Yes, Virginia, there really are paraconsistent logics

Journal of Philosophical Logic

Volumn 28, Issue 5, 1999, Pages 489-500

  Brown, Bryson ^a  

^a UNIVERSITY OF LETHBRIDGE   (Canada)

Author keywords

Aggregation; Dialefheism; Paraconsistent logic; Preservationism

Indexed keywords

EID: 52549127030     PISSN: 00223611     EISSN: 15730433     Source Type: Journal    
DOI: 10.1023/A:1004390309035     Document Type: Review

References (12)

1. B. H. Slater, "Paraconsistent Logics?" Journal of Philosophical Logic 24 (1995), 451-454.
2. By 'dialethec' I mean logics which achieve paraconsistency by allowing both "A" and "-?A" to receive designated values, for at least some formulae A.
3. See P. K. Schotch and R. E. Jennings, "On Detonating," in Priest, Routley, Norman (eds.), Paraconsistent Logic: Essays on the Inconsistent (München: Philosophia Verlag, 1989).
4. As P. K. Schotch has pointed out ("Paraconsistent Logic: The view from the right," PSA, 1992, Vol. 2) it's reasonable to regard a tautologous set as requiring less dividing up than a merely consistent one, since a tautologous set can be freely combined with any consistent set salva degree of incoherence, while a consistent set cannot.
5. First, do no harm. See P. K. Schotch, "... The view from the right."
6. See "Paradox Tolerant Logic," Logique et Analyse 26 (1983), 291-308.
7. The modal logics that are so closely related to this paraconsistent logic would allow a modal account of satisfaction that divides the satisfaction of inconsistent sets amongst tuples of worlds.
8. Priest, Tanaka "Paraconsistent Logic," in Stanford's online philosophy encyclopedia, at http://plato.stanford.edu/entries/logic- paraconsistent/logic-paraconsistent.html
9. Incidentally, in my view at least, the dialethic reading of this logic is not (completely) indefensible. LP's "-?" really is a lot like classical negation. To cite just a few points, theorems of the system are the classical tautologies; whenever the variables in A are assigned classical values, A is assigned the appropriate classical value; whenever A is assigned "both" some of its variables are as well, and in particular, contradictions can only be designated by receiving the paradoxical value "both". But whether or not this sort of defense finally succeeds, this logic remains a perfectly good paraconsistent logic, as the more conservative reading to follow shows.
10. Note that this process of treating sentence letters as ambiguous can be carried out by substituting unique new letters for some but not all instances of the letters to be treated as ambiguous. Of course the resulting set of sentences, Γ, can be consistent when Γ is not.
11. Γ's members can be LP-satisfied by assigning 'r' the value T, so long as we assign 'p' and 'q' the value both. Thus '-r' is not satisfied by every valuation satisfying Γ.
12. By "minimal" here I mean only that no proper subset of any member of L is such that treating its members as ambiguous will allow Γ to be rendered consistent.