Alternative Axiomatizations of the Conditional System VC
The central result of the paper is an alternative axiomatization of the conditional system VC which does not make use of Conditional Modus Ponens: (A > B) ⊃ (A ⊃ B) and of the axiom-schema CS: (A ∧ B) ⊃ (A > B). Essential use is made of two schemata, i.e. X1: (A ∧ ♢A) ⊃ (♢A >< A) and T:...
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| Format: | Article |
| Language: | ces |
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Institute of Philosophy of the Slovak Academy of Sciences
2019-08-01
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| Series: | Organon F |
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| Online Access: | https://doi.org/10.31577/orgf.2019.26305 |
| _version_ | 1818910301367566336 |
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| author | Claudio E. A. Pizzi |
| author_facet | Claudio E. A. Pizzi |
| author_sort | Claudio E. A. Pizzi |
| collection | DOAJ |
| description | The central result of the paper is an alternative axiomatization of the conditional system VC which does not make use of Conditional Modus Ponens: (A > B) ⊃ (A ⊃ B) and of the axiom-schema CS: (A ∧ B) ⊃ (A > B). Essential use is made of two schemata, i.e. X1: (A ∧ ♢A) ⊃ (♢A >< A) and T: □A ⊃ A, which are subjoined to a basic principle named Int: (A ∧ B) ⊃ (♢A > ♢B). A hierarchy of extensions of the basic system V called VInt, VInt1, VInt1T is then construed and submitted to a semantic analysis. In Section 3 VInt1T is shown to be deductively equivalent to VC. Section 4 shows that in VC the thesis X1 is equivalent to X1∨: (♢A >< A) ∨ (♢¬A >< ¬A), so that VC is also equivalent to a variant of VInt1T here called VInt1To. In Section 6 both X1 and X1∨ offer the basis for a discussion on systems containing CS, in which it is argued that they cannot avoid various kinds of partial or full trivialization of some non truth-functional operators. |
| first_indexed | 2024-12-19T22:40:38Z |
| format | Article |
| id | doaj.art-23ab3d3f4fac4f039aa0e4a18f6b65d5 |
| institution | Directory Open Access Journal |
| issn | 1335-0668 2585-7150 |
| language | ces |
| last_indexed | 2024-12-19T22:40:38Z |
| publishDate | 2019-08-01 |
| publisher | Institute of Philosophy of the Slovak Academy of Sciences |
| record_format | Article |
| series | Organon F |
| spelling | doaj.art-23ab3d3f4fac4f039aa0e4a18f6b65d52022-12-21T20:03:04ZcesInstitute of Philosophy of the Slovak Academy of SciencesOrganon F1335-06682585-71502019-08-0126342744510.31577/orgf.2019.26305Alternative Axiomatizations of the Conditional System VCClaudio E. A. Pizzi0University of SienaThe central result of the paper is an alternative axiomatization of the conditional system VC which does not make use of Conditional Modus Ponens: (A > B) ⊃ (A ⊃ B) and of the axiom-schema CS: (A ∧ B) ⊃ (A > B). Essential use is made of two schemata, i.e. X1: (A ∧ ♢A) ⊃ (♢A >< A) and T: □A ⊃ A, which are subjoined to a basic principle named Int: (A ∧ B) ⊃ (♢A > ♢B). A hierarchy of extensions of the basic system V called VInt, VInt1, VInt1T is then construed and submitted to a semantic analysis. In Section 3 VInt1T is shown to be deductively equivalent to VC. Section 4 shows that in VC the thesis X1 is equivalent to X1∨: (♢A >< A) ∨ (♢¬A >< ¬A), so that VC is also equivalent to a variant of VInt1T here called VInt1To. In Section 6 both X1 and X1∨ offer the basis for a discussion on systems containing CS, in which it is argued that they cannot avoid various kinds of partial or full trivialization of some non truth-functional operators.https://doi.org/10.31577/orgf.2019.26305Conditional logiccentering conditiontrivializationmodal collapse |
| spellingShingle | Claudio E. A. Pizzi Alternative Axiomatizations of the Conditional System VC Organon F Conditional logic centering condition trivialization modal collapse |
| title | Alternative Axiomatizations of the Conditional System VC |
| title_full | Alternative Axiomatizations of the Conditional System VC |
| title_fullStr | Alternative Axiomatizations of the Conditional System VC |
| title_full_unstemmed | Alternative Axiomatizations of the Conditional System VC |
| title_short | Alternative Axiomatizations of the Conditional System VC |
| title_sort | alternative axiomatizations of the conditional system vc |
| topic | Conditional logic centering condition trivialization modal collapse |
| url | https://doi.org/10.31577/orgf.2019.26305 |
| work_keys_str_mv | AT claudioeapizzi alternativeaxiomatizationsoftheconditionalsystemvc |