David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Philosophical Logic 25 (1):101 - 108 (1996)
A four-valued matrix is presented which validates all theorems of the implicational fragment, IF, of the classical sentential calculus in which at most two distinct sentence letters occur. The Wajsberg/Diamond-McKinsley Theorem for IF follows as a corollary: every complete set of axioms (with substitution and detachment as rules) must include at least one containing occurrences of three or more distinct sentence letters. Additionally, the matrix validates all IF theses built from nine or fewer occurrences of connectives and letters. So the classic result of Jagkovski for the full sentential calculus -that every complete axiom set must contain either two axioms of length at least nine or else one of length at least eleven-can be improved in the implicational case: every complete axiom set for IF must contain at least one axiom eleven or more characters long. Both results are "best possible", and both apply as well to most subsystems of IF, e.g., the implicational fragments of the standard relevance logics, modal logics, the relatives of implicational intutionism, and logics in the Lukasiewicz family
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
Arthur N. Prior (1962). Formal Logic. Oxford, Clarendon Press.
Jan Kalicki (1950). Note on Truth-Tables. Journal of Symbolic Logic 15 (3):174-181.
M. D. Gladstone (1970). On the Number of Variables in the Axioms. Notre Dame Journal of Formal Logic 11 (1):1-15.
Citations of this work BETA
No citations found.
Similar books and articles
Richard Tursman (1968). The Shortest Axioms of the Implicational Calculus. Notre Dame Journal of Formal Logic 9 (4):351-358.
Zachary Ernst, Branden Fitelson, Kenneth Harris & Larry Wos (2002). Shortest Axiomatizations of Implicational S4 and S. Notre Dame Journal of Formal Logic 43 (3):169-179.
Branden Fitelson, Vanquishing the XCB Question: The Methodological Discovery of the Last Shortest Single Axiom for the Equivalential Calculus.
T. Thacher Robinson (1968). Independence of Two Nice Sets of Axioms for the Propositional Calculus. Journal of Symbolic Logic 33 (2):265-270.
Ivo Thomas (1970). Final Word on a Shortest Implicational Axiom. Notre Dame Journal of Formal Logic 11 (1):16-16.
Larry Wos, Dolph Ulrich & Branden Fitelson, Vanquishing the XCB Question: The Methodological Discovery of the Last Shortest Single Axiom for the Equivalential Calculus.
Branden Fitelson & Larry Wos (2001). Finding Missing Proofs with Automated Reasoning. Studia Logica 68 (3):329-356.
Diderik Batens (1987). Relevant Implication and the Weak Deduction Theorem. Studia Logica 46 (3):239 - 245.
Tadeusz Prucnal (1974). Interpretations of Classical Implicational Sentential Calculus in Nonclassical Implicational Calculi. Studia Logica 33 (1):59 - 64.
Branden Fitelson (2002). Shortest Axiomatizations of Implicational S4 and S5. Notre Dame Journal of Formal Logic 43 (3):169-179.
Added to index2009-01-28
Total downloads19 ( #189,896 of 1,792,154 )
Recent downloads (6 months)5 ( #170,928 of 1,792,154 )
How can I increase my downloads?