Logic and aggregation

Journal of Philosophical Logic

Volumn 28, Issue 3, 1999, Pages 265-288

  Brown, Bryson ^a   Schotch, Peter ^a  

^a UNIVERSITY OF LETHBRIDGE   (Canada)

Author keywords

Adjunction; Aggregation; Colouring; Graph; Hypergraph; Implication; Inconsistency; Paraconsistent logic

Indexed keywords

EID: 43149090124     PISSN: 00223611     EISSN: 15730433     Source Type: Journal    
DOI: 10.1023/a:1004365302825     Document Type: Article

References (30)

1. The term is due to F. Miro Quesada, who used it in 1976 to refer to an inconsistency tolerating group of logics including those of Newton da Costa's (one of the pioneer of the subject). One of us (Schotch) first heard it in the presentation of a paper by Wolf and da Costa at the Pittsburgh meeting of the Society for Exact Philosophy in 1978.
2. See P. K. Schotch and R. E. Jennings, "On detonating," in Priest, Routley and Norman, eds., Paraconsistent Logic: Essays on the Inconsistent (München: Philosophia Verlag, 1989),
3. "Inference and necessity," J. Philos. Logic,
4. P. Apostoli and B. Brown, ("A solution to the completeness problem for non-normal modal logics", (forthcoming in J. Symbolic Logic)),
5. "A solution to the completeness problem for weakly aggregature modal logic", J. Symbolic Logic 60 (3) (September 1995) 832-842.
6. A personal favorite is: (A ⊃ B) ∨ (B ⊃ C). It is mildly ironic that this is required to be a tautology in order for the Lewis modal logic S1 to have a semantics (or at least the semantics that it in fact has).
7. The logic S2. The logic S1 has only that version of this 'paradox' in which the main connective is material rather than strict implication.
8. The drawing board in question was getting rather worn out by this time anyway. Lewis' first attempts to formalize strict implication had collapsed into classical logic and once that problem had been solved he discovered that his logic had the 'wrong' form of transitivity principle.
9. Both W. T. Parry and R. Barcan-Marcus.
10. In this group belong not only Russell but also Quine. Strange bedfellows indeed.
11. In this connection, Judith Pelham's recent paper "Russell's early philosophy of logic" in Russell and Analytic Philosophy, forthcoming University of Toronto Press, is most illuminating.
12. The phrase is Wittgenstein's. He gives an account of going to visit Russell over a period of several days before the first war, with the express purpose of puzzling out what this "sign" meant. The two of them finally decided that it was meaningless. This may be shocking but it isn't surprising given what Russell has to say in the Principles concerning assertion.
13. Quine uses this mode of attack (among many others of course) but he does not follow through as we do in the next sentence.
14. In the usual sense that a set of sentences is true provided all of its member sentences are true. One might take the position here (starting a competing radical school) that this betrays deeply embedded classical thinking. We really ought to construct an alternative to the classical theory of true sets, such that individual sentences inherit their truth from the fact that they are members of true sets rather than the contrary way of doing things. It might even be the case, on this alternative radical proposal, that one need not embrace true contradictions. To the best of my knowledge, this new school is without members.
15. In other places this sort of predicate is defined in terms of partitions or covering families. This works, and is to some extent more intuitive, but it fails to generalize to the entire class of level functions. In addition, it fails to account for those Γ such that CON(Γ, 0) which must be dealt with by a 'convention'.
16. Such, for example, is Henry Kyburg's diagnosis of what goes wrong in the lottery paradox. See "Conjunctivitis," in Marshall Swain, ed., Induction, Acceptance and Rational Belief, 55-82 (D. Reidel, Dordrecht, 1970).
17. It is, to my mind, rather unfortunate that paraconsistent logic in our style has come to be called "the non-adjunctive approach".
18. See Schotch and Jennings, "On detonating," 1989.
19. In most introductory logic books what is called the rule of reiteration, is an amalgam of this rule and the rule of monotonicity or dilution.
20. R. Sylvan has remarked in a survey of non-monotonic logics that we can comfortably take "non-monotonic" as a synonym for "non- deductive". If forcing really is to be regarded as non-monotonic then, we trust we have made this view rather more uncomfortable.
21. An n-colouring maybe defined as a function C ∈∪H→{1, . . . , n}.
22. This sense of containment is straightforward: T contains T′ iff every member of T′ is a subset of some member of T.
23. See Brown, and Apostoli, "A solution to the completeness problem for weakly aggregative modal logic," J. Symbolic Logic 60(3) (September 1995), 832-842.
24. There is a bit more at stake here than just Hippocraticality; we might wish to concentrate on properties Φ which satisfy certain 'nice' conditions e.g. compactness. It can also be shown that this is 'transmitted' by the construction in the sense that if Φ is compact, then its corresponding level function is level-compact. By this is meant that if ℓ(Γ) = n then ℓ(Δ) = n for Δ some finite subset of Γ (where ℓ is a level function). This result is due to Blaine d'Entremont, Inference and Level (M. A. Thesis, Dalhousie University, 1982).
25. This is quite important for the adjudication of a suggestion of D. Lewis in "Logic for equivocators", Nous 16, 431-441. He suggests in that essay that his approach is akin to ours, but it is easy to see that the logic he proposes has the classical dilution rule, meets the conditions referred to, and hence fails to be Hippocratic.
26. In "On detonating" this species of forcing is called A-forcing, and its presentation there is rather different. This is largely because the reformulation of level functions in terms of the predicate ψ had not then appeared.
27. See B. Brown, "A paraconsistent approach to old quantum theory", PSA 1992, Vol. 2 (Philosophy of Science Association, 1993),
28. D. Braybrooke, B. Brown, and P.K. Schotch, Logic on the Track of Social Change (Oxford University Press, 1995).
29. k, simply by assigning those two vertices the same colour and every other vertex a unique colour.
30. The present form of this proof derives from a long correspondence between one of the authors (Brown), who first discovered it, and P. Apostoli, whose extensive contributions are here gratefully acknowledged.