2

  1. $(A\Leftrightarrow B)$ means that $A$ is true only when $B$ is true, and $B$ is true only when $A$ is true. So, does $(A\Leftrightarrow B)$ mean that the truth value of $A$ solely depends on the truth value of $B$ and vice-versa?

  2. $(A\Rightarrow B)$ means that if $A$ is true then $B$ is true and that if $B$ is true then $A$ might not be true. So, does $(A\Rightarrow B)$ mean that the truth value of $B$ does not depend on only the truth value of $A$ ?

ryang
  • 44,428
Vedant Rana
  • 202
  • 4
    Not exactly "depends on"... It means that the circumstance that one is TRUE and the other is FALSE is not possible. – Mauro ALLEGRANZA Oct 25 '22 at 11:42
  • Then can you tell me the correct meaning of $A\rightarrow B$? – Vedant Rana Oct 25 '22 at 12:08
  • 2
    It means that if $A$ is true then $B$ is true. Here is an example... "If it is currently raining outside, then my grass is wet." Here in our example when rain is currently falling the rain causes my grass to become wet. It is possible however for my grass to be wet for a different reason such as me using my hose and sprinkler setup to water my lawn. It is also possible for it to not be raining and my grass be dry. In terms of math and logic, $A\to B$ is equivalent to saying (and indeed is defined as saying) $\neg A \vee B$ – JMoravitz Oct 25 '22 at 12:22
  • 1
    The important things here are to be able to recognize the difference between the statements $A\to B$, $A\leftarrow B$, $A\leftrightarrow B$, and so on... written in words as "If $A$ then $B$", "$A$ if $B$", and "$A$ if and only if $B$" respectively. Yet another way of writing these are with the words "necessary conditions" and "sufficient conditions" – JMoravitz Oct 25 '22 at 12:26

1 Answers1

-1
  1. does $(A\Leftrightarrow B)$ mean that the truth value of $A$ solely depends on the truth value of $B$ and vice-versa?

Depends on what you mean by "solely depends". If $(A\Leftrightarrow B\Leftrightarrow C),$ then does A's truth value "solely depend" on B's?

  1. does $(A\Rightarrow B)$ mean that the truth value of $B$ does not depend on only the truth value of $A$ ?

Suppose that $A$ and $B$ are propositional variables such that $(A⇒B).$ (For example, $A$ and $B$ might represent "There are 17 hours in an Earth day" and "A square has 4 sides", respectively, with no direct connection between them.)

Then $(A⇒B)$ isn't an assertion of logical truth (truth regardless of what atomic propositions are assigned to $A$ and $B),$ but still tells us the 3 possible permutations of the truth values of $A$ and $B:$ we know that

  • If $A$ is false, then $B$ could either be true or be false
  • If $A$ is true, then $B$ is true.

Thus, $B'$s value is not a function of $A'$s, even as there are dependencies between them. (On the other hand, $(A\Rightarrow B)'$s value is a function of $A'$s and $B'$s.)

Is it instructive to frame $A'$s and $B'$s truth values in terms of dependencies, or is it an overcomplication that's potentially ambiguous, confusing or misleading?

ryang
  • 44,428