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I am trying to find a reasoning for the truth table of $A \rightarrow B$ .Here is what I came to so far.

$1.$ We have to first assume that if something can't be proven wrong , then it is true.

$2.$ $A \rightarrow B$ means that "If $A$ is true , then $B$ will be true as well".

$3.$ If $A$ is true and $B$ is true, then clearly $A \rightarrow B$ is true as well.

$4.$ If $A$ is true and $B$ is false then clearly $A \rightarrow B$ is false because it directly contradicts "If $A$ is true , then $B$ will be true as well".

$5.$ If $A$ is false and $B$ is false/true , then it can't directly contradict "If $A$ is true , then $B$ will be true as well" because we first assumed $A$ to be false , not true . So from "If $A$ is false and $B$ is false/true" , we dont know anything about what will happen if $A$ is true . Then we will use the principal "if something can't be proven wrong , then it is true." to conclude that $A \rightarrow B$ is true.

Is this reasoning valid?

Kripke Platek
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    Yes your analysis nailed it. Another way of saying the same thing is that $$(A \implies B) \iff {[\text{not}~ A] \vee B}.$$ – user2661923 Dec 19 '20 at 06:42
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    I just want to add that 5 is usually the part where people tend to have misunderstandings. But yeah, basically it means that from a false statement you can deduce "anything" you want. – nicomezi Dec 19 '20 at 06:56
  • I saw another post about this with the same reasoning but explained much better:link – Kripke Platek Dec 19 '20 at 08:12
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    Two comments. You say that if $A$ and $B$ are both true then 'clearly' $A \to B$ is true. Hmm, that to me isn't clear at all. Take 'If grass is green, then snow is white'. Sure, grass is green and snow is white, but I think one can reasonably question the truth of the conditional, which suggests some kind of relation between the color of grass and snow. So, I think that in this case, you also need to appeal to assumption 1: if $A$ and $B$ are both true, we can't show $A \to B$ to be false, and so by assumption 1 it is true. – Bram28 Dec 19 '20 at 13:55
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    My second comment pertains to that very Assumption 1. Note that this Assumption is crucial for your proof to work. But, is it a reasonable assumption? Is something true just because we don't know or can't prove it is false? We don't know that is is false that there life on other planets .. so therefore it is true that there is life on other planets?We also can't seem to prove that it is false that God exists (how do you prove the non-existence of something?) ... but does it follow that therefore it is true that God exists? In sum, I find Assumption 1 to be highly suspect. – Bram28 Dec 19 '20 at 13:59
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    In the end, it all boils down to this. We can mathematically define any truth-function any way we want. But the question is: does the mathematically defined truth-function of the material implication 'capture' our intuitive concept of the conditional? And here, there is all kinds of controversy. See, for example: https://en.wikipedia.org/wiki/Paradoxes_of_material_implication for the conceptual issues (and counter-intuitions) that arise when declaring the material implication to be 'the correct' definition of the conditional. – Bram28 Dec 19 '20 at 15:35
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    Indeed, based on issues like this, plenty of logicians/philosophers will say that the English conditional simply isn't truth-functional in nature at all: there is just no classical truth-function that captures the everyday conditional at all. Yes, if you had to pick one, the material implication comes closest, which is why we do end up using it when doing argument analysis. As such, it is really not any different from any mathematical tool (or scientific model): they are often approximations of reality. Not perfect ... but often 'good enough'. – Bram28 Dec 19 '20 at 15:39
  • reply to your first comment: That is absolutely true. It also came into my mind a couple of minutes after posting it. Why did I miss that ? – Kripke Platek Dec 19 '20 at 15:40
  • reply to your second comment : yep. My whole argument is standing upon this weird assumption that "We have to first assume that if something can't be proven wrong , then it is true." – Kripke Platek Dec 19 '20 at 15:43
  • Right ... again I would recommend looking into the Paradox of Material Implication. But if you have to pick a truth-table, then here is an argument for why you may want to pick the one that we typically use: https://math.stackexchange.com/questions/2208612/logical-conditional-truth-table-rationale – Bram28 Dec 19 '20 at 15:52

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