Consider an example

A

B > A
It seems logically vaild .

Now assume A is being apple and B is banana .

So its equivalent

**It's Apple

It's apple or not banana

If it's Banana then it's apple **

Clearly it's not valid .

So what kind of fallacy I did here ? This addition step is always confusing for me ? I am unable to distinguish when this step is vaild or not ?

• You seem to be confusing VALID with being TRUE in the real world. Valid does not mean true in the real world. Consult a logic text or a professional to clarify. The step of addition is always valid. An argument can be logically valid but false in reality. You are having issues with that it seems. Commented Feb 4, 2021 at 19:03
• @Logikal - A valid argument can only be false in reality if one or more of the premises is false in reality, but I think Rishi is assuming a situation where the object under consideration really is an apple, in which case the premises "it's an apple" and "its an apple or a non-banana" would both be true. The real problem here is that Rishi is confusing the material conditional in logic, which has a specialized meaning, with an if-then statement in ordinary English, i.e. an indicative conditional (I commented on the difference here). Commented Feb 4, 2021 at 23:23
• @hynosifl, the point I was making was that truth is an entirely different matter to validity in Mathematical logic. Whether there is a false premise is irrelevant to the argument validity. As such validity has no direct correlation to reality. Sure you can sometimes make valid arguments that apply to reality but for each one you make another person can show an impractical valid argument in reality. What do you have left: a system of thinking that at beat gives you 50 - 50 chances on applying to reality. One would have to state that logic is more than being about validity which is taught. Commented Feb 4, 2021 at 23:54
• @Logikal - I agree truth and validity are different, and that an argument can be valid even with false premises, but do you agree that all logically valid arguments that have premises that are true in reality also have conclusions that are true in reality, assuming no semantic ambiguity in the meaning of the propositions expressing the premises and conclusions? Commented Feb 5, 2021 at 3:40
• @hynosifl, YES but the term soundness is usually not mentioned right away but people focus on validity & are taught that way. So when I was taught the focus was NOT VALIDITY but soundness so that the reasoning DOES APPLY TO REALITY. So yes we can agree on if the premises are true in reality the conclusion and reasoning will apply. Focusing on soundness instead of validity is more of an epistemological concern. When I went to school we just said LOGIC. Today epistemology is a distinct field due to the rise of Mathematical logic. So the term logic is still ambiguous due to the other ttpes. Commented Feb 5, 2021 at 4:18

No fallacy at all: the argument is valid.

It is the way the Conditional works in classical logic.

If we assume A, our assumption amounts to saying that we accept it as true.

Thus, also B → A will be true, because a conditional with true consequent is true.

Obviously, you have not proved that B → A is a tautology (i.e. valid, always true); you have proved that A implies B → A.

It's apple. Therefore, If it's banana then it's apple.

I think you misunderstand the difference between the (formal) logical implication and the material implication.

Material implication is a binary connective that can be used to create new sentences; so 𝜙→𝜓 is a compound sentence using the material implication symbol →. Alternatively, in some contexts, material implication is the truth function of this connective.

Logical implication is a relation between two sentences 𝜙 and 𝜓, which says that any model that makes 𝜙 true also makes 𝜓 true. This can be written as 𝜙⊨𝜓, or sometimes, confusingly, as 𝜙⇒𝜓, although some people use ⇒ for material implication.