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.

Inference by complementary elimination

1957 ◽  
Vol 22 (3) ◽  
pp. 233-236
Author(s):  
Bernard K. Symonds ◽  
Roderick M. Chisholm
Keyword(s):  
Modus Ponens ◽  
The Other ◽  
Second Step ◽  
Modus Tollens ◽  
Disjunctive Syllogism ◽  
Hypothetical Syllogism

The inferences countenanced by the traditional rules of modus ponens, modus tollens, disjunctive syllogism, hypothetical syllogism, and the complex types of dilemma may be regarded as single applications of one rule of inference, “the rule of complementary elimination”. In the present paper, we shall discuss this rule informally and illustrate it in application to expressions written in the language of Principia Mathematica. Our illustrations will contain no connectives except for those for conjunction, disjunction, and negation; we use parentheses in place of dots; and we allow disjunction and conjunction to have any number of operands more than two.In applying complementary elimination to a set of premises, we take the following three steps, (i) We form, merely by disjoining the premises, an expression which we shall call a premise disjunction, (ii) If we have n premises, we eliminate n minus one (or fewer) pairs of the following sort from our premise disjunction: each pair is such that one of its members is the negation of the other and both members are specific occurrences of disjuncts of our premise disjunction. We shall call such pairs complementary pairs, (iii) The formula obtained by means of our second step is one that may be made well-formed merely by eliminating parentheses or connectives other than negation; we make such elimination, and any formula we thus obtain is a consequence of our premises.

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 Probability logic: A model-theoretic perspective

2020 ◽  
Author(s):  
M Pourmahdian ◽  
R Zoghifard
Keyword(s):  
Characterization Theorem ◽  
Expressive Power ◽  
Probability Logic ◽  
Theoretic Analysis ◽  
Compactness Property ◽  
Probability Models ◽  
Finitely Additive Probability ◽  
Additive Probability ◽  
Basic Probability ◽  
Abstract Logics

Abstract This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.

Basic probability theory

1992 ◽  
pp. 21-53
Author(s):  
Roy Billinton ◽  
Ronald N. Allan
Keyword(s):  
Probability Theory ◽  
Basic Probability

Basic probability theory

1983 ◽  
pp. 5-35
Author(s):  
Roy Billinton ◽  
Ronald N. Allan
Keyword(s):  
Probability Theory ◽  
Basic Probability
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