Related papers: Boole's Chapter XV: Syllogism Details
In modern algebra it is well-known that one cannot, in general, apply ordinary equational reasoning when dealing with partial algebras. However Boole did not know this, and he took the opposite to be a fundamental truth, which he called the…
In the present paper we aim to provide a thoughtful and exegetical account of the fundamental ideas at the basis of Boole's theory, with the goal of developing our investigation strictly within the conceptual structure originally introduced…
This article explores the ideas that went into George Boole's development of an algebra for logical inference in his book The Laws of Thought. We explore in particular his wife Mary Boole's claim that he was deeply influenced by Indian…
This work presents an operational and geometric approach to logic. It starts from the multilinear elective decomposition of binary logical functions in the original form introduced by George Boole. A justification on historical grounds is…
Autoepistemic logic extends propositional logic by the modal operator L. A formula that is preceded by an L is said to be "believed". The logic was introduced by Moore 1985 for modeling an ideally rational agent's behavior and reasoning…
Generalizations of the Monty Hall problem are studied according to George Boole's (1853) "An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities"
Aristotelian assertoric syllogistic, which is currently of growing interest, has attracted the attention of the founders of modern logic, who approached it in several (semantical and syntactical) ways. Further approaches were introduced…
This paper is an informal survey of some of the deep connections between logic and optimization. It covers George Boole's probability logic, decision diagrams, logic and cutting planes, first order predicate logic, default and nonmonotonic…
A new computational method is presented to implement the system of deductive logic described by Aristotle in Prior Analytics. Each Aristotelian problem is interpreted as a parametric probability network in which the premises give…
We propose a new modal logic endowed with a simple deductive system to interpret Aristotle's theory of the modal syllogism. While being inspired by standard propositional modal logic it is also a logic of terms that admits a (sound)…
Gottfried Leibniz embarked on a research program to prove all the Aristotelic categorical syllogisms by diagrammatic and algebraic methods. He succeeded in proving them by means of Euler diagrams, but didn't produce a manuscript with their…
An examination of George Boole's mysterious use of the Algebra of Numbers to create an Algebra of Logic, and subsequent research connected to this.
The computational method of parametric probability analysis is introduced. It is demonstrated how to embed logical formulas from the propositional calculus into parametric probability networks, thereby enabling sound reasoning about the…
A formal axiomatic mathematical framework for Boolos' Hardest Logic Puzzle Ever is presented and two theorems about its solvability are proved. By strictly following Boolos' instructions (in particular, the requirement that all gods are…
A rigorous, modern version of Boole's algebra of logic is presented, based partly on the 1890s treatment of Ernst Schroder.
Aristotle's discussions on modal syllogistic have often been viewed as error-prone and have garnered significant attention in the literature due to historical and philosophical interests. However, from a contemporary standpoint, they also…
The effect of Richard T. Cox's contribution to probability theory was to generalize Boolean implication among logical statements to degrees of implication, which are manipulated using rules derived from consistency with Boolean algebra.…
This paper revisits the foundations of mathematical proof through the lens of Aristotle's threefold conception of truth: sensory evidence, axiomatic definition, and syllogistic deduction. I argue that modern mathematics has too often…
We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form: Some A are R-related to some B; Some A are R-related to all B; All A are R-related to some B; All A are…
The language of probability is used to define several different types of conditional statements. There are four principal types: subjunctive, material, existential, and feasibility. Two further types of conditionals are defined using the…