If statements A and B are true, how do we know that statements A and B are both true?
Is all logic just made up?
>inb4 classical logic is axiomatic
How do we know that we can suppose things?
For example, let's say I have an axiom that says A is true. I then suppose that if A is true, then B is true. Logically, B should be true.
Why? Because you've never seen or heard of a situation where given the statements "A is true" and "if A is true, B is true" in which B was not true? That doesn't mean there couldn't ever be one.
Note that this is different from saying "if A implies B and B implies C, then A implies C." That is axiomatic.
What I'm saying is "given A is true, and A implies B is true, how do we know that B is true?" This is, as far as I know, not specified axiomatically.
tl;dr: How do we know that just because an axiom is true, any and all implications of said axiom are also true? This is not and cannot be an axiom in itself.
>>7985886
It's called a thought experiment.
Regardless of whether in actuality the premises of an argument are true or not, you assume they are true for the sake of entertaining the *validity*, not soundness, of said argument. Questioning otherwise is a whole separate (perhaps, metaphysical) exercise all together.
Hope this helps! (Unless you're just another semantic troll - in which case, fuck you.)
>>7985886
B-B-BUT WHAT IF MODUS PONENS IS WRONG GUISE
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>>7985926
I'm not a troll. I understand what you're saying, but I'm asking "why assuming the premise of the argument is assumed to be true, and the argument is itself (an axiom of identity), does the implication of the argument assumed to be true?" You could say this is simply the definition of an implication, but where do you get the idea that an implications exist? Unless, maybe, we assume that, while implicitly assuming the argument of our thought experiment exists, so do all its associated implications. Did that make sense?
>>7985929
This is what I'm getting at.
Modus Ponens may be assumed to be true, then.
Why is it not included in every list of axioms?
>>7985954
Rules of inference aren't axioms you idiot. God fucking dammit take a logic 101 course.
https://en.wikipedia.org/wiki/Rule_of_inference
>>7985960
>https://en.wikipedia.org/wiki/Rule_of_inference
>Axioms are usually regarded as starting points for applying rules of inference and generating a set of conclusions
You can't apply a single axiom without assuming what I've said already to be true. Otherwise, any rule of inference may or may not be true in the system. Even assuming you could, that wouldn't mean any implications of that statement would necessarily be true.
>>7985886
>How do we know that we can suppose things?
proof by contradiction: suppose we can't.
Cuz it says so, cracka
>>7985886
Think.
Order your thoughts cleanly and you'll see right away, that if the logical conclusions of "if A is true, then B is true" are "non true", then the premise must be "non true".
If the premise is "non true", your argument is not sound.
