Abstract
This chapter continues the consideration of the potential for interpreting ‘nonsense’ values as catastrophic faults in computational processes, focusing on the particular case in which Nuel Belnap’s ‘artificial reasoner’ is unable to retrieve the semantic value assigned to a variable. This leads not only to a natural interpretation of Graham Priest’s semantics for the \(\vdash \)-Parry system \(\mathsf {S}^{\star }_{\mathtt {fde}}\) but also a novel, many-valued semantics for Angell’s \(\mathsf {AC}\), completeness of which is proven by establishing a correspondence with Correia’s semantics for \(\mathsf {AC}\). These many-valued semantics have the additional benefit of allowing the application the material in Chap. 2 to the case of \(\mathsf {AC}\), thereby defining natural intensional extensions of \(\mathsf {AC}\) in the spirit of Parry’s \(\mathsf {PAI}\).
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Notes
- 1.
That we are employing the value \(\mathfrak {u}\in \mathcal {V}_{\mathsf {\Sigma }_{0}}\), i.e., an infectious nonsense value , is not by accident.
- 2.
We will see that this inference in a sense characterizes the single address account, as the proof theory for \(\mathsf {S}^{\star }_{\mathtt {fde}}\) is equivalent to the addition of this inference to the logic determined by the two address case.
- 3.
Angell asserts the existence of a semantics for ‘analytic equivalence’ by employing ‘analytic truth tables’ in the abstract [2]. Possibly due to the severe constraints on space, however, Angell’s definition of an analytic truth table is not entirely clear.
- 4.
Note that as the conditions for \(\Vdash _{\mathsf {v}}\) provide no means of eliminating instances of formulae from a pseudosequent, whenever a pseudosequent \(\varnothing \Vdash _{\mathsf {v}}A\) is derivable, it is derivable after a finite number of manipulations of a finite initial pseudosequent \(\varGamma \Vdash _{\mathsf {v}}\varDelta \). Hence, it is always sufficient to consider finite Correia models, justifying our assumption of the finitude of Correia models \(\mathsf {v}\).
- 5.
Constancy might suggest that \(\mathsf {PS}_{\mathtt {fde}}^{\star }\) would be a more appropriate name for the Parry-like extension of \(\mathsf {S}^{\star }_{\mathtt {fde}}\). But the system has been introduced in print as \(\mathsf {PFDE}_{\varphi }\) and we will retain that nomenclature now.
References
Angell, R.B.: Three systems of first degree entailment. J. Symb. Log. 42(1), 147 (1977)
Angell, R.B.: Analytic truth-tables. J. Symb. Log. 46(3), 677 (1981)
Angell, R.B.: Deducibility, entailment and analytic containment. In: Norman, J., Sylvan, R. (eds.) Directions in Relevant Logic, Reason and Argument, pp. 119–143. Kluwer Academic Publishers, Boston, MA (1989)
Belnap Jr., N.D.: How a computer should think. In: Ryle, G. (ed.) Contemporary Aspects of Philosophy, pp. 30–56. Oriel Press, Stockfield (1977)
Belnap Jr., N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-valued Logic, pp. 8–37. Reidel, Dordrecht (1977)
Correia, F.: Semantics for analytic containment. Stud. Log. 77(1), 87–104 (2004)
Correia, F.: Grounding and truth functions. Log. et Anal. 53(211), 251–279 (2010)
Daniels, C.: A story semantics for implication. Notre Dame J. Form. Log. 27(2), 221–246 (1986)
Daniels, C.: A note on negation. Erkenntnis 32(3), 423–429 (1990)
Deutsch, H.: The completeness of S. Stud. Log. 38(2), 137–147 (1979)
Deutsch, H.: Paraconsistent analytic implication. J. Philos. Log. 13(1), 1–11 (1984)
Dunn, J.M.: A modification of Parry’s analytic implication. Notre Dame J. Form. Log. 13(2), 195–205 (1972)
Epstein, R.L.: Relatedness and dependence in propositional logics. J. Symb. Log. 46(1), 202–203 (1981)
Fine, K.: Analytic implication. Notre Dame J. Form. Log. 27(2), 169–179 (1986)
Fine, K.: Angellic content. J. Philos. Log. 45(2), 199–226 (2016)
Kapsner, A.: Logics and Falsifications. Springer, Cham (2014)
Kielkopf, C.F.: Formal Sentential Entailment. University Press of America, Washington, D.C. (1977)
Kleene, S.C.: Introduction to Metamathematics. North-Holland Publishing Company, Amsterdam (1952)
McCarthy, J.: A basis for a mathematical theory of computation. In: Braffort, P., Hirschberg, D. (eds.) Computer Programming and Formal Systems, pp. 33–70. North-Holland Publishing Company, Amsterdam (1963)
Parry, W.T.: Ein Axiomensystem für eine neue Art von Implikation (analytische Implikation). Ergeb. eines Math. Kolloqu. 4, 5–6 (1933)
Pohlers, W.: Proof Theory. Springer, Berlin (2008)
Routley, R.: Relevant Logics and their Rivals, vol. 1. Ridgeview Publishing, Atascadero, CA (1982)
Shramko, Y., Wansing, H.: Some useful 16-valued logics: How a computer network should think. J. Philos. Log. 35(2), 121–153 (2005)
Woodruff, P.: On constructive nonsense logic. In: Modality. Morality, and Other Problems of Sense and Nonsense, pp. 192–205. GWK Gleerup Bokforlag, Lund (1973)
Zinov’ev, A.A.: Foundations of the Logical Theory of Scientific Knowledge (Complex Logic). Boston Studies in the Philosophy of Science. D. Reidel Publishing Company, Dordrecht, Holland (1973)
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Ferguson, T.M. (2017). Faulty Belnap Computers and Subsystems of \(\mathsf{E}_{\texttt {fde}}\) . In: Meaning and Proscription in Formal Logic. Trends in Logic, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-319-70821-8_5
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