Hilbert-style proof Calculus for Propositional Logic in ABC notation

2019 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Propositional Logic ◽  
Inference Rule ◽  
Modus Ponens ◽  
Axiomatic System ◽  
First Contact

All nine axioms and a single inference rule of logic (Modus Ponens) within the Hilbert axiomatic system are presented using capital letters (ABC) in order to familiarize the beginner student in hers/his first contact with the topic.

scholarly journals On Basic Probability Logic Inequalities

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1409
Author(s):  
Marija Boričić Joksimović
Keyword(s):  
Probability Theory ◽  
Inference Rule ◽  
Modus Ponens ◽  
Probability Logic ◽  
Modus Tollens ◽  
Short Overview ◽  
Basic Probability ◽  
Probabilistic Version ◽  
Hypothetical Syllogism

We give some simple examples of applying some of the well-known elementary probability theory inequalities and properties in the field of logical argumentation. A probabilistic version of the hypothetical syllogism inference rule is as follows: if propositions A, B, C, A→B, and B→C have probabilities a, b, c, r, and s, respectively, then for probability p of A→C, we have f(a,b,c,r,s)≤p≤g(a,b,c,r,s), for some functions f and g of given parameters. In this paper, after a short overview of known rules related to conjunction and disjunction, we proposed some probabilized forms of the hypothetical syllogism inference rule, with the best possible bounds for the probability of conclusion, covering simultaneously the probabilistic versions of both modus ponens and modus tollens rules, as already considered by Suppes, Hailperin, and Wagner.

Principal type-schemes and condensed detachment

1990 ◽  
Vol 55 (1) ◽  
pp. 90-105 ◽  
Author(s):  
J. Roger Hindley ◽  
David Meredith
Keyword(s):  
Type Theory ◽  
Propositional Logic ◽  
Modus Ponens ◽  
The Other ◽  
Combinatory Logic ◽  
Minimal Amount ◽  
Principal Type ◽  
Condensed Detachment

The condensed detachment rule, or ruleD, was first proposed by Carew Meredith in the 1950's for propositional logic based on implication. It is a combination of modus ponens with a “minimal” amount of substitution. We shall give a precise detailed statement of rule D. (Some attempts in the published literature to do this have been inaccurate.)The D-completeness question for a given set of logical axioms is whether every formula deducible from the axioms by modus ponens and substitution can be deduced instead by rule D alone. Under the well-known formulae-as-types correspondence between propositional logic and combinator-based type-theory, rule D turns out to correspond exactly to an algorithm for computing principal type-schemes in combinatory logic. Using this fact, we shall show that D is complete for intuitionistic and classical implicational logic. We shall also show that D is incomplete for two weaker systems, BCK- and BCI-logic.In describing the formulae-as-types correspondence it is common to say that combinators correspond to proofs in implicational logic. But if “proofs” means “proofs by the usual rules of modus ponens and substitution”, then this is not true. It only becomes true when we say “proofs by rule D”; we shall describe the precise correspondence in Corollary 6.7.1 below.This paper is written for readers in propositional logic and in combinatory logic. Since workers in one field may not feel totally happy in the other, we include short introductions to both fields.We wish to record thanks to Martin Bunder, Adrian Rezus and the referee for helpful comments and advice.

scholarly journals Forward and Backward Chaining with P Systems

2014 ◽  
pp. 149-158
Author(s):  
Sergiu Ivanov ◽  
Artiom Alhazov ◽  
Vladimir Rogojin ◽  
Miguel A. Gutiérrez-Naranjo
Keyword(s):  
Computer Science ◽  
Propositional Logic ◽  
Membrane Computing ◽  
Modus Ponens ◽  
The Other ◽  
P Systems ◽  
Backward Chaining ◽  
Other Hand

One of the concepts that lie at the basis of membrane computing is the multiset rewriting rule. On the other hand, the paradigm of rules is profusely used in computer science for representing and dealing with knowledge. Therefore, establishing a “bridge” between these domains is important, for instance, by designing P systems reproducing the modus ponens-based forward and backward chaining that can be used as tools for reasoning in propositional logic. In this paper, the authors show how powerful and intuitive the formalism of membrane computing is and how it can be used to represent concepts and notions from unrelated areas.

Analytic cut

1969 ◽  
Vol 33 (4) ◽  
pp. 560-564 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Propositional Logic ◽  
Inference Rule ◽  
The Other ◽  
Inference Rules ◽  
Quantification Theory ◽  
The Real ◽  
Gentzen Systems ◽  
Real Importance ◽  
Per Se ◽  
Axiom Systems

The real importance of cut-free proofs is not the elimination of cuts per se, but rather that such proofs obey the subformula principle. In this paper we accomplish this latter objective in a different manner.In the usual formulations of Gentzen systems, there is only one axiom scheme; all the other postulates are inference rules. By contrast, we consider here some Gentzen type axiom systems for propositional logic and Quantification Theory in which there is only one inference rule; all the other postulates are axiom schemes. This admits of an unusually elegant axiomatization of logic.

What is an inference rule?

1992 ◽  
Vol 57 (3) ◽  
pp. 1018-1045 ◽  
Author(s):  
Ronald Fagin ◽  
Joseph Y. Halpern ◽  
Moshe Y. Vardi
Keyword(s):  
Propositional Logic ◽  
General Framework ◽  
Inference Rule ◽  
Order Logic ◽  
Classical Propositional Logic ◽  
Semantic Framework ◽  
First Order ◽  
Extremal Points ◽  
The Relationship ◽  
General Semantic

AbstractWhat is an inference rule? This question does not have a unique answer. One usually finds two distinct standard answers in the literature; validity inference (σ ⊦vφ for every substitution τ, the validity of τ[σ] entails the validity of τ[φ]), and truth inference (σ⊦l φ if for every substitution τ, the truth of τ[σ] entails the truth of τ[φ]). In this paper we introduce a general semantic framework that allows us to investigate the notion of inference more carefully. Validity inference and truth inference are in some sense the extremal points in our framework. We investigate the relationship between various types of inference in our general framework, and consider the complexity of deciding if an inference rule is sound, in the context of a number of logics of interest: classical propositional logic, a nonstandard propositional logic, various propositional modal logics, and first-order logic.

Completeness of an ancient logic

1972 ◽  
Vol 37 (4) ◽  
pp. 696-702 ◽  
Author(s):  
John Corcoran
Keyword(s):  
Present Article ◽  
Propositional Logic ◽  
Natural Deduction ◽  
Deductive System ◽  
Axiomatic System ◽  
Valid Argument ◽  
Deduction System ◽  
Strong Completeness ◽  
Natural Deduction System ◽  
System Review

In previous articles ([4], [5]) it has been shown that the deductive system developed by Aristotle in his “second logic” (cf. Bochenski [2, p. 43]) is a natural deduction system and not an axiomatic system as previously had been thought [6]. It was also pointed out that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument formable in the language of the system is demonstrable by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotélian system.

Implication with possible exceptions

2001 ◽  
Vol 66 (2) ◽  
pp. 517-535
Author(s):  
Herman Jurjus ◽  
Harrie de Swart
Keyword(s):  
Propositional Logic ◽  
Modus Ponens ◽  
The Other ◽  
Classical Propositional Logic ◽  
Consequence Relation ◽  
Other Hand

AbstractWe introduce an implication-with-possible-exceptions and define validity of rules-with-possible-exceptions by means of the topological notion of a full subset. Our implication-with-possible-exceptions characterises the preferential consequence relation as axiomatized by Kraus, Lehmann and Magidor [Kraus, Lehmann, and Magidor, 1990]. The resulting inference relation is non-monotonic. On the other hand, modus ponens and the rule of monotony, as well as all other laws of classical propositional logic, are valid-up-to-possible exceptions. As a consequence, the rules of classical propositional logic do not determine the meaning of deducibility and inference as implication-without-exceptions.

The Development of Modus Ponens in Antiquity : From Aristotle to the 2nd Century AD

Phronesis ◽  
2002 ◽  
Vol 47 (4) ◽  
pp. 359-394 ◽  
Author(s):  
Susanne Bobzien
Keyword(s):  
20Th Century ◽  
Propositional Logic ◽  
Late Antiquity ◽  
Modus Ponens ◽  
Classical Propositional Logic ◽  
Aristotelian Logic ◽  
Gradual Development ◽  
Stoic Logic ◽  
Short Presentation

Abstract'Aristotelian logic', as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classified as 'hypothetical syllogisms'. However, Aristotle did not discuss such arguments, nor did he call any arguments 'hypothetical syllogisms'. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them 'hypothetical syllogisms'; nor did they describe them as ponendo ponens, etc. The tradition of the four argument forms and the classification of the arguments as hypothetical syllogisms hence need some explaining. In this paper, I offer some explanations by tracing the development of certain elements of Aristotle's logic via the early Peripatetics to the logic of later antiquity. I consider the questions: How did the four argument forms arise? Why were there four of them? Why were arguments of these forms called 'hypothetical syllogisms'? On what grounds were they considered valid? I argue that such arguments were neither part of Aristotle's dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle's logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some 20th century theories.

scholarly journals Forward and Backward Chaining with P Systems

2011 ◽  
Vol 2 (2) ◽  
pp. 56-66 ◽  
Author(s):  
Sergiu Ivanov ◽  
Artiom Alhazov ◽  
Vladimir Rogojin ◽  
Miguel A. Gutiérrez-Naranjo
Keyword(s):  
Computer Science ◽  
Propositional Logic ◽  
Membrane Computing ◽  
Modus Ponens ◽  
The Other ◽  
P Systems ◽  
Backward Chaining ◽  
Other Hand

One of the concepts that lie at the basis of membrane computing is the multiset rewriting rule. On the other hand, the paradigm of rules is profusely used in computer science for representing and dealing with knowledge. Therefore, establishing a “bridge” between these domains is important, for instance, by designing P systems reproducing the modus ponens-based forward and backward chaining that can be used as tools for reasoning in propositional logic. In this paper, the authors show how powerful and intuitive the formalism of membrane computing is and how it can be used to represent concepts and notions from unrelated areas.

scholarly journals Forward and Backward Chaining with P Systems

2012 ◽  
pp. 1522-1531
Author(s):  
Sergiu Ivanov ◽  
Artiom Alhazov ◽  
Vladimir Rogojin ◽  
Miguel A. Gutiérrez-Naranjo
Keyword(s):  
Computer Science ◽  
Propositional Logic ◽  
Membrane Computing ◽  
Modus Ponens ◽  
The Other ◽  
P Systems ◽  
Backward Chaining ◽  
Other Hand

One of the concepts that lie at the basis of membrane computing is the multiset rewriting rule. On the other hand, the paradigm of rules is profusely used in computer science for representing and dealing with knowledge. Therefore, establishing a “bridge” between these domains is important, for instance, by designing P systems reproducing the modus ponens-based forward and backward chaining that can be used as tools for reasoning in propositional logic. In this paper, the authors show how powerful and intuitive the formalism of membrane computing is and how it can be used to represent concepts and notions from unrelated areas.

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