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Bolzano’s Logic

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Editor: Admir Skodo. Other Logics: Alternatives to Formal Logic in the History of Thought and Contemporary Philosophy challenges the widespread idea of formal logic as inherently monolithic, universal, and ahistorical. Written by both leading and up-and-coming scholars, and edited by Admir Skodo, Other Logics offers a wide variety of historical and philosophical alternatives to this idea, all arguing that logic is a historical, concrete, and multi-dimensional phenomenon. To name a few examples, Frank Ankersmit lays down a representationalist logic, Alessandra Tanesini forcefully argues for the possibility of logical aliens, Christopher Watkin analyzes how leading contemporary French philosophers view the idea of logic, and Aaron Wendland unearths Heidegger's critique of formal logic.

In Other Logics readers will find provocative interventions in a highly contested field in contemporary philosophy. Marx's Concept of the Alternative to Capitalism.

The Logical Significance of Assertion: Frege on the Essence of Logic

What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics in the form of classes of algebras that constitute the algebraic semantics for these deductive systems and connected problems like representation and duality.

Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic Czelakowski Works in the more recent abstract algebraic logic AAL focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator Czelakowski Whether a given relation holds for two individuals is one bit of information, so relations are studied with Boolean arithmetic.

Elements of the power set are partially ordered by inclusion , and lattice of these sets becomes an algebra through relative multiplication or composition of relations. The conversion refers to the converse relation that always exists, contrary to function theory. A given relation may be represented by a logical matrix ; then the converse relation is represented by the transpose matrix.

Transactions of the Charles S. Peirce Society: A Quarterly Journal in American Philosophy

A relation obtained as the composition of two others is then represented by the logical matrix obtained by matrix multiplication using Boolean arithmetic. The description of the key binary relations has been formulated with the calculus of relations.

The rise and fall and rise of logic | Aeon Essays

But a univalent relation is only a partial function , while a univalent total relation is a function. The relation algebra structure, based in set theory, was transcended by Tarski with axioms describing it. Then he asked if every algebra satisfying the axioms could be represented by a set relation. The negative answer [5] opened the frontier of abstract algebraic logic.

Algebraic logic treats algebraic structures , often bounded lattices , as models interpretations of certain logics , making logic a branch of order theory. In the table below, the left column contains one or more logical or mathematical systems, and the algebraic structure which are its models are shown on the right in the same row. Some of these structures are either Boolean algebras or proper extensions thereof. Modal and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators.

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Algebraic formalisms going beyond first-order logic in at least some respects include:. Algebraic logic is, perhaps, the oldest approach to formal logic, arguably beginning with a number of memoranda Leibniz wrote in the s, some of which were published in the 19th century and translated into English by Clarence Lewis in Parkinson and Loemker translated selections from Couturat's volume into English.

Modern mathematical logic began in , with two pamphlets whose respective authors were George Boole and Augustus De Morgan. The "calculus of relations" [8] is arguably the culmination of Leibniz's approach to logic. Whitehead all shared Leibniz's dream of combining symbolic logic , mathematics , and philosophy.

According to Helena Rasiowa , "The years saw, in particular in the Polish school of logic, researches on non-classical propositional calculi conducted by what is termed the logical matrix method.

Since logical matrices are certain abstract algebras, this led to the use of an algebraic method in logic.