classical logic, intuitionistic logic and similar logical systems, the
principle of explosion (Latin: ex falso [sequitur] quodlibet, 'from
falsehood, anything [follows]'; or ex contradictione [sequitur]
quodlibet, 'from contradiction, anything [follows]'), or the principle
of Pseudo-Scotus, is the law according to which any ...
As I said in a comment, the = sign is not normally used in basic truth-functional logic of the kind discussed in this book chapter, so I'll assume you want to show there's a contradiction in assuming that (A v B) and (~A * ~B) are both true simultaneously, i.e. assuming (A v B) * (~A * ~B) leads to a contradiction.
As @k-wasilewski showed one basic way to ...
A ∨ B = ¬A ∧ ¬B
A ∨ B → ¬A ∧ ¬B
Assume that A is true.
If A is true that means that A ∧ ¬A, then A must not be the case.
Assume that B is true.
If B is true then that means that B ∧ ¬B, then B must not be the case.
Since it is not the case that A or B are true then A ∨ B must be false.
Therefore the argument is invalid.
So the question is:
WILL LOGIC STILL WORK WITHOUT DISJUNCTION INTRODUCTION?
The answer is "work for what purpose?"
What I mean by this is that what you're trying to do here (more broadly), namely "get rid" of the principle of explosion is a traveled path (well in the latter half of the 20th century) broadly known as paraconsistent ...
Material conditonal belongs to the language of propositional logic; you use it to build sentences.
Logical implication belongs to the meta-language ; you use this notion in order to talk about the logical relations between a set of sentences ( the premises) and a sentence called the conclusion.
Logical implication has no truth table. But when you use it, ...
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 ...
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 ...