I understand Implication, as follows:

p = rain stopped.
q = i go out.

p->q =  if "rain stopped" then "i go out".

p                       q                       p->q
F                       F                       T
(rain is not stopped)   (i don't go out)        (if rain is not stopped, I don't go out)
F                       T                       F
(rain is not stopped)   (i go out)              (if rain is not stopped, I go out)
T                       F                       F
(rain is stopped)       (i don't go out)        (if rain is stopped, I don't go out)
T                       T                       T
(rain is stopped)       (i go out)              (if rain is stopped, I go out)

But, this video says that $F \rightarrow T = T$

  • 2
    $\begingroup$ Just because you go out, even when it is raining, does not signify that the statement $(p \implies q)$ is false. In fact, this situation is actually focusing on the critical difference between the statements $(p \implies q)$ and $(p \iff q)$. $\endgroup$ Aug 5 '21 at 10:28
  1. The fact that $$P\to Q$$ is true whenever its antecedent $P$ is false (principle of explosion; vacuous truth) is actually so by definition:

          $P\to Q\,$ is the truth function that is tautologically equivalent to $\,\lnot P\lor Q.$

    So, $P\to Q$ is false precisely when $P$ is true but $Q$ false.

  2. To be clear: when $P$ is false, $\,P\to Q\,$ (is always true so) never gives any information about $Q.$

  3. Summarising these two explanations of the motivation for the above definition:

    if we insist, to the contrary, that  False$\to$True  be false, then, unfortunately, these violations of natural deduction arise: $$\text{$A$ is true and $B$ is false $\implies\bigg(\big[(A\land B)\to A\big]\;$is false}\bigg)$$ and $$\big[\forall n\in\mathbb Z \,\big(n \text{ is a multiple of }4\, \to \,n \text{ is even}\big)\big]\;\text{is false}.$$

  4. It is worth noting that in logic/mathematics, $P$ need not cause $Q$ for $P$ to imply $Q$ (i.e., for the material conditional $P\to Q$ to actually be true); after all, $\to$ is defined truth-functionally rather than based on interpretation (of the meanings of $P$ and $Q$).


Implication $P\Rightarrow Q$ is only false if $P$ is true and $Q$ is false (true premise and wrong conclusion). The video is right.


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