The Logic of the Trilemma – Symbolization & Proof

The Premises and Conclusion of the Trilemma in English
The intermediate conclusion–Either Jesus was a liar, or Jesus was mentally ill, or Jesus was God– is not itself a required assumption, because it can supposedly be deduced from the first three premises. So, McDowell’s Trilemma can be stated in terms of five premises and the main conclusion:

1. Jesus claimed to be God.
2. If Jesus claimed to be God, and Jesus was not God, and Jesus knew that he was not God, then Jesus was a liar.
3. If Jesus claimed to be God, and Jesus was not God, and Jesus did not know that he was not God, then Jesus was mentally ill.
4. It is not the case that Jesus was a liar.
5. It is not the case that Jesus was mentally ill.

Conclusion: Jesus was God.
Propositional Logic Symbolization of McDowell’s Trilemma
L: Jesus was a liar.
M: Jesus was mentally ill.
G: Jesus was God.
C: Jesus claimed to be God.
K: Jesus knew that he was not God.

1. C
2. [C ∙ (~G ∙ K)]
É L
3. [C ∙ (~G ∙ ~K)] É M
4. ~L
5. ~M
Conclusion: G
Derivation of the Conclusion from the Premises

The following logic proof shows that the above formulation of McDowell’s Trilemma is a valid deductive argument:

1. C
2. [C ∙ (~G ∙ K)] É L
3. [C ∙ (~G ∙ ~K)] É M
4. ~L
5. ~M …………………………………………../ G
6. C É [(~G ∙ K) É L] ........................2, Exportation
7. C É [(~G ∙ ~K) É M] .....................3, Exportation
8. (~G ∙ K) É L .................................1, 6, Modus ponens
9. (~G ∙ ~K) É M ..............................1, 7, Modus ponens
10. ~G É (K É L) ...............................8, Exportation
11. ~G É (~K É M) ............................9, Exportation
-----12. ~G .............................Assumption for Conditional Proof
-----13. K É L ....................................10, 12, Modus ponens
-----14. ~K É M ................................11, 12, Modus ponens
-----15. K v ~K ..................................Tautology
-----16. L v M .........................13, 14, 15, Constructive dilemma
17. ~G É (L v M) ...............................12 -16, Conditional Proof
18. ~~G v (L v M) ..............................17, Material implication
19. G v (L v M) ..................................18, Double negation
20. (L v M) v G ..................................19, Commutativity
21. ~L ∙ ~M .......................................4, 5, Conjunction
22. ~ (L v M) .....................................21, DeMorgan’s rule
23. G .......................................20, 22, Disjunctive syllogism

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