How is it possible to deduce $A$ from $A\wedge B$?

Question

If I know that $A\wedge B = T$ then I can say that surely $A$ and $B$ separately are true, but if $A\wedge B = F$ then how can I infer the truth value of $A$ or the one of $B$. It seems impossible to infer the result : I can say just that $¬A∨¬B = T$. What am I missing? Am i misuderstanding the process of making deductions?

EDIT:

Is it linked to the fact that $A\wedge B\implies A$ is a tautology? Because this is the reason given in my book. Should I consider just the case $A\wedge B = T$ ?

Answer 1

If (A $\land$ B) is false, then inferring either A or B is not valid. In general, inferring anything from a false premise is not valid, though there do exist exceptions such as if (A <-> B) is false, and B is false, then A is true.

Often enough though, in formal systems, it's required that a formula holds true for inferences to get made from it (and the rule of inferences used always preserve truth also). That's one reason why the axiomatic method has something to recommend to it, and sometimes gets regarded as more rigorous.

Answer 2

Looking at the truth-table is helpful for $A \wedge B$ $$ \begin{matrix} A & B & A \wedge B \\ \hline F & F & F \\ F & T & F \\ T & F & F \\ T & T & T \end{matrix} $$ If you know that $A \wedge B=F$, it could be any of the first three cases. So you cannot infer $A$'s state or $B$'s state, but you can infer that one of them is false.

Answer 3

Using the Morgan Laws, all that you can deduce from knowing $A\wedge B=F$ is that $$\neg (A\wedge B)=\neg F=T \Longrightarrow \boxed{(\neg A)\vee (\neg B)=T}$$

Answer 4

Now the rules of inference are based on intuition, and their application and development dates long back.

Now if you see a proposition as $A ∧ B$ then at the back of our mind we shall be working that that this said proposition is true.( So it's the convention followed in many textbooks, if $p$ is a proposition, we prove a theorem: $p$, where the extension "that $p$ is true is implicit")

Now if $A ∧ B$ it means that in our hand both $A$ and $B$ are true and again going by the convention, "are true" is implicit

So intuitively we have:

$(A ∧ B) \implies A$

Now we can use hardcore mathematics such as boolean algebra and truth-table proof as the other answers.

Answer 5

Look at the very definition of deductive validity: An argument or inference is deductively valid if and only if it is the case that if the premise(s) is/are true, then the conclusion is true. And indeed, if $A \land B$ is true, then $A$ has to be true as well, and so you can indeed infer $A$ from $A \land B$. The case where $A \land B$ is false is simply not of interest in asking whether $A$ follows from $A \land B$ or not.