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# Material implication paradox

Ischaramoochie
Moderator PEx Veteran ⭐⭐

I'm having a little trouble grasping material implication and the connexion between the consequent and antecedent in terms of the implication operaion in symbolic logic.

consider that

p implies q

t | T | t

t | F | f

f | T | t

f | T | f *

i'm having trouble establishing how the absence or falsity of p could imply the truth and falsity of q. as far as i know, the third and fourth occurences have nothing to do with the initial proposition and that the inclusion of them is purely by accident. If anyone of you can help me, i'd greatly appreciate it.

consider that

p implies q

t | T | t

t | F | f

f | T | t

f | T | f *

i'm having trouble establishing how the absence or falsity of p could imply the truth and falsity of q. as far as i know, the third and fourth occurences have nothing to do with the initial proposition and that the inclusion of them is purely by accident. If anyone of you can help me, i'd greatly appreciate it.

## Comments

Ischaramoochie:I was intrigued by this post of yours and did some research.

I believe it's read as

P(t/f) | implies (T/F) | Q (t/f)so the first occurence is read as 'P Implies Q' or 'true Implies true' and the last occurence is read as 'false Implies false'.Btw, I made a simplified (more structured) adaptation of the solutions that I saw (examples using intuitive implication instead since it's a paradox):

Letter assignment for P condition is assigned to X while Q to Y.

___________________________X precedes Y?

___________X___________Y_(alphabetical order)

P is TRUE = M > Q is TRUE = N___TRUE

P is TRUE = M > Q is FALSE =B___FALSE

P is FALSE = A > Q is TRUE = N___TRUE

P is FALSE = A > Q is FALSE = B___TRUE

in this case, or with using ordinary language instead of intuitive implication, "p(f) implies q(f) is not always false". however, for the purpose of simplification, since logic only dealy with possibilities of truth and falsity, by the very fact that it can be true in some aspect, it is given the truth value of "t"

there are two rules governing this, in implication:

1. negative, an implication proposition is false if and only if the antecedent is true and the consequent is false

2. positive, an implication proposition is only certainly/always true in two aspects, if the antecedent is true and the consequent is true, and if the antecedent is false and the consequent is true.