The shortest possible length of the longest implicational axiom

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)
DOI 10.1007/BF00357843
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
Download options
PhilPapers Archive

Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 16,707
External links
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.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

Monthly downloads

Added to index


Total downloads

14 ( #184,535 of 1,726,249 )

Recent downloads (6 months)

2 ( #289,836 of 1,726,249 )

How can I increase my downloads?

My notes
Sign in to use this feature

Start a new thread
There  are no threads in this forum
Nothing in this forum yet.