Theism and Dialetheism

ABSTRACT

The divine attributes of omniscience and omnipotence have faced objections to their very consistency. Such objections rely on reasoning parallel to semantic paradoxes such as the Liar or to set-theoretic paradoxes like Russell's paradox. With the advent of paraconsistent logics, dialetheism—the view that some contradictions are true—became a major player in the search for a solution to such paradoxes. This paper explores whether dialetheism, armed with the tools of paraconsistent logics, has the resources to respond to the objections levelled against the divine attributes.

KEYWORDS:

• theism
• dialetheism
• omnipotence
• omniscience
• paradox
• paraconsistency

Notes

¹ See Rudavsky [1984] for mediaeval Islamic, Jewish, and Christian perspectives.

² That is not to say that there isn't interesting work to be done in this area, but I will not attempt to do it here. I hope to address some of these questions in future work.

³ Standard frames for normal modal logics have an accessibility relation on worlds . For modal operators, I am presupposing ‘universal access’, and so I drop for simplicity.

⁴ The approach here is a combination of the many-valued first-order logics given by Priest [2008a: ch. 21] and the modal semantics given by Priest [2008b: 193].

⁵ Let us stipulate, for simplicity, that every object in the domain has a name. We have taken a fixed-domain approach to the semantics, so that all of our names are rigid: for any and in , .

⁶ is the th Cartesian product of the domain, so -many times. Thus, gives the extensions and anti-extensions of our predicates. The extension, , is the set of objects in the domain of which the predicate is true. The anti-extension, , is the set of objects in the domain of which the predicate is false. The extensions and anti-extensions are relative to a world: that is, need not be the same as . Like classical logic, the union of with is the entirety of the domain, thus we have no truth-value ‘gaps’. This latter assumption is not required, and may even be disadvantageous. See Beall and Cotnoir [forthcoming].

⁷ Here we allow for the possibility that and have a non-empty intersection, resulting in truth-value ‘gluts’.

⁸ abbreviates the result of substituting some name for all free occurrences of in . This takes the place of variable assignments, since we have stipulated that all objects in the domain have a name. These semantics for quantifiers in many-valued logic are fairly typical (see Priest [2008a]).

⁹ The conditional does not have a truth table, but is sound with respect to the truth table of the conditional in , given below.

Table

Since anything invalid by the truth tables is also -invalid, this can be a useful way of checking for inferences that break down.

¹⁰ There are many different versions of the paradox, but perhaps the best version is due to Homer Simpson, who asked, ‘Can Jesus microwave a burrito so hot that he himself could not eat it?’ [Vitti and Marcatel 2002].

¹¹ See LaCroix [1977: 187] and Plantinga [1990: 70]. The main point behind these examples dates back to Ockham.

¹² Notably, Swinburne [1994] has argued for this view, while rejecting that God is essentially or necessarily omnipotent.

¹³ This point is due to Restall [1997].

¹⁴ Assume that is designated; so, , which can only happen if . So, . But then . Hence, , which is designated.

¹⁵ In fact, Milne's stated conclusion is not merely that any theist must be a dialetheist, but also that God himself must be a dialetheist. After all, any omniscient being must know that is contradictory, and know that he knows it.

¹⁶ This holds in , by axioms I and VI (see Appendix).

¹⁷ More precisely, it is directly analogous to Russell's paradox of propositions.

¹⁸ See Priest [2006: chs 2, 10].

¹⁹ Others, such as Beall [2000], reject premise (ii).

²⁰ Bringsjord's actual definition is this: But this clearly cannot be right, since it would bring about failures of the factivity of knowledge: let be some contingent but actually false proposition. It's possible that God knows : in fact, at any world in which ‘’ is true, God knows it. But that cannot entail that God actually knows , since is actually false by supposition. The definition below solves this problem.

²¹ See, e.g., Rayo and Uzquiano [2006]. Simmons [1993] argues that if we accept that the concepts of ‘set’ or ‘truth’ are indefinitely extensible, then we can save a version of omniscience.

²² We have not given semantic clauses for ‘=’. So, ‘’ is treated as an abbreviation for .

²³ See also Weber [2010, 2012] for developments.

²⁴ For a proof, see Restall [1992: 426].

²⁵ Again see Restall [1992: 426].

²⁶ This follows in from contraposition.

²⁷ Beall [2004], Kroon [2004], Mares [2004], and Bobenreith [2007] are self-avowed semantic dialetheists. See Priest's [2006: 299–302] discussion on whether he is a metaphysical dialetheist.

²⁸ For another motivation stemming from Christian theology, see Beall [forthcoming].

²⁹ For a good selection of objections to a related view called universal possibilism, see Plantinga [1980]; see McCann [2012] for discussion. Some have suggested that dialetheism will make the Problem of Evil more difficult for the theist: see Conee [1991: sec. VI].

³⁰ Thanks to David Aiken, Don Baxter, Tim Baylor, Colin Caret, Filippo Casati, Mark Gedney, Alexus McLeod, Hitoshi Omori, Andrew Parisi, Greg Restall, Dave Ripley, Kevin Scharp, Patrick Todd, Alan Torrance, and Zach Weber (among others whom I am surely forgetting) for discussion of these and related ideas. Special thanks go to Jc Beall for discussions on these topics spanning a decade. Thanks to three anonymous referees for this journal for detailed comments on earlier versions of this paper.

³¹ This presentation follows Weber and Cotnoir [2015].