Defining implication relation for classical logic
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arXiv
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| Format: | Preprint |
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2013
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| _version_ | 1866912357153767424 |
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| author | Fu, Li |
| author_facet | Fu, Li |
| contents | In classical logic, "P implies Q" is equivalent to "not-P or Q". It is well known that the equivalence is problematic. Actually, from "P implies Q", "not-P or Q" can be inferred ("Implication-to-Disjunction" is valid), whereas from "not-P or Q", "P implies Q" cannot be inferred in general ("Disjunction-to-Implication" is not generally valid), so the equivalence between them is invalid in general. This work aims to remove the incorrect Disjunction-to-Implication from classical logic (CL). The logical system (the logic IRL) this paper proposes has the expected properties: (a) CL is obtained by adding Disjunction-to-Implication to IRL, and (b) Disjunction-to-Implication is not derivable in IRL; while (c) fundamental laws in classical logic, including law of excluded middle (LEM) and principle of double negation, law of non-contradiction (LNC) and ex contradictione quodlibet (ECQ), conjunction elimination and disjunction introduction, and hypothetical syllogism and disjunctive syllogism, are all retained in IRL. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1312_7832 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Defining implication relation for classical logic Fu, Li Logic Artificial Intelligence Computation and Language Logic in Computer Science In classical logic, "P implies Q" is equivalent to "not-P or Q". It is well known that the equivalence is problematic. Actually, from "P implies Q", "not-P or Q" can be inferred ("Implication-to-Disjunction" is valid), whereas from "not-P or Q", "P implies Q" cannot be inferred in general ("Disjunction-to-Implication" is not generally valid), so the equivalence between them is invalid in general. This work aims to remove the incorrect Disjunction-to-Implication from classical logic (CL). The logical system (the logic IRL) this paper proposes has the expected properties: (a) CL is obtained by adding Disjunction-to-Implication to IRL, and (b) Disjunction-to-Implication is not derivable in IRL; while (c) fundamental laws in classical logic, including law of excluded middle (LEM) and principle of double negation, law of non-contradiction (LNC) and ex contradictione quodlibet (ECQ), conjunction elimination and disjunction introduction, and hypothetical syllogism and disjunctive syllogism, are all retained in IRL. |
| title | Defining implication relation for classical logic |
| topic | Logic Artificial Intelligence Computation and Language Logic in Computer Science |
| url | https://arxiv.org/abs/1312.7832 |