Strictly speaking, this proof is directed only towards Protestants who concede and maintain the infallibility of Scripture. Reliability here is the key word to understanding infallibility: The Catholic Church's teachings in matters of faith and morals can be trusted ... you can rely on them. If you, Mr./Ms. Christian, don't concede or maintain the reliability of either Scripture or Tradition ... then on what do you rely?
- R = Holy Spirit is reliable (Rom 3:3-4; 2 Tim 2:13).
- G = Holy Spirit guides the Church (Jn 14:26, 16:13).
- C = Church is reliable
Method: Reductio ad absurdum
1. R P
2. G P
3. R • G ⊃ C CP
4. ~C AP
5. ~( R • G) 3 & 4 MT
6. ~R ∨ ~G 5 DeM
7. R • ~G or ~R • G 6 MImp[*]
8. ~G or ~R 7 Simp
9. ~G • G or ~R • R 1 & 8 or 2 & 8 Conj (Contradiction)
10. R • G ⊃ C 3-9 RAA
11. ∴ C 1,2 & 10 MP (QED)
Long explanation for the confused:
Reductio ad absurdum, or the Rule of Conditional Proof, is a method of argument that seeks to prove the truth of a conditional argument (“If A, then B”) indirectly, by showing a contradiction derived by assuming that the consequent (the B statement) is false.
The conditional argument is the premiss in Step 3: “If Statements R and G are both true, then Statement C is true.” Without direct Scriptural backup of Statement C for its truth, we must first prove the conditional statement true before we can prove Statement C true.
By assuming the opposite conclusion in Step 4 (“It is not the case that the Church is reliable”), the rule of Modus Tollens (if the consequent of a conditional argument is false, then the antecedent — the A statement of “If A, then B” — is false) tells us that “It is not the case that Statements R and G are both true”. According to DeMorgan’s Theorem, this allows us to infer that either Statement R or Statement G is false.
By the rule of Material Implication, we can go either of two directions, both of which are demonstrated: Either R is true and G is false, or R is false and G is true. If G is false, then we contradict our premiss in Step 2. However, if R is false, then we contradict our premiss in Step 1. Since either direction leads to a contradiction, then the conditional premiss must be true; given the truth of Statements R and G, by Modus Ponens (“If the antecedent of a conditional is true, then the conclusion must be true”) Statement C must also be true, quod erat demonstrandum (which was to be demonstrated).
[*] Technically, the rule of Material Implication requires the negation of one of the already-negated disjuncts of Step 6, with a further step later to convert the double-negative into a positive if necessary. I’ve eliminated this symbolic redundancy to keep the long explanation from being more cumbersome than it must be.