Monday, December 5, 2011

Formal proof: the reliability of the Church

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.