field of logic from what it had remained since the days of Aristotle. Regarded as the founder of modern logic and much of modern philosophy, Frege laid the foundations of predicate logic, first-order predicate calculus and quantificational logic – formal systems central to computer science and mathematics. Frege was not satisfied with the ambiguity and imprecision of ordinary language. He created a new ‘formula language’ with elaborate symbols and definite rules, focused on conceptual content rather than rhetorical style, which he called Begriffsschrift – a formal language for 'pure thought'. Before Frege, George Boole (1815-1864) created what later became known as ‘Boolean logic’ which is fundamental to operations of computer science today. An application of Wittgensteinian logic could help filter authentic information from information disorder (non-information, off-information, mal-information and mis-information). Wittgensteinian logic applied in natural language processing technology (NLP), if possible and via automation, could transform the quality of information online. Many challenges remain.

Concepts

At the start Frege had not decided what a concept was to be, but knew well what a concept was supposed to do: to relate to objects which ‘fell under it’ in a fundamentally different way than a name relates to what it names. Such was to be the foundation of his account of quantificational logic. In the end two notions of concept emerge, one on which a concept has representational ‘intent’, another on which it does not. Some notions of unsaturation are discussed. The most natural of these fit what does have predicative intent. (This not all to the good.)

An analysis of Existential Graphs–part 2: Beta

This paper provides an analysis of the notational difference between Beta Existential Graphs, the graphical notation for quantificational logic invented by Charles S. Peirce at the end of the 19th century, and the ordinary notation of first-order logic. Peirce thought his graphs to be “more diagrammatic” than equivalently expressive languages (including his own algebras) for quantificational logic. The reason of this, he claimed, is that less room is afforded in Existential Graphs than in equivalently expressive languages for different ways of representing the same fact. The reason of this, in turn, is that Existential Graphs are a non-linear, occurrence-referential notation. As a non-linear notation, each graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic that are obtained by permuting those elements (sentential variables, predicate expressions, and quantifiers) that in the graphs lie in the same area. As an occurrence-referential notation, each Beta graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic in which the identity of reference of two or more variables is asserted. In brief, Peirce’s graphs are more diagrammatic than the linear, type-referential notation of first-order logic because the function that translates the latter to the graphs does not define isomorphism between the two notations.

The Modalities of Judgment

This chapter is an exposition of Frege’s theory of modality before adopting the sense/reference distinction. The background for understanding this theory is Kant’s conception of judgment and of the classification of judgments in the Table of Judgments. Frege agrees with Kant that modality is not an aspect of content. However, Frege’s discovery of modern quantificational logic leads him to reject Kant’s theory of logically significant structure. Moreover, Frege insists on a sharp distinction between judging and assuming, which leads him to reject Kant’s position that modalities mark distinct types of judgment. As a result, modality has no logical significance. Frege takes discourse in which we seem to ascribe necessity or possibility to contents to effect various sorts of implicature, and his accounts of these implicatures provides a reductionist and epistemic conception of modality.

Gödel’s theorems

Utilizing the formalization of mathematics and logic found in Whitehead and Russell’s Principia Mathematica (1910), Hilbert and Ackermann (1928) gave precise formulations of a variety of foundational and methodological problems, among them the so-called ‘completeness problem’ for formal axiomatic theories – the problem of whether all truths or laws pertaining to their subjects are provable within them. Applied to a proposed system for first-order quantificational logic, the completeness problem is the problem of whether all logically valid formulas are provable in it. In his doctoral dissertation (1929), Gödel gave a positive solution to the completeness problem for a system of quantificational logic based on the work of Whitehead and Russell. This is the first of the three theorems that we here refer to as ‘Gödel’s theorems’. The other two theorems arose from Gödel’s continued investigation of the completeness problem for more comprehensive formal systems – including, especially, systems comprehensive enough to encompass all known methods of mathematical proof. Here, however, the question was not whether all logically valid formulas are provable (they are), but whether all formulas true in the intended interpretations of the systems are. For this to be the case, the systems would have to prove either S or the denial of S for each sentence S of their languages. In his first incompleteness theorem, Gödel showed that the systems investigated were not complete in this sense. Indeed, there are even sentences of a simple arithmetic type that the systems can neither prove nor refute, provided they are consistent. So even the class of simple arithmetic truths is not formally axiomatizable. The idea behind Gödel’s proof is basically as follows. Let a given system T satisfy the following conditions: (1) it is powerful enough to prove of each sentence in its language that if it proves it, then it proves that it proves it, and (2) it is capable of proving of a certain sentence G (Gödel’s self-referential sentence) that it is equivalent to ‘G is not provable in T’. Under these conditions, T cannot prove G, so long as T is consistent. For suppose T proved G. By (1) it would also prove ‘G is provable in T’, and by (2) it would prove ‘G is not provable in T’. Hence, T would be inconsistent. Under slightly stronger conditions – specifically, (2) and (1′) every sentence of the form ‘X is provable in T’ that T proves is true – it can be shown that a consistent T cannot prove ‘not G’ either. For if ‘not G’ were provable in T it would follow by (2) that ‘G is provable in T’ would also be provable in T. But then by (1′) G would be provable. Hence, T would be inconsistent. The proof of Gödel’s second incompleteness theorem essentially involves formalizing in T a proof of a formula expressing the proposition that if T is consistent, then G. The second incompleteness theorem (that is, the claim that if T is consistent it cannot prove its own consistency) then follows from this and the first part of the proof of the first incompleteness theorem. The two incompleteness theorems have been applied to a wide variety of concerns in philosophy. The best known of these are critical applications to Hilbert’s programme and logicism in the philosophy of mathematics and to mechanism in the philosophy of mind.

Logic, medieval

Medieval logic is crucial to the understanding of medieval philosophy, for every educated person was trained in logic, as well as in grammar, and these disciplines provided techniques of analysis and a technical vocabulary that permeate philosophical, scientific and theological writing. At the practical level, logic provided the training necessary for participation in the disputations that were a central feature of medieval instruction, and whose structure – with arguments for and against a thesis, followed by a resolution – is reflected in many written works. At the theoretical level, logic, like other subjects, involved the study of written texts through lectures and written commentaries. The core of the logic curriculum from the twelfth century onwards was provided by the logical works of Aristotle. These provided the material for the study of types of predication, the analysis of simple propositions and their relations of inference and equivalence, the analysis of modal propositions, categorical and modal syllogisms, fallacies, dialectical Topics, and scientific reasoning as captured in the demonstrative syllogism. Comprehensive as this list might seem, medieval logicians realized that other logical subjects needed to be investigated, and, again from the twelfth century onward, new techniques and new genres of writing appeared. The main new technique involved the use of ‘sophismata’, or puzzling cases intended to draw attention to weaknesses and difficulties in logical definitions and rules. The new genres of writing especially included works on ‘supposition theory’, which concerned the types of reference that the subjects and predicates of propositions have in different contexts, and works on ‘syncategoremata’, which concerned the effect on sense and reference produced by the presence and placing of such logical terms as ‘all’, ‘some’, ‘not’, ‘if…then’, ‘except’, and so on. Other important topics for investigation include ‘insolubles’, or semantic paradoxes, and ‘consequences’, or valid inference forms. These new developments were seen as providing a supplement to Aristotelian logic, rather than an alternative. The only context in which people occasionally suggested that Aristotelian logic was inapplicable was that of Trinitarian theology, and the only logician who deliberately set out to reform logic as a whole was Ramon Llull. The study of medieval logic involves two kinds of difficulty. In the first place, few texts are available in translation, and indeed, many are not even available in printed form. In the second place, there is a problem of interpretation. For a very long time, the specifically medieval contributions to logic were ignored or despised, and when people began to take them more seriously, there was a strong tendency to look at them through the spectacles of modern formal logic. More recently, scholars have come to realize that medieval interests cannot be mapped precisely onto modern interests, and that any attempt, for example, to make a sharp distinction between propositional and quantificational logic is misleading. The first task of the modern reader is to try to understand what the medieval logician was really concerned with.