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Material implication paradox

IscharamoochieIscharamoochie 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.

Comments

  • IscharamoochieIscharamoochie Moderator PEx Veteran ⭐⭐
    errr, anybody there? freakster? yuri? anybody?
  • caporegimecaporegime ilpadrino lite™ PExer
    Ischaramoochie:
    i'm having trouble establishing how the absence or falsity of p could imply the truth and falsity of q.
    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
  • IscharamoochieIscharamoochie Moderator PEx Veteran ⭐⭐
    yes, ut as i see it, the fourth occurence simply suggests an accidental relation since implication only deals with the relarion of the antecedent and consequent so long as there is an antecedent to link with it. if however, there is no antecedent, or that it is false, there is no established relation between the two propositions. if p then implies q, then when p is true, q must be true. but since if the absence of p can also imply q, then it must be noted that the truth-value of q is not wholly dependent of p. therefore, to say that the absence of p would be a necessary condition for the absence or falsity of q would sill be a hypothetical statement and would remain to be proven. all that the fourth occurence suggests is that in the absence or falsity of p, it is true that q can also be absent or false.

    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.

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