Intuitive notion of negation: implication example

Question

I am bit puzzled with intuitive notion of negating an implication. Say we have an implication. If A is true, then B is true. Say we want to negate it. Not analytically, rather intuitively.

Now, why is following NOT a negation of implication?

If A is true, then B is either true or not (we don't know).

Answer 1

The conditional $A \to B$ does not mean :

"If A is true, then B is true".

The truth table for the conditional has four cases, and only one of them has FALSE as "output".

Thus, considering the negation of $A \to B$, we want that it is TRUE exactly when the original one is FALSE.

I.e. $\lnot (A \to B)$ must be TRUE exactly when $A$ is TRUE and $B$ is FALSE.

This means that the negation of "If A is true, then B is true" is equivalent to :

"A and not B".

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Another approach is : consider that $A \to B$ is TRUE either when $A$ is FALSE, or when $A$ is TRUE also $B$ is.

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There are many discussion about the use of conditional in natural languages and its counterpart in logic; see e.g. the so-called Paradoxes of material implication.

The Material implication of classical propositional calculus is defined through its truth table and thus it is a "simplified model" of the way natural language works.

Its usefulness in formalizing many mathematical (and not only) arguments is the only reason to use it in formal contexts.

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Quite different is the case when "implication" means Logical consequence.

We have that $A \vDash B$ when there is no interpretations that makes $A$ True and $B$ False.

In this case, we have that if $A$ is True, we can assert that also $B$ is.

On the other hand, if $A$ is False, then $B$ can be either True or False.

If we follow this approach, what is the negation of $A \vDash B$ ?

It is :

there is at least one interpretation that makes $A$ True and $B$ False.

Answer 2

What does $A$ implies $B$ mean in classical logic? In the context of a logical proof, it means only that, at the moment, it is not the case that both A is true and B is false. Symbolically:

$A \implies B \space \equiv \space \neg (A \land \neg B)$

This is often given as a definition in textbooks, but it can also be derived using other rules of natural deduction including other elementary properties of logical implication.

Note that there is no suggestion here of cause and effect, or the passage of time. It may help to think of it is as a single snap shot of reality. This is true of classical logic in general, and nearly all of modern mathematics. For the most part, causality and the passage of time are in the realms of science, not mathematics.

Now, the negation of $A\implies B$ is obviously $A\land \neg B$.

In words: $A$ is true and $B$ is false.