Gabbay Dov M. - Handbook of the history of logic. Volume 11, Logic, a history of its central concepts

The present volume marks the conclusion of the Handbook of the History of Logic series. This capstone volume addresses central topics in the history of logic, showing how logicians, philosophers, mathematicians and others understood these topics over the years and how they guided their development down to the present century.

Certainly the most central topic in logic is the notion of logical consequence. Asmus and Restall start with Aristotles definition of a syllogism as an argument in which, certain things having been assumed, something other than these follows of necessity from their truth, without needing any term from outside and carry the explanation of this conception through the middle ages and into the twenty-first century. Any account of logical consequence must determine the type of entities that can be premises and conclusion, must explain what ways premises can combine, and crucially, must explain the types of connection that that are allowed to hold between premises and conclusion in order for it to really be a consequence.

A part of this explanation will involve certain connected concepts: the quantifiers and the connectives. Bonevac traces the notion of quantification from Aristotle through modern generalized quantifiers. It is important to note, he says, that there is no theory-neutral way of defining quantification or even of delineating the class of quantifiers, and so a history of quantification has to trace the development of both what is to be explained along with how it is to be explained. Alongside the account of quantifiers and quantification needs to be an account of the logical particles the connectives. Bonevac and Dever discuss the implicit treatment of propositional connectives in Aristotle before moving on to the explicit theory of them developed by the Stoics. The development of an understanding of the connectives took a winding path from the Stoics through the medieval logica vetus (Old Logic) and the revolutionary logica nova (New Logic) of the 13th century, through the under-appreciated algebraic understanding of Leibniz, and to the Modern-Era logicians of the 19th and 20th century.

As Bonevac and Dever remark, the history of the connectives is marked by an ambivalence between the attitude that the connectives are operators on the content of the items being connected and an attitude that they are operators on the speech-act force of the items (say, in a presentation of an argument). And again, there is the ambivalence between the view that negation is a propositional operator (It is not the case that -) and that it is a term operator (- is notpale). The 20th century saw the latter issue decided in favor of the propositional approach for negation. However, the question of whether negation is a content or a speech-act operator (for instance, denial) is disputed. Speranza and Horn start with Paul Grices account of negation, using it as a springboard to discuss the ways this difference has been in effect over the history of logic.

Aristotles notion of following of necessity from the truth of the premises is often described in terms of the truth-values of the premises and the conclusion: It is not possible for the truth-value of every premise to be true while the truth-value of the conclusion is false. But of course, the history of logic has seen accounts where there are more than two truth-values, and furthermore where there are gaps and gluts of truth-value. Indeed, the notion of truth-value permeates much of the broader realms of philosophy, linguistics, mathematics and computer science, and Beziau undertakes a very broad-ranging discussion of the mathematical conception of truth-value to show how it underlies many of the more familiar conceptions that are associated with that concept.

Modality is yet another central concept in logic. Not only is it employed in Aristotles definition of a correct argument, but also it features in the characterization of modalized sentences, and thereby into metaphysics and language (de re and de dicto modalities). Knuuttila explores the ancient and medieval traditions in modality distinguishing modality as extensional (all possibilities will be actualized) from modality as alternativity and showing where these two conceptions emerge in more recent accounts of modality. These differing accounts also are manifested in the modal syllogistic and logics that were developed in ancient and medieval times, Knuuttila shows, as were interpretations of the modalities in terms of epistemic operators like knows and believes.

One version of logic employs no independently-claimed-to-be-necessarily-true statements (axioms) but instead employs only rules. Although some have claimed that Aristotles syllogistic is such a system, the more modern version traces its history to 1934. Despite this very recent invention, most logic that is currently taught in philosophy is of this nature natural deduction. Pelletier and Hazen discuss the history (since 1934) of this development, its relationship with other conceptions of logic, and the metatheoretic facts that allow it to have such a prominent position in modern logic.