A Case Against Omniscience: Fallibility

4. Formal Symbolic Argument

The argument following is a symbolic representation in first order predicate logic of the argument presented in the previous section using a formalized natural language. This formulation makes clear the logical and non-logical elements and the formal entailments linking the premises.

Definitions:
 Ox: x is omniscient Gx: x has attributes posited by classical theism Py: y is a true proposition Txy: x knows the truth of y Cxy: x has no epistemically relevant reasons for doubting the truth of y a: 'x is omniscient'
Domains:
 x: the set of thinkers that thinks itself God y: the set of sentences in natural languages
1. x(Ox → ∀y(Py → Txy))
1. x(Ox → Pa)
1. x(Ox → Txa)
1. x(Txa → Cxa)
1. x¬Cxa
1. x¬Txa
1. x¬Ox
1. x(Gx → Ox)
1. x¬Gx
1. ¬∃xGx
1. [P]
1. [P]
1. [1,2 US,TH, UG]
1. [P]
1. [P]
1. [4,5 MT]
1. [3,6 MT]
1. [P][def]
1. [7,8 MT]
1. [9 T]

Support for Premises

(1) is true by definition.[7]

(2) is supported by the fact that if a thinker is omniscient, then a well-formed sentence ascribing omniscience to that thinker expresses a true proposition.

(3) is derived from application to (1) and (2) of universal specification [US], theorem [TH] and universal generalization [UG] as per the following proof:

1.
1.
1. {1}
1. {1}
1. {2}
1. {1,2}
1. {1,2}
1. x(Ox → ∀y(Py → Txy))
1. x(Ox → Pa)
1. Ob → ∀y(Py → Tby)
1. Ob → (Pa → Tba)
1. Ob → Pa
1. Ob → Tba
1. x(Ox → Txa)
1. [P]
1. [P]
1. [1 US]
1. [3 US]
1. [2 US]
1. [4,5 TH]
1. [6 UG]

(4) is supported by the principle that 'X knows Y to be true' if and only if 'X has epistemically relevant reasons for believing Y and no epistemically relevant reasons for doubting Y'.

(5) is supported by the fact that it is logically possible that at least one proposition believed by Thinker X is false. Two scenarios in which Thinker X believes a false proposition are that Thinker X is a brain in a vat (see Putnam [1999]) being manipulated by other more powerful beings or is being deceived by Descartes' demon [Descartes 1641]. A third scenario is that Thinker X is experiencing delusions or some other serious mental dysfunction. This third scenario in fact plays out with persons suffering from psychosis. Some psychotic patients actually believe that they are God (see, for example, Moralis [2008: 260] and Rudalevičienė et al [2008]).[8] Note that the truth of (5) does not need to meet the stronger requirement that it is possible that an omniscient thinker is mistaken. It only needs to meet the weaker requirement that Thinker X cannot epistemically justify the proposition that it is impossible that it is a brain in a vat or is suffering delusions or some similar epistemic distortion.[9]

(6) is derived from application to (4) and (5) of modus tollens [MT].

(7) is derived from application to (3) and (6) of modus tollens [MT].

(8) is true by definition of 'God' under classical theism.

(9) is derived from application to (7) and (8) of modus tollens [MT].

(10) is derived from application to (9) of theorem [T] De Morgan's law:

x¬Fx ⊢ ¬∃xFx

Footnotes

1. [7] 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. [8] Pearce [2013] has further suggestions for how a thinker can be mistaken, such as that he may not be aware of another inaccessible dimension run by another God, that he is plugged into the Matrix and that he is part of an experiment programming him to think that he is omniscient.
3. [9] For a discussion of the conditions under which agents know that they know and under which they don't know that they know, see Feldman [1981].