## Weaker D-Complete Logics

Venue: | University of Wollongong Department |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Megill_weakerd-complete,

author = {Norman Megill},

title = {Weaker D-Complete Logics},

booktitle = {University of Wollongong Department},

year = {}

}

### OpenURL

### Abstract

BB 0 IW logic (or T! ) is known to be D-complete. This paper shows that there are infinitely many weaker D-complete logics and it also examines how certain D-incomplete logics can be made complete by altering their axioms using simple substitutions. Keywords: condensed detachment 1 Introduction The condensed detachment rule, first proposed by C. A. Meredith in Lemmon et al [5], is a form of modus ponens preceded by `just enough' substitution to make the modus ponens possible. The substitution mechanism, for implicational formulas, was a precursor to Robinson's unification algorithm [8]. Roughly, a system of implicational logic is D-complete if the system with the same axioms, but with condensed detachment (D) instead of modus ponens and substitution, has the same theorems. To show that a logic is D-complete it is sufficient to show that all the substitution instances of its axioms are deducible in the corresponding condensed logic (i.e. the logic with rule D only). It is well know...

### Citations

945 |
1965] ‘A machine-oriented logic based on the resolution principle
- Robinson
(Show Context)
Citation Context ...m of modus ponens preceded by `just enough' substitution to make the modus ponens possible. The substitution mechanism, for implicational formulas, was a precursor to Robinson's unification algorithm =-=[8]-=-. Roughly, a system of implicational logic is D-complete if the system with the same axioms, but with condensed detachment (D) instead of modus ponens and substitution, has the same theorems. To show ... |

77 |
Basic Simple Type Theory
- Hindley
- 1995
(Show Context)
Citation Context ...nd Meredith [3] and Kalman [4].) Meyer and Bunder [6] showed that the system based on (B), (B 0 ), (I) and (W) (a ! a ! b) ! a ! b, which we will call BB 0 IW logic (or T! ), is D-complete. (See also =-=[2]-=- and [7]). Here we show that D-completeness can be shown for a weaker logic, which we will call M . This is based on (I) and the following: (A1) (a ! a) ! (c ! b ! b) ! c ! (a ! b) ! a ! b (A2) (b ! b... |

12 |
Condensed detachment as a rule of inference, Studia Logica 42
- Kalman
- 1983
(Show Context)
Citation Context ...axioms only from the list: (I) a ! a (B) (a ! b) ! (c ! a) ! c ! b (B 0 ) (a ! b) ! (b ! c) ! a ! c (C) (a ! b ! c) ! b ! a ! c (K) a ! b ! a is D-incomplete. (See Hindley and Meredith [3] and Kalman =-=[4]-=-.) Meyer and Bunder [6] showed that the system based on (B), (B 0 ), (I) and (W) (a ! a ! b) ! a ! b, which we will call BB 0 IW logic (or T! ), is D-complete. (See also [2] and [7]). Here we show tha... |

11 |
Principal type-schemes and condensed detachment
- Hindley, Meredith
- 1990
(Show Context)
Citation Context ...ery logic with axioms only from the list: (I) a ! a (B) (a ! b) ! (c ! a) ! c ! b (B 0 ) (a ! b) ! (b ! c) ! a ! c (C) (a ! b ! c) ! b ! a ! c (K) a ! b ! a is D-incomplete. (See Hindley and Meredith =-=[3]-=- and Kalman [4].) Meyer and Bunder [6] showed that the system based on (B), (B 0 ), (I) and (W) (a ! a ! b) ! a ! b, which we will call BB 0 IW logic (or T! ), is D-complete. (See also [2] and [7]). H... |

10 |
Calculi of pure strict implication
- LEMMON, MEREDrrH, et al.
- 1969
(Show Context)
Citation Context ...can be made complete by altering their axioms using simple substitutions. Keywords: condensed detachment 1 Introduction The condensed detachment rule, first proposed by C. A. Meredith in Lemmon et al =-=[5]-=-, is a form of modus ponens preceded by `just enough' substitution to make the modus ponens possible. The substitution mechanism, for implicational formulas, was a precursor to Robinson's unification ... |

6 |
Condensed detachment is complete for relevance logic: A computer-aided proof
- Mints, Tammet
- 1991
(Show Context)
Citation Context ...ith [3] and Kalman [4].) Meyer and Bunder [6] showed that the system based on (B), (B 0 ), (I) and (W) (a ! a ! b) ! a ! b, which we will call BB 0 IW logic (or T! ), is D-complete. (See also [2] and =-=[7]-=-). Here we show that D-completeness can be shown for a weaker logic, which we will call M . This is based on (I) and the following: (A1) (a ! a) ! (c ! b ! b) ! c ! (a ! b) ! a ! b (A2) (b ! b) ! (c !... |

4 |
Condensed detachment and combinators
- Meyer, Bunder
- 1988
(Show Context)
Citation Context ...st: (I) a ! a (B) (a ! b) ! (c ! a) ! c ! b (B 0 ) (a ! b) ! (b ! c) ! a ! c (C) (a ! b ! c) ! b ! a ! c (K) a ! b ! a is D-incomplete. (See Hindley and Meredith [3] and Kalman [4].) Meyer and Bunder =-=[6]-=- showed that the system based on (B), (B 0 ), (I) and (W) (a ! a ! b) ! a ! b, which we will call BB 0 IW logic (or T! ), is D-complete. (See also [2] and [7]). Here we show that D-completeness can be... |

2 |
A simplified version of condensed detachment
- Bunder
(Show Context)
Citation Context ...(I) and (W). 2 Condensed detachment The formulation we give here for condensed detachment is equivalent to the more standard one of, for example, Hindley and Meredith [3] but is simpler to state (see =-=[1]-=-). In all the work below, if P is a formula of logic, oe i (P ) will represent the result of a simultaneous substitution of formulas for propositional variables in P . (Rule D). From P ! Q and R concl... |