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I'm at the end of my doctoral degree in mathematics. As far as I'm aware, I've been using the symbol $\implies$ effectively for 10-years of higher education (for example, effectively enough to get research papers in mathematical physics through peer review). I thought I would brush up on my formal logic, and it turns out that my intuitive understanding of what $\implies$ means is not enough to understand the logic table of $\implies$;

$$ \begin{matrix} A & B & A \implies B\\ T & T & T \\ T & F & F \\ F & T & T \\ F & F & T \end{matrix} $$

Suppose I have the following three propositions:

$$ A: 2^2 = 4 $$ $$B: \sin(n\pi) = 0 \;\forall \; n \in \mathbb{Z}$$ $$C: \sin(0) = 0$$

All of these statements are true. B is a more general version of C, and A is unrelated to B and C.

I see that B is true, and C is true, and if I know B then I will know C. So I agree with the truth table if I apply it to B and C.

However, A and B are both true, but if I know A it doesn't tell me anything about B. So I don't see why I would want to say that A $\implies$ B is true in this case. My intuitive understanding of $\implies$ in relation to A and B is then that A doesn't imply B, and I'd say that A $\implies$ B is false.

Can anybody give me some intuitive explanation about why I would want to assign this truth table to $\implies$, preferably with some examples? The lecture notes I've been reading seem to think that "snow is black implies grass is red" is a good example for understanding this truth table. Apparently from the fact that "snow is not black" and "grass is not red", it should be intuitively obvious to me that "snow is black implies grass is red" is true. I run into the same problem here; as far as I can see the colour of snow doesn't tell me anything about the colour of grass.

I think my confusion is coming from the fact that I'm assigning some kind of causality to $\implies$, when I'm not supposed to do that? That's what my reading told me anyway, but I don't understand it.

Any explanation about what operation what my intuitive understanding may be referring to would also be helpful for me. I think this is probably some concept from probability or information theory (if you guessed that my research so far hasn't been too closely related to these fields, you'd be correct).

Edit; I'm looking for some explanation about what concept from probability or information theory captures the intuitive understanding of the word implies that I've tried to express here, additionally to the explanation of the formal logic

Jojo
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  • Does this answer your question? In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False? – Joe Sep 13 '20 at 18:38
  • This is probably going to sound unsatisfying, but the ultimate answer down to how we define $\implies$. In classical logic, every statement has to be true or false. By convention, $P \implies Q$ is vacuously true if $P$ is false. On an intuitive level, there are some reasons why this makes sense (they are outlined in the link above). However, it all comes back to definitions. – Joe Sep 13 '20 at 18:43
  • Because snow isn't black, we cannot come up with any counter-examples to the statement 'snow is black implies grass is red'. It is not possible to say that the statement is false. And so it is true. – Joe Sep 13 '20 at 18:46
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    "if..., then..." in propositional logic does not mean that there is a "causal link" between the two statements. – Mauro ALLEGRANZA Sep 13 '20 at 18:52
  • Maybe the question we have to ask is: why can't we derivate a contradiction given that truth table? and also are the other intepretations of the truth table of → consistent? – Mauro curto Sep 13 '20 at 18:54
  • Hi, I've read quite a lot of different explanations like those in the question that was linked to me (by another Joe). I still haven't understood this, I've edited my question to explain what I think would help me understand. Please can it be unlocked as a duplicate now. – Jojo Sep 13 '20 at 19:03
  • For any logical propositions A and B, whether or not there is any connection between them, A implies B means only that it is not the case that A is true and B is false. A bit like a correlation in statistics. I think you will find my blog posting on this topic to be useful. http://www.dcproof.com/IfPigsCanFly.html There, among other things, I justify each line of the truth table from first principles. – Dan Christensen Sep 13 '20 at 20:17
  • If you feel intuitively that "if P then Q" or "P implies Q" must imply some sort of causal relationship, then I feel like one option is to say "wow, mathematicians gave $\implies$ a name/reading that doesn't fit." and then continue on with your understanding of the truth table and modus ponens as normal. – Mark S. Sep 13 '20 at 20:52
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    Dan's linked article is bogus. – user21820 Jun 27 '21 at 12:22
  • @user21820: What is bogus about it? – Joe Nov 29 '23 at 19:28
  • @Joe: Come to the Basic Math chat and I can explain there. – user21820 Nov 30 '23 at 06:29

1 Answers1

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The $\implies$ symbol’s meaning is if A then B.
I.e. if $sin(n\pi =0),\forall n \in \mathbb Z$ then $sin(0)=0$.
Your problem is that you are assuming, what you intended to say is that if non-else is known, does there exist a possibility that A is true while B false? In which case, as long as neither A or B are assumed your intuition holds.
The more formal(and less helpful) way to phrase it is:
For some base axiom system $P$, does adjoining the axiom $A \land \lnot B$ Result in a contradiction?

razivo
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  • Hi thanks for your answer. By 'non-else' do you mean 'nothing else?' Or is 'non-else' some concept I don't know? – Jojo Sep 13 '20 at 19:24
  • Nothing else, it is a little vague, what I meant is nothing else specific to these statements. – razivo Sep 13 '20 at 19:26