A Case Against Omniscience: Fallibility

6. Objection: Non-discursive Reasoning

Objection 8: It is possible that an omniscient thinker does not use discursive/inferential reasoning. In classical theism, the very need to think discursively/inferentially, that is, in terms of distinct propositions linked together into arguments, is a feature of embodied intellects only and not of God.

Reply: Ask yourself this question: How does Thinker X know infallibly that it is such a thinker? Without a good reason for thinking that it is a non-discursive thinker, how does Thinker X know that it is not simply mistaken in thinking itself so? From Thinker X's perspective as a being who thinks itself God, how does it know that it is a being who knows non-discursively/non-inferentially—and that it is not simply delusional or part of an experiment by some more powerful and intelligent being who puts that idea into Thinker X?

In addition, it appears that the notion that a knower can know the truth value of every proposition non-discursively/non-inferentially is incoherent. Understanding any formal logical rule of inference requires thinking discursively. Otherwise, the rule of inference escapes the knower's understanding. Take any disjunction of the form: 'P OR Q' and the additional premise: 'NOT P'. For any knower to know that 'P OR Q' and 'NOT P' entails 'Q' requires applying the rule of inference known as 'Disjunctive Syllogism' to the two premises to entail the conclusion. Without applying this discursive rule of inference, any knower, including an omniscient knower, does not know the truth of the theorem:

'P OR Q' and 'NOT P' entails 'Q'

Book cover: Summa Theologica by Thomas Aquinas

For a knower to know what 'entailment' is, they must apprehend and understand that entailment is a discursive process. To claim otherwise is akin to claiming that a knower can know the meaning of 'red' without ever having the experience of redness. Even understanding the meaning of the logical operators ('AND', 'OR', etc.) requires discursive thinking. Each logical operator is defined by the iterative assignment of possible truth values (as set out in the logical operator's truth table). If a knower doesn't know the truth value of 'P AND Q' for each iterative assignment of truth values to P and Q, then they haven't understood 'P AND Q'.

What I'm pointing out here is that it's a mistake to presume that the need to think discursively/inferentially is a feature only of embodied intellects. Thinking discursively is a necessary requirement for knowing what logical entailment is whatever the metaphysical nature of the thinker.

What I said above about understanding logical inference applies also to mathematical knowledge. From the 1800s, work in the axiomatization of mathematics has laid bare the understanding of mathematical objects as the iterative application of a small number of rules to a minimal set of axioms. These axiomatizations reveal how each number generated is the result of the application of the 'successor' function.[11] So, again, a knower cannot know what a number is unless they recognize the recursive application of the rule that generates them. In mathematics, the application of a small number of rules of inference generates an infinite number of natural numbers. Likewise, using the rules of logical inference generates an infinite number of theorems in logic.[12]

Even understanding the word 'non-recursive' itself requires recursive thinking. Understanding the word requires applying the rule: whenever you see the prefix 'non-' added to the beginning of an English word, understand the meaning of the composite word to be the denial of whatever follows the prefix.

A classical theist may concede all of the above, admitting that an omniscient thinker must understand fully the necessary recursive and discursive elements of mathematics and logic. However, they may say, an omniscient God does not understand the truths of mathematics and logic successively, one thing at a time. God understands all at once, they may say. In his Summa Theologica, Aquinas likens God's non-discursive understanding to the way we limited humans understand some discursions all at once. As Aquinas puts it:

For many things, which we understand in succession if each is considered in itself, we understand simultaneously if we see them in some one thing; if, for instance, we understand the parts in the whole, or see different things in a mirror.

[ST 1.14.7]

I concur that we sometimes get a flash of insight into understanding a proof or theorem at an intuitive level. I recall a number of times when studying logic and working through a proof using the truth table method line by line. Some time afterwards, the proof just 'clicked' with me. However, that intuitive insight I had did not replace my line-by-line reasoning. Nor did it legitimate my line-by-line understanding. Without that line-by-line analysis, that insight of mine can no more count as genuine knowledge as the 'insight' of the pseudo-scientist who proclaims boldly that Einstein's four-dimensional space-time model is fundamentally mistaken. That moment of insight is the reward that comes after many, many hours working discursively through the proofs. That moment of insight is not a short-cut alternative to reasoned, discursive analysis, but is grounded in it. If there is an omniscient thinker, its in-an-instant 'grasp' of a whole similarly counts as knowledge only if grounded in a prior proper analysis of the parts of the proof.


  1. [11] In Peano's systematization, for example, each successive natural number results from the application of the 'successor' function S. So, the number 1 is defined as the successor of 0 or S(0), 2 is defined as the successor of the successor of 0 or S(S(0)), and so on for all of the natural numbers. In this way, the successive application of the single S function generates the infinite number of natural numbers.
  2. [12] For example, the Addition rule generates 'A → A∨B'; 'A → A∨B∨C'; 'A → A∨B∨C∨D' and so on and so on to a theorem with infinitely many elements.

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