Preservation, Commutativity and Modus Ponens: Two Recent Triviality Results

Jake Chandler

doi:10.1093/mind/fzv154

PLACING PROBABILITIES OF CONDITIONALS IN CONTEXT

The Review of Symbolic Logic ◽

10.1017/s1755020314000173 ◽

2014 ◽

Vol 7 (3) ◽

pp. 415-438

Author(s):

RONNIE HERMENS

Keyword(s):

Conditional Probability ◽

Modus Ponens ◽

Bayes Rule ◽

Context Sensitive ◽

Intuitive Rules ◽

Triviality Results

AbstractIn this paper I defend the tenability of the Thesis that the probability of a conditional equals the conditional probability of the consequent given the antecedent. This is done by adopting the view that the interpretation of a conditional may differ from context to context. Several triviality results are (re-)evaluated in this view as providing natural constraints on probabilities for conditionals and admissible changes in the interpretation. The context-sensitive approach is also used to re-interpret some of the intuitive rules for conditionals and probabilities such as Bayes’ rule,Import-Export, and Modus Ponens. I will show that, contrary to consensus, the Thesis is in fact compatible with these re-interpreted rules.

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Hilbert-style proof Calculus for Propositional Logic in ABC notation

10.31219/osf.io/jd3gp ◽

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.

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Pythagorean fuzzy full implication multiple I method and corresponding applications

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-210527 ◽

2021 ◽

pp. 1-15

Author(s):

TaiBen Nan ◽

Haidong Zhang ◽

Yanping He

Keyword(s):

Decision Making ◽

Modus Ponens ◽

Pythagorean Fuzzy Set ◽

Implication Operator ◽

Continuity Properties ◽

Fuzzy Implication ◽

Decision Making Problem ◽

Fuzzy Implication Operator ◽

Degree Of Similarity ◽

Fuzzy Modus Ponens

The overwhelming majority of existing decision-making methods combined with the Pythagorean fuzzy set (PFS) are based on aggregation operators, and their logical foundation is imperfect. Therefore, we attempt to establish two decision-making methods based on the Pythagorean fuzzy multiple I method. This paper is devoted to the discussion of the full implication multiple I method based on the PFS. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO), Pythagorean fuzzy biresiduum, and the degree of similarity between PFSs based on the Pythagorean fuzzy biresiduum. In addition, the full implication multiple I method for Pythagorean fuzzy modus ponens (PFMP) is established, and the reversibility and continuity properties of the full implication multiple I method of PFMP are analyzed. Finally, a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication multiple I method in a decision-making problem. The advantages of the new method over existing methods are also explained. Overall, the proposed methods are based on logical reasoning, so they can more accurately and completely express decision information.

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On Basic Probability Logic Inequalities

Mathematics ◽

10.3390/math9121409 ◽

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.

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The sentential calculus using rule of inference Re

Journal of Symbolic Logic ◽

10.2307/2964209 ◽

1960 ◽

Vol 25 (2) ◽

pp. 143-143 ◽

Cited By ~ 1

Author(s):

R. B. Angell

Keyword(s):

Modus Ponens ◽

Sentential Calculus

Axiomatizations of the sentential calculus which use Rmp (modus ponens), have been shown equivalent to axiomatizations similar in all respects except that Rmp is replaced by the less restricted rule Re (rule of excision)1:Re. If S and (…(S⊃S') …), then (… S' …).

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Post completeness in modal logic

Journal of Symbolic Logic ◽

10.2307/2272418 ◽

1972 ◽

Vol 37 (4) ◽

pp. 711-715 ◽

Cited By ~ 8

Author(s):

Krister Segerberg

Keyword(s):

Modal Logic ◽

Upper Bound ◽

Modus Ponens ◽

Proper Subset ◽

Proper Extension ◽

Image Position

Let ⊥, →, and □ be primitive, and let us have a countable supply of propositional letters. By a (modal) logic we understand a proper subset of the set of all formulas containing every tautology and being closed under modus ponens and substitution. A logic is regular if it contains every instance of □A ∧ □B ↔ □(A ∧ B) and is closed under the ruleA regular logic is normal if it contains □⊤. The smallest regular logic we denote by C (the same as Lemmon's C2), the smallest normal one by K. If L and L' are logics and L ⊆ L′, then L is a sublogic of L', and L' is an extension of L; properly so if L ≠ L'. A logic is quasi-regular (respectively, quasi-normal) if it is an extension of C (respectively, K).A logic is Post complete if it has no proper extension. The Post number, denoted by p(L), is the number of Post complete extensions of L. Thanks to Lindenbaum, we know thatThere is an obvious upper bound, too:Furthermore,.

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