by
Robin Hirsch
,
Ian Hodkinson

Citations: | 1 - 0 self |

@MISC{Hirsch98gamesin,

author = {Robin Hirsch and Ian Hodkinson},

title = {Games in Algebraic Logic},

year = {1998}

}

11.52> A is A, to non-contradiction, Nothing both A and not-A, and to excluded middle, Everything either A or not-A. These three principles, it is affirmed, dictate all the forms of inference, and evolve all the canons of syllogism. I am not prepared to deny the truth of either of these propositions, at least when A is not self-contradictory, but I cannot see how, alone, they are competent to the functions assigned. I see that they distinguish truth from falsehood: but I do not see that they, again alone, either distinguish or evolve one truth from another. Every transgression of these laws is an invalid inference: every valid inference is not a transgression of these laws. But I cannot admit that every thing which is not a transgression of these laws is a valid inference. And I cannot make out how just the only propositions which are true

187 | Boolean algebras with operators - Jónsson, Tarski - 1951 |

71 | Cylindric Algebras (Part I - Henkin, Monk, et al. - 1971 |

65 |
The representation of relational algebras
- Lyndon
- 1950
(Show Context)
Citation Context ...ediate that any field of binary relations must obey these axioms and is therefore a relation algebra, it turned out that not all relation algebras are isomorphic to genuine fields of binary relations =-=[Lyn50]-=- (or to put it another way, not all relation algebras are representable as fields of binary relations). Worse, there could be no finite axiomatisation that exactly defined the representable relation a... |

45 |
On representable relation algebras
- Monk
- 1964
(Show Context)
Citation Context ...put it another way, not all relation algebras are representable as fields of binary relations). Worse, there could be no finite axiomatisation that exactly defined the representable relation algebras =-=[Mon64]-=-. On the other hand, the class of representable relation algebras (or cylindric algebras) was shown, by methods of universal algebra, to form a variety with a recursive (equational) axiomatisation [JT... |

37 |
The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning
- Boole
(Show Context)
Citation Context ...e that the syllogism was inadequate for dealing with relations of rank greater than one. The other decisive influence was George Boole, who had put forward a highly successful algebra of propositions =-=[Boo51]-=-. De Morgan wrote When the ideas thrown out by Mr Boole shall have borne their full fruit, algebra, though only founded on ideas of number in the first instance, will appear like a sectional model of ... |

28 | Step by step — building representations in algebraic logic - Hirsch, Hodkinson - 1997 |

24 |
Representation problems for relation algebras
- Jónsson, Tarski
- 1192
(Show Context)
Citation Context ...64]. On the other hand, the class of representable relation algebras (or cylindric algebras) was shown, by methods of universal algebra, to form a variety with a recursive (equational) axiomatisation =-=[JT48]-=-. This led to the problem of finding a nice, recursive axiomatisation of the representable algebras. Lyndon and Monk did produce recursive axiomatisations, but these were rather complex. A game-theore... |

22 | On the syllogism, no. iv, and on the logic of relations - Morgan |

8 | Axiomatising various classes of relation and cylindric algebras - Hirsch, Hodkinson - 1997 |

7 |
The Vienna Logic
- Kant
- 1992
(Show Context)
Citation Context ... discovering the principles of everyday thinking and of mathematical argument. Two authors influenced him greatly: Aristotle and Boole. Aristotle's syllogism had held sway for 2000 years. Indeed Kant =-=[Kan72]-=- had argued that Since Aristotle's time Logic has not gained much in extent, as indeed nature forbids it should. . . . Aristotle has omitted no essential point of the understanding; we have only becom... |

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