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God of the gaps: a neglected reply to God’s stone problem
Abstract
Traditional monotheism has long faced logical puzzles (omniscience, omnipotence, and more). We argue that such puzzles rest on the assumed logical truth of the Law of Excluded Middle, which we suggest there is little theological reason to accept. By way of illustration we focus on God's alleged stone problem, and present a simple but plausible ‘gappy’ framework for addressing this puzzle. We assume familiarity with the proposed (subclassical) logic but an appendix is offered as a brief review.
... However, Beall and Cotnoir (2017) argue that rather than resting on a tautological premise, the paradox rests on an untrue one. Since "a key feature of God's essential omnipotence (or omni-anything) is that in some cases, one can no more falsely attribute limits to God's omnipotence than one can truly do so," it is not true that God can or cannot create a stone He cannot lift. ...
... According to Beall and Cotnoir (2017), omni-problems may not necessarily be problems at all. Instead, they evidence the true nature of divine reality. ...
... Otherwise, L is false. (Beall & Cotnoir, 2017) To illustrate how (G) works, consider the limit sentence, "God cannot create on a Friday." This sentence is false given (G) since there is a scenario where God creates on a Friday, and there is no scenario where God's omnipotence is limited. ...
In “God of the gaps: A neglected reply to God’s stone problem,” Jc Beall and A. J. Cotnoir offer a gappy solution to the paradox of the stone – a paradox that involves God’s omnipotence. This paper shows that their solution extends to a puzzle concerning God’s impeccability or inability to sin. This latter puzzle not only involves God’s omnipotence but also His omnibenevolence.
... (2) From (1), by universal elimination (or instantiation): God can make it the case that C. (3) From (2), by (C): God can make it the case that C iff God can make it the case that C only if ⊥. (4) From (3), by biconditional elimination: If God can make it the case that C, then God can make it the case that C only if ⊥. (5) From (2) and (4), by modus ponens: God can make it the case that C only if ⊥. (6) From (2) and (5), by modus ponens: ⊥. Tedder and Badia (2018) argue that Beall and Cotnoir (2017)'s gappy solution does not have the resources to address this Curry-like paradox. Thus, it does not present a satisfactory solution to all types of paradoxes of unrestricted omnipotence. 1 ...
... Sed contra, we show that Beall and Cotnoir (2017)'s solution has the resources to address Tedder and Badia (2018)'s Curry-like paradox. In particular, given Beall and Cotnoir (2017)'s resources, the Curry-like paradox is unsound. ...
... Sed contra, we show that Beall and Cotnoir (2017)'s solution has the resources to address Tedder and Badia (2018)'s Curry-like paradox. In particular, given Beall and Cotnoir (2017)'s resources, the Curry-like paradox is unsound. We prove this in two steps. ...
In ‘Currying omnipotence: A reply to Beall and Cotnoir’, Andrew Tedder and Guillermo Badia argue that Jc Beall and A. J. Cotnoir’s gappy solution to the traditional paradox of unrestricted omnipotence does not extend to a Curry-like version of the paradox. In this paper, we show that it does extend to it.
... i. Unary predicates: 'ܲ', 'ܳ', 'ܴ' with or without numerical subscripts (unary: they take one name to make a sentence) 12 Anderson and Belnap 1975;Anderson, Belnap, and Dunn 1992;Dunn 1966;1976. 13 For arguments towards this conclusion see Beall 2017. 14 A generalization to the full stock of (standard) first-order vocabulary is not difficult but, again, is not necessary for purposes of this paper. ...
... Logic, as above, doesn't demand as much; but systematic theorizing motivates the methodological goal. And with that goal, one is quickly motivated to the view that the ticked sentence is false and also true -a contradiction (see Beall 2017). ...
... Indeed, Priest's career and large body of work, much like the late Sylvan's (née Routley) career and work, has focused largely on advocating and defending the spread of contradictory theories beyond this limited area. 25 In order for logic itself to force the issue one needs something like the law of excluded middle, which is not valid according to logic (viz., FDE) as I've advanced it here (see Beall 2017). 418 complementary natures. ...
The fundamental problem of Christology (as Richard Cross famously coined it) is the apparent contradiction of Christ as recorded at Chalcedon. Christ is human (with everything entailed thereby) and Christ is divine (with everything entailed thereby). Being divine entails (among many other of God’s properties) being immutable. Being human entails (among many other of our essential properties) being mutable. Were Christ two different persons (viz., a human person, a divine person) there’d be no apparent contradiction. But Chalcedon rules as much out. Were Christ only partly human or only partly divine there’d be no apparent contradiction. But Chalcedon rules as much out. Were the very meaning of ‘mutable’ and/or ‘immutable’ (or other such predicates) other than what they are, there’d be no apparent contradiction. But the meaning is what it is, and changing the meaning of our terms to avoid the apparent contradiction of Christ is an apparent flight from reality.What, in the end, is the explanation of the apparent contradiction of Christ? Theologians and philosophers have long advanced many consistency-seeking answers, all of which increase the metaphysical or semantical complexity of the otherwise strikingly simple but radical core of Christianity’s GodMan. In this paper, I put the simplest explanation on the theological table: namely, Christ appears to be contradictory because Christ is contradictory (i.e., some predicate is both true and false of Christ, and hence some logical contradiction is true of Christ). This explanation may sound complicated to the many who are steeped in the mainstream account of logic according to which logic precludes the possibility of true contradictions. But the mainstream account of logic can and should be rejected. Ridding theology of the dogma of mainstream logic illuminates the simple though striking explanation of the apparent contradiction of Christ — namely, that Christ is a contradictory being. Just as the simplest explanation to the apparent roundness of the earth has earned due acceptance, so too should the simplest explanation of the apparent contradiction of Christ.
... My aim in this section is to lay out Beall's and Cotnoir's model and extend it to the TFC without getting too hung up on the nuanced logical details (hopefully, making the exposition more accessible to the general philosophical community). For much more on the underlying logical ingredients/mechanics seeBeall and Cotnoir (2017). ...
... See the appendix ofBeall and Cotnoir (2017) for some of the mechanics of K3. ...
The recent work of logician Jc Beall marks a paradigm shift within the fields of analytic theology and philosophy of religion. Thanks to Beall’s work, the long held (and generally unquestioned) assumption that theology is governed by (or closed under) the classical account of logic, is no longer free for the assumption. More importantly, by dropping this unquestioned commitment to the classical account, Beall’s work has uncovered natural and well-motivated solutions to some of monotheistic theologies’ most difficult and longstanding problems. That said, much of Beall’s work (and the work of others who have followed his lead) has been paraconsistent, utilizing glut-theoretic (contradictory) models to solve theologies problems. In this essay, my plan is to go paracomplete, with the aim of exploring a yet to be explored solution to the infamous foreknowledge and freedom problem. My solution finds its roots in the recent work Jc Beall and Aaron Cotnoir (‘God of the Gaps’, Analysis , 2017). Specifically, in this essay I will explore a gap-theoretic solution to the foreknowledge and freedom problem; one in which it is neither true nor false that God has foreknowledge. By utilizing Beall’s and Cotnoir’s model — which sees limit claims on God’s omni-properties as either just false or gappy — a natural and well-motivated solution to the foreknowledge and freedom problem emerges. Moreover, by utilizing the Beall-Cotnoir gap-theoretic model, not only is the foreknowledge and freedom problem circumvented, but an interesting and novel account of divine omniscience emerges.
... There are of course other existing works. To note a few: Beall [15], Beall and Cotnoir [17], Cotnoir [31], Göcke [40] and Anderson [12]. 11 See Ahsan [6]. ...
... ( Translated by Bayrak and Harris in Renard [14, p. 182]). 17 For early , in line with Plotinus (d.270), God was the unknowable absolute One who can be neither comprehended by reason nor accurately described. Their doctrine removed all the attributes, including "being," from God, and unlike the majority of the , they kept Her essence utterly unknowable and ineffable [50, p. 26]. ...
The application of paraconsistent logics to theological contradictions is a fascinating move. Jc Beall’s (J Anal Theol, 7(1): 400–439, 2019) paper entitled ‘Christ—A Contradiction: A Defense of ‘Contradictory Christology’ is a notable example. Beall proposes a solution to the fundamental problem of Christology. His solution aims at making the case, and defending the viability of, what he has termed, ‘Contradictory Christology’. There are at least two essential components of Beall’s ‘Contradictory Christology’. These include the dogmatic statements of Chalcedon and FDE logic. The first is the theological contradiction in question. The second is the type of paraconsistent logic. Both components are integral to a contradictory theology in general. I argue that there can be no such thing as an Islamic contradictory theology. I make the case by establishing two points. These points correspond to both integral components of a contradictory theology in general. The first is that an Islamic theological contradiction does not entail an actual (logical) contradiction. The second is that FDE logic, including alternative sub-classical systems of logic, are not adequate in tolerating an Islamic theological contradiction.
... 1 Beall and DeVito (2023) provides a helpful catalogue of other contradictions in Christian doctrines. See also Beall (2021) and Beall and Cotnoir (2017). 2 See Vaidya (2023) for discussion of the logic of the non-duality claim in Advaita. 3 Not every dialetheist must be committed to seeing the doctrine of the trinity as expressing a contradiction. ...
Many authors show how useful logic can be as a tool for building theories that can account for problems in the philosophy of religion, such as paradoxical assertions. As a consequence, one's philosophy of logic is crucial as well, since it determines which logics, from the set of available and constructible logics, one can use to build a theory. In this paper, we present the relatively recent debate between logical pluralism and monism because the positions in this debate determine which logic(s) can, with justification, be applied to build a theory that addresses problems in the philosophy of religion. We begin by presenting the problem of paradoxical assertions and the debate over logical pluralism that bears on the addressing paradoxical assertions. We then canvass strategies for arguing in favor of logical monism, and pluralism; ultimately, we conclude that the Western tradition has reached a stalemate on this issue. We then turn our attention to the potential for Indian religious traditions to contribute to the debate. We present the five‐step‐syllogism from Nyāya‐Hindu philosophy, the four corners of reasoning from Buddhist philosophy, and the seven‐fold theory of predication from Jaina philosophy. The upshot of our presentation is to lay the groundwork for cross‐traditional logical debate by identifying the ways in which Indian discussions of debate and dialogue relate to modern approaches to logic and the philosophy of logic.
... Cognitive logic is an important branch of philosophical logic that has developed quite maturely so far, but it also has its share of problems and challenges. In addition to the traditional problem of logical omnipotence, there are other philosophical aspects [4][5]. Despite the problems faced by cognitive logic, it has made great progress, gradually moving from static to dynamic and from single-subject to multi-subject directions while diversifying the research methods [6]. ...
This paper analyzes the research object of cognitive logic in the context of information technology and philosophical reflection and constructs a model of cognitive logic for contextual dynamics. Contextual recognition of philosophical cognitive logic is achieved through contextual sequence feature extraction, and realistic, intelligent interaction is analyzed through probabilistic potential semantic analysis, while philosophical cognitive logic deepening is achieved using PELP grammar. The analysis of the deepening model showed that the interaction degree of the mental dimension increased from 0.79 to 0.91, and the interaction degree of the thinking dimension increased from 0.88 to 0.97. Applying the model to learning, the difference between learning methods without the model was up to 15 points, and the deepening of philosophical cognitive logic based on intelligent interaction could improve thinking ability.
... (Additionally, the first-order extension follows the pattern above, with quantifiers mimicking conjunction and disjunction in the usual ways.) 20 Notes 1 See, (Beall 2021;Beall Forthcoming;Beall and Cotnoir 2017;Chowdhury 2021). In addition, A. J. Cotnoir's (2018) was an early advocate of the exploration of glut-theoretic theology. ...
This essay marks the first steps towards a viable glut-theoretic (contradictory) solution to the longstanding foreknowledge and free will dilemma. Specifically, I offer a solution to the dilemma that accommodates omniscience (foreknowledge) and human freedom (as the ability to do otherwise) in a simple, flat-footed way. This goal is accomplished via viewing the theological fatalist argument not as a problem, but as a sound argument: omniscience and human free will are contradictory and by dropping to a weaker underlying account of logical consequence, we can embrace them in their full-throated, robust (though contradictory) interpretations. That said, the primary aim of this paper is one of exploration: how does a subclassical solution to the foreknowledge and free will dilemma stack up in comparison to the traditional solutions on offer in the literature. This essay represents the beginning of such an exploration.
... Milne's conclusion is substantive only if dialetheism is correct, and the views discussed herein presuppose classical logic. For a discussion of omni-properties within non-classical frameworks, such as dialetheism, seeCotnoir (2017) as well asBeall and Cotnoir (2017).2 Swinburne (1977) andAbbruzzese (1997) are among those who say this.Jago (2018, pp. ...
The coherence of omniscience is sometimes challenged using self-referential sentences like, “No omniscient entity knows that which this very sentence expresses,” which suggest that there are truths which no omniscient entity knows. In this paper, I consider two strategies for addressing these challenges: The Common Strategy, which dismisses such self-referential sentences as meaningless, and The Conciliatory Strategy, which discounts them as quirky outliers with no impact on one’s status as being omniscient. I argue that neither strategy succeeds. The Common Strategy fails because it is both unmotivated and impotent. The Conciliatory Strategy fails because it leads to embarrassing situations in which omniscient entities are epistemically inferior to non-omniscient entities: we can, for example, devise trivia-based drinking games that force omniscient entities into an intoxicated state; and, given plausible closure principles for belief, such entities are unable to have the sorts of beliefs that give them reason to refuse to play (e.g., they are unable to believe that they can lose the game).
... It isn't clear how, given that Beall's theological consequence relation will also validate the De Morgan laws. On a purely gappy approach to theology (Beall and Cotnoir 2017), one never asserts sentences of either form, since they will never be true, and so there is no pressure to distinguish them. On a purely glutty approach (Cotnoir 2018), we deny that there are any gaps, and thus we can interpret sentences like (2) as asserting the falsity of a disjunction of gluts (which is also glutty). ...
What is the proper role of logic in analytic theology? This question is thrown into sharp relief when a basic logical principle is questioned, as in Beall’s ‘Christ – A Contradiction.’ Analytic philosophers of logic have debated between exceptionalism and anti-exceptionalism, with the tide shifting towards anti-exceptionalism in recent years. By contrast, analytic theologians have largely been exceptionalists. The aim of this paper is to argue for an anti-exceptionalist view, specifically treating logic as a modelling tool. Along the way I critically engage with Beall on the role of logic in theology, maintaining that theological inquiry is in some ways disanalogous with other theoretical enterprises.
Beall and Cotnoir propose that ‘God can create an unliftable stone’ is a truth-value gap (neither true nor false). However, this yields a revenge paradox on whether God can eschew gaps. Can God avoid gappy ascriptions of power? Either way, God’s power seems to have limits. In response, it may be said that ascribing God the power to avoid gaps is itself gappy – it concerns a power that God neither has nor lacks. Yet this ends up being inconsistent, for it implies that God definitely lacks that power. Following Aquinas, perhaps Beall and Cotnoir could accept this lack and still uphold omnipotence, suggesting that the power to avoid gaps is impossible for God. Yet the Aquinian stratagem is enough to block the original paradox, which saps the motivation to proffer truth-value gaps in addition. I conclude that the gappy solution is either inadequate or insufficiently motivated.
I examine two related arguments for the claim that if God is omnipotent, God cannot lack abilities such as the ability to do evil or to act irrationally. Both arguments concern the idea that omnipotence is inconsistent with being dominated with respect to abilities. I raise new issues in the formulation of such dominance principles about ability, and attempt to solve them. I also discuss and reject existing objections to these arguments. I conclude that these arguments are promising but not conclusive, and that important work remains to be done in formulating the relevant dominance principles.
Classical theists hold that God is omnipotent. But now suppose a critical atheologian were to ask: Can God create a stone so heavy that even he cannot lift it? This is the dilemma of the stone paradox. God either can or cannot create such a stone. Suppose that God can create it. Then there's something he cannot do – namely, lift the stone. Suppose that God cannot create the stone. Then, again, there's something he cannot do – namely, create it. Either way, God cannot be omnipotent. Among the variety of known theological paradoxes, the paradox of the stone is especially troubling because of its logical purity. It purports to show that one cannot believe in both God and the laws of logic. In the face of the stone paradox, how should the contemporary analytic theist respond? Ought they to revise their belief in theology or their belief in logic? Ought they to lose their religion or lose their mind?
Apparent contradiction is common in traditional monotheism, and perhaps especially so in standard Christian theology given central doctrines such as the incarnation and trinity. This Element aims to chart out a very elementary but abstract framework through which such contradictions may be approached. This Element does not attempt to address the many options for thinking about contradictions in the face of logical entailment; it charts only a few salient abstract options.
Can God change the past? The standard Aquinas line answers this question negatively: God cannot change the past since such an act implies a contradiction; thus is not within the purview of God's omnipotence. While the Aquinas line is well-known, there are other, non-standard solutions to this question. In this paper, I look into such answers. In particular, I explore those answers that employ the resources of gappy and glutty logics. I show how these solutions are motivated and how each offers an alternative conception of what it is for God to be omnipotent. Finally, I consider and reply to potential issues that may be raised against these non-standard answers.
In “God of the Gaps: A Neglected Reply to God’s Stone Problem”, Jc Beall and A. J. Cotnoir offer a gappy solution to the paradox of (unrestricted) omnipotence that is typified by the classic stone problem. Andrew Tedder and Guillermo Badia, however, have recently argued that this solution could not be extended to a more serious Curry-like version of the paradox. In this paper, we show that such a gappy solution does extend to it
Dialetheism is the view that some contradictions are true. Resorting to either metaphysical dialetheism or semantic dialetheism may seem like an appropriate resolve to certain theological contradictions. At least for those who concede to theological contradictions, and take dialetheism seriously. However, I demonstrate that neither of these types of dialetheism would serve to be amenable in resolving an Islamic theological contradiction. This is a theological contradiction that I refer to as ‘the paradox of an unknowable and ineffable God’. As a result of this, I propose an alternative type of dialetheism which aims to resolve the paradox of an unknowable and ineffable God. I call this type of dialetheism, ‘mystical dialetheism’.
In ‘Theism and Dialetheism’, Cotnoir explores the idea that dialetheism (true contradictions) can help with some puzzles about omnipotence in theology. In this note, I delineate another aspect of this project. Dialetheism cannot help with one big puzzle about another classic ‘omni’ property, omnibenevolence—the famous problem of evil. For someone (including a dialetheist) who thinks that the existence of evil is a knock-down argument against traditional theism, it is a knock-down argument against dialetheic theism, too.
Beall and Cotnoir (2017) argue that theists may accept the claim that God's omnipotence is fully unrestricted if they also adopt a suitable nonclassical logic. Their primary focus is on the infamous Stone problem (i.e., whether God can create a stone too heavy for God to lift). We show how unrestricted omnipotence generates Curry‐like paradoxes. The upshot is that Beall and Cotnoir only provide a solution to one version of the Stone problem, but that unrestricted omnipotence generates other problems which they do not adequately address.
This paper is a sequel to Beall (2011), in which I both give and discuss the philosophical import of a ‘classical collapse’ result for the propositional (multiple-conclusion) logic LP+. Feedback on such ideas prompted a spelling out of the first-order case. My aim in this paper is to do just that: namely, explicitly record the first-order result(s), including the collapse results for K3+ and FDE+.
I believe that, for reasons elaborated elsewhere (Beall, 2009; Priest, 2006a, 2006b), the logic LP (Asenjo, 1966; Asenjo & Tamburino, 1975; Priest, 1979) is roughly right as far as logic goes.1 But logic cannot go everywhere; we need to provide nonlogical axioms to specify our (axiomatic) theories. This is uncontroversial, but it has also been the source of discomfort for LP-based theorists, particularly with respect to true mathematical theories which we take to be consistent. My example, throughout, is arithmetic; but the more general case is also considered.
Anselmian theists, for whom God is the being than which no greater can be thought, usually infer that he is an omniscient, omnipotent and omnibenevolent being. Critics have attacked these claims by numerous distinct arguments, such as the paradox of the stone, the argument from God's inability to sin, and the argument from evil. Anselmian theists have responded to these arguments by constructing an independent response to each. This way of defending Anselmian theism is uneconomical. I seek to establish a new defence which undercuts almost all the existing arguments against Anselmian theism at once. In developing this defence, I consider the possibility that the Anselmian God is not an omniscient, omnipotent and omnibenevolent being.
Analetheism and dialetheism