What is the true value of $A(\lnot\varphi)$ in $A(\lnot\varphi\rightarrow(\theta\land\lnot\theta))=1$

Question

$A(\lnot\varphi\rightarrow(\theta\land\lnot\theta))=1$

I know $(\theta\land\lnot\theta)$ is a falsehood. Can I say $A(\lnot\varphi)=1$? Why?

Answer 1

We map T on $1$ and F on $0$.

As you correctly say : $\theta \land \lnot \theta$ is always F (it is a contradiction) and thus $A(\theta \land \lnot \theta)=0$.

We have to consider the truth table for the logical conncetive $\to$:

in order that $p \to $ F is T we need that $p$ is F.

Thus, in order that $A(¬φ→(θ∧¬θ))=1$ we need that $A(\lnot \varphi)=0$ [ $\lnot \varphi$ must be F ], and thus $A(\varphi)=1$ [ i.e. $\varphi$ must be T].