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A Case Against Omniscience:
Infinite Regress

Two arguments for why an omniscient being is impossible as the concept leads to an infinite regress and internal contradiction.

Citation Information

Allan, Leslie 2022. A Case Against Omniscience: Infinite Regress, URL = <https://www.rationalrealm.com/philosophy/metaphysics/omniscience-infinite-regress.html>.

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DRAFT FOR REVIEW (draft release 10 August 2022)

1. Introduction

There is a rich literature on the philosophical problems facing classical theism's attachment to perfection-making divine attributes, such as omnipotence and omnibenevolence. (For an abbreviated list of outstanding problems, see my Allan [2022a]). In this essay, I will address two problems that arise with ascribing the attribute of omniscience to God. I will try to show that attributing this perfection-making property to God throws the theist into an inescapable infinite regress of knowing states. (I flesh out another quite separate problem with omniscience based on fallibility in my Allan [2022b]).

What are the two ways in which attributing omniscience leads to an infinite regress? It is these:

  1. Omniscience leads to an infinite chain of justification that can never be completed.
  2. Omniscience leads to an infinite set of known propositions that can never be closed.

Now, a concept encompassing an infinite regress is not necessarily fatal. However, I will try to show that the resulting infinite regress in both cases leads to a self-contradiction. And that is fatal to any coherent notion of omniscience.

I will begin by developing the first Argument (A), deal with some objections, and then move on to developing the second Argument (B) and close by responding to objections. Before I present the formal statement of my first Argument (A), I will present a more informal version of it. I do this in the hope that it gives a flavour of its intuitive plausibility. This form takes the first person perspective.

2. Informal Statement of Argument A

  1. I know everything.
  2. I know that I know everything.
  3. I have a good reason for believing that I know everything.
  4. I know that I have a good reason for believing that I know everything.
  5. I have a good reason for believing that I have a good reason for believing that I know everything.
  6. I know that I have a good reason for believing that I have a good reason for believing that I know everything.

This proposition (vi) that I purportedly know to be true in turn requires its own good reason to believe. This chain of required reasons for my belief that I know everything leads to an infinite regress of required reasons. Without at least an in principle possibility of closure to this chain of reasons, it seems impossible for me to know everything. Therefore, proposition (i) is necessarily false.

Now translate this first person perspective to the more general perspective examining the plausibility of any being capable of knowing everything. The same principles of logic and what it is to know a proposition to be true apply. From the perspective of any reflective knower, it seems impossible for any being to be omniscient.

For completeness, I now present the more formal statement of my argument.

3. Formal Statement of Argument A

  1. For Being K, if K is omniscient, then K knows the truth value of every proposition.
  2. 'Being K is omniscient' is a proposition.
  3. Sentences of the form, 'Being K has a justificatory reason for believing X', are propositions.
  4. For Being K, K knows a proposition to be true if and only if K has a justificatory reason for believing the proposition.
  5. Being K is omniscient.
  6. Being K knows (5) is true.
    [viz.: 'Being K is omniscient']
  7. Being K has a justificatory reason for believing (5).
  8. Being K knows (7) is true.
    [viz.: 'Being K has a justificatory reason for believing (5) "Being K is omniscient"'] [first loop]
  9. Being K has a justificatory reason for believing (7).
    [viz.: 'Being K has a justificatory reason for believing (5) "Being K is omniscient"'.]
  10. Being K knows (9) is true.
    [viz.: "Being K has a justificatory reason for believing (7) 'Being K has a justificatory reason for believing (5) "Being K is omniscient"'". [second loop]
Book cover: Metaphysics: A Very Short Introduction by Stephen Mumford

Before analysing and drawing a conclusion from Argument A, I provide the following support for each of the premises in the argument.

(1) is analytically true.[1]

(2) and (3) are uncontroversial.

(3) follows logically from (1) and (2).

(4) is supported by the commonly held distinction between knowledge that proposition X is true and mere belief that X is true. While the latter only requires a state of belief, the former requires a justificatory reason for believing X.

(5) is assumed for the sake of argument (ex hypothesi)

(6) follows logically from (1), (2) and (5).

(7) follows logically from (4) and (6).

(8) follows logically from (1), (3) and (7).

(9) follows logically from (4) and (8).

(10) follows logically from (1), (3) and (9).

Now that I've shown how each of the premises is reasonable, what ought we conclude from Argument A? Firstly, notice how the argument shows how an omniscient being's omniscient knowledge of his own omniscience is recursive. The argument traverses two loops of the recursion through K knowing a proposition and K knowing that he has a good reason for knowing that proposition. The argument ends at (10) with what K knows as an omniscient being. Beyond (10), the loop then continues with K having a justificatory reason for that belief, and so on ad finitum.

There appear to be two ways to escape this infinite loop. Consider the proposition that K knows in (10) as an omniscient being:

  1. (P)
    Being K has a justificatory reason for believing (7) 'Being K has a justificatory reason for believing (5) "Being K is omniscient"'.

Either K is justified in believing (P) or he is not. If he is not justified in believing P, then he has a mere belief and not knowledge. In that case, K is not omniscient.

On the other hand, if K is justified in believing (P) and is omniscient, then he must be justified in believing that he is justified in believing (P). If that justification is that his knowledge that he is justified in believing (P) is self-justifying, then that knowledge presumes the epistemological principle that at least one proposition is self-justifying. If K is omniscient, then K must know the truth of that principle. However, to know that principle is true, K must have a justificatory reason for believing it. Otherwise, K is entertaining a mere belief. K's knowledge of that principle, then, spawns a new infinite loop of justifications. Or it terminates in another foundational epistemological principle that itself requires justification if it is to be more than mere belief.

So, the first way out of the loop is to declare that K is not omniscient. The second way out of the loop only starts a new infinite loop. In that case, K's belief in his own omniscience is never justified, robbing K of omniscience. In either case, assuming that K is omniscient leads to the self-contradiction that K is not omniscient. It's impossible, then, for a being to be omniscient.

4. Reply to Objections to Argument A

In this section, I will deal with some objections that have been raised against my first argument.

Objection 1: Premises (1) and (4) are false. God's knowledge is 'knowledge by identity' where the knower is identical with its object. At the limit, the distinction between knower and known vanishes. What is known by an omniscient being is not knowledge in our usual sense of 'justified true belief'.

Reply: According to this objection, God's knowledge is not propositional at all. Aurobindo gives an example of this type of knowledge as when we experience an 'uprush of wrath' in which 'we lose sight of the thinker and become the thought and the thinking' [1940: 544]. If all of God's knowledge were of this kind, then theists must abandon all thought that God knows mathematical truths that are propositional in nature, such as '2+2=4'. They must also give up on the notion that God knows any of things we ordinarily know, such as 'Paris is the capital of France'. If God does not know many of things that mere mortals know, then how can he be omniscient?

Secondly, if God's knowledge was of this immediate kind, then his omniscience conflicts with his moral perfection. Consider, for example, what it's like to torture an innocent child for sheer pleasure. If God knows what this is like, then he literally is the torturing of an innocent child for sheer pleasure. Knowledge of this kind also conflicts potentially with his immutability. If God knows it is raining now, then he changes his state when it stops raining.

5. Formal Statement of Argument B

  1. God knows all true propositions.
  2. 'God knows all true propositions' is a proposition.
  3. God knows 'God knows all true propositions' is true.
  4. 'God knows "God knows all true propositions"' is a proposition.
  5. God knows 'God knows "God knows all true propositions"' is true.

and so on ad finitum.

The above argument demonstrates the fact that God's knowing is recursive. For any set of 'all true propositions' that an omniscient being knows, there is always one more true proposition that is not included in the set. But then it can't be the set of 'all true propositions'. If it be thought that it really is the set of 'all true propositions', then the fact of God's knowing that he knows this set cannot be in the set. In that case, there is a true proposition that God does not know. But given God's omniscience, that is not possible. Either way leads to a contradiction. If God knows the set of 'all true propositions', then it's impossible for it to be the set of 'all true propositions'. If it is the set of 'all true propositions', then there is a fact that God does not know—which is also impossible. So, omniscience is impossible.[2]

6. Reply to Objections to Argument B

Is Atheism Dead? by Eric Metaxas

Objection 1: God's knowing the truths of arithmetic is not a problem, even though the truths of arithmetic are infinite in number. So why should there be a problem in God knowing the infinite number of true propositions?

Reply: Even if God can know the infinite number of the truths of arithmetic, his knowledge of what he knows is recursive. That's because, as Argument B shows, his knowledge of what he knows is self-referential and therefore cannot be bounded. Attempts in bounding all of God's knowledge of what he knows about his knowledge leads to unavoidable contradictions. So even if God can know the infinite set of the truths of arithmetic, he can never know the complete set of propositions he knows. Whenever you try to conceptualize the set of all propositions that an omniscient being knows, there is necessarily at least one true proposition that is not in the set—which is a contradiction.

Secondly, even for the truths of arithmetic (and any other set of truths, for that matter), Cantor's theorem demonstrates that there is no such knowable object as 'every proposition' or 'every truth' that is available for God to know. And that is what is required for 'omniscience' to be instantiated; that God know the truth of every proposition (see f.n. 2).

But let's grant for the sake of argument that God can know an infinite number of true propositions. An additional cost for the classical theist is that this contradicts other attributes of God; namely his timelessness and immutability. To know that one knows X requires attending to what one knows and evaluating and judging that one knows that one knows X. This reflective activity entails a sequence of events: first the initial knowing and then the reflection on what is known and then the judgment on what is known of the knower. For God to know what he knows requires a temporal sequence in which he changes his state: from knower to knower of his knowing. This necessarily requires God existing in time and the changing of God's state of knowing.

References

  • Allan, Leslie 2022a. Problems with God's Attributes, URL = <https://www.rationalrealm.com/philosophy/metaphysics/problems-god-attributes.html>.
  • Allan, Leslie 2022b. A Case Against Omniscience: Fallibility, URL = <https://www.rationalrealm.com/philosophy/metaphysics/omniscience-fallibility.html>.
  • Aurobindo, Sri 1940 (2005). The Life Divine in The Complete Works of Sri Aurobindo, Volumes 21 and 22, India: Ashram Publication Department.
  • Cameron, Ross 2018. Infinite Regress Arguments, The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), ed. Edward N. Zalta, URL = <https://plato.stanford.edu/archives/fall2018/entries/infinite-regress/>.
  • Duniho, Fergus 2019. The Impossibility of Omniscience, For the Love of Wisdom, URL = <https://fortheloveofwisdom.net/144/religion/the-impossibility-of-omniscience/> (Updated May 26, 2019).
  • Feldman, Richard 1981. Fallibilism and Knowing That One Knows, Philosophical Review 90/2: 266–82.
  • Grim, Patrick 1983. Some Neglected Problems of Omniscience, URL = <https://www.academia.edu/45184582/Some_Neglected_Problems_of_Omniscience> American Philosophical Quarterly 20/3: 265–76.
  • Grim, Patrick 1984. There Is No Set of All Truths, URL = <https://philpapers.org/archive/GRITIN-5.pdf> Analysis 44/4: 206–8.
  • Grim, Patrick 1985. Against Omniscience: The Case from Essential Indexicals, URL = <https://www.academia.edu/37634900/Against_Omniscience_The_Case_from_Essential_Indexicals> Noûs 2/19: 151–80.
  • Grim, Patrick 2013. Problems with Omniscience, in Debating Christian Theism, eds J. P. Moreland, C. V. Meister and K. A. Sweis, Oxford University Press: 169–180.
  • Kretzmann, Norman 1973. Omniscience and Immutability, in Philosophy of Religion: Selected Readings, eds W. L. Rowe and W. J. Wainwright, New York: Harcourt Brace Jovanovich: 60–70.
  • Malpass, Alex 2016. Craig's List – Omniscience and Actually Existing Infinities, UseOfReason, URL = <https://useofreason.wordpress.com/2016/03/06/craigs-list-omniscience-and-actually-existing-infinities/>.
  • Pearce, Jonathan M. S. 2013. God Cannot Know He Is Omniscient, A Tippling Philosopher, URL=<https://www.skepticink.com/tippling/2013/03/20/god-cannot-know-he-is-omniscient/> (Retrieved: July 11, 2022).
  • Wierenga, Edward 2021. Omniscience, The Stanford Encyclopedia of Philosophy, (Summer 2021 Edition), ed. Edward N. Zalta, URL = <https://plato.stanford.edu/archives/sum2021/entries/omniscience/>.

Footnotes

  1. [1] For alternate definitions of 'omniscience', see Wierenga [2021] and Grim [1983: §1]. These alternate definitions do not, I think, impact the force of my argument here.
  2. [2] My Argument B may be seen as a cut-down version of Grim's objection that there is no finite set of all truths for God to know [Grim 1983: §§5–7, 1985, 2013: §§VIII–IX]. Grim applies Cantor's theorem to the posited set of all truths, whereas my argument is a less technical formulation that applies to only one kind of truth. My argument has a reduced scope in only applying to the truths about what God can know about his own omniscience. In stating it this way, my aim is to make the argument more accessible.

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