To: |
<cmenzel@xxxxxxxx>, "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx> |
---|---|

From: |
"Patrick Cassidy" <pat@xxxxxxxxx> |

Date: |
Wed, 3 Feb 2010 19:59:08 -0500 |

Message-id: |
<007a01caa535$417f9350$c47eb9f0$@com> |

Chris, Thanks, that is getting closer to specifics, but I am still unclear exactly where the logical inconsistencies lie. The issue that I would consider a problem is not just that people can create logically incompatible theories - that is absolutely certain and trivial to demonstrate. The question is whether, if those theories are actually logically incompatible, they can or cannot always be expressed with some common set of primitives. I will illustrate the trivial case. (01) Theorem 1: P(x) Theorem 2: not P(x) (02) Both of these inconsistent theories use the same primitives (P, x, not, and the logical meaning of the predicate relation), and therefore we can describe the relation between them (and prove the inconsistency). Neither theorem would be part of the ontological commitment of the FO, but a description of the predicate P and a definition of x, along with the logical primitives could be - and the theorems could be *described* by that FO, as theorems whose truth value is only asserted in some context that is not the FO's ontological commitment. Those primitives (P, x, not) would not be inconsistent with each other. The basic principle is that we should be able, in an ontology, to describe a theory without asserting that it is necessarily true. The things that are necessarily true would be the ontological commitment of the set of primitives chosen. Thus, we can have primitives in the FO such as object, time interval, spatial interval, mass, energy, real numbers, and force, and generate from these (and probably some others as well) the Newtonian mechanics and Einsteinian space-time theory. Neither of those theories would be part of the ontological commitment (not asserted to be necessarily true or false), but they can be described by the FO and one can make assertions regarding the circumstances under which the predictions of those theories conform (within experimental error) to the results of measurements. People (some at least) can grasp the difference between describing a model and asserting that it is useful versus asserting that a particular model is necessarily true in all possible worlds, and I feel confident that we can structure our ontologies so that they can represent that difference too. I also have no doubt that mathematicians can describe fascinating theories that axiomatize inconsistent *models* of something or other. The question I do not have the background to answer is whether, given any two theories that are demonstrably logically inconsistent, can both always be described using the same set of elements, including the logical primitives? In the case of the example you mentioned, where the axiom of choice is replaced by a contradictory axiom, would it not be possible to *describe* both the axiom of choice and its opposite with a common set of elements, without asserting that either axiom is true? The axioms themselves need not be in the FO, just those elements required to state the axioms. Then the contradictory axioms can be asserted, one in each of the incompatible theories, using those primitives. Then the different set theories, having the contradictory axioms which are not themselves in the FO, would (if I understand correctly) be logically inconsistent, but still *describable* with the same set of primitives (including the descriptions of the axiom of choice and its contradictory axiom). (03) The reason I suspect that the composability holds generally, aside from not having seen examples showing otherwise, is that it seems to me, if one can prove a logical inconsistency of two theories, they must be described by the same set of elements (plus the logical elements), or else the logical contradiction could not be proved. The set of elements that can describe those models would be the set that *could* be in the FO (unless they prove inconsistent with something else in the FO) Now I freely admit that with my impoverished mathematical background, I may well have missed some very important point(s), obvious to mathematicians. I would sincerely be happy to be disabused of this error, since I would thereby learn something that I think is important for an understanding of the question of conceptual primitives - to learn the limitations of that tactic. If there are limitations, this will indicate the choices that must be made to in order to apply the tactic of conceptual primitives to the problem of semantic interoperability. (04) Could you show me provably logically inconsistent theories whose component elements must themselves be necessarily inconsistent? If inconsistent theories have contradictory axioms, the question is whether those contradictory axioms can or cannot be composed of elements with a consistent meaning. (05) Pat (06) Patrick Cassidy MICRA, Inc. 908-561-3416 cell: 908-565-4053 cassidy@xxxxxxxxx (07) > -----Original Message----- > |

Previous by Date: | Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Christopher Menzel |
---|---|

Next by Date: | Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, doug foxvog |

Previous by Thread: | Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Christopher Menzel |

Next by Thread: | Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, John F. Sowa |

Indexes: | [Date]
[Thread]
[Top]
[All Lists] |