On Basic Probability Logic Inequalities
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 <i>A</i>, <i>B</i&...
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MDPI AG
2021-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/12/1409 |
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| author | Marija Boričić Joksimović |
| author_facet | Marija Boričić Joksimović |
| author_sort | Marija Boričić Joksimović |
| collection | DOAJ |
| description | 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 <i>A</i>, <i>B</i>, <i>C</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo>→</mo><mi>C</mi></mrow></semantics></math></inline-formula> have probabilities <i>a</i>, <i>b</i>, <i>c</i>, <i>r</i>, and <i>s</i>, respectively, then for probability <i>p</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>→</mo><mi>C</mi></mrow></semantics></math></inline-formula>, we have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>g</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula>, for some functions <i>f</i> and <i>g</i> 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. |
| first_indexed | 2024-03-10T10:19:30Z |
| format | Article |
| id | doaj.art-5e54cf461b7841449cde7134e17f2e32 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T10:19:30Z |
| publishDate | 2021-06-01 |
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| spelling | doaj.art-5e54cf461b7841449cde7134e17f2e322023-11-22T00:32:31ZengMDPI AGMathematics2227-73902021-06-01912140910.3390/math9121409On Basic Probability Logic Inequalities Marija Boričić Joksimović0Faculty of Organizational Sciences, University of Belgrade, Jove Ilića 154, 11000 Belgrade, SerbiaWe 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 <i>A</i>, <i>B</i>, <i>C</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo>→</mo><mi>C</mi></mrow></semantics></math></inline-formula> have probabilities <i>a</i>, <i>b</i>, <i>c</i>, <i>r</i>, and <i>s</i>, respectively, then for probability <i>p</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>→</mo><mi>C</mi></mrow></semantics></math></inline-formula>, we have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>g</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula>, for some functions <i>f</i> and <i>g</i> 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.https://www.mdpi.com/2227-7390/9/12/1409inequalityprobability logicinference rule |
| spellingShingle | Marija Boričić Joksimović On Basic Probability Logic Inequalities Mathematics inequality probability logic inference rule |
| title | On Basic Probability Logic Inequalities |
| title_full | On Basic Probability Logic Inequalities |
| title_fullStr | On Basic Probability Logic Inequalities |
| title_full_unstemmed | On Basic Probability Logic Inequalities |
| title_short | On Basic Probability Logic Inequalities |
| title_sort | on basic probability logic inequalities |
| topic | inequality probability logic inference rule |
| url | https://www.mdpi.com/2227-7390/9/12/1409 |
| work_keys_str_mv | AT marijaboricicjoksimovic onbasicprobabilitylogicinequalities |