Truth table and the validation of a relation between two statements.

Question

If $a$ has a property $b$ and having the property $b$ implies a property $c$, does it mean that $a$ has the property $c$? How can the truth table be use to validate that?

Answer 1

This idea is best represented by predicate logic. To say that something has the properties $b$ and $c$, in predicate logic, is represented by formulas as:

• $Bx$: $x$ has the property $b$
• $Cx$: $x$ has the property $c$
• $Ba$: a has the property $b$

Given these formulas, you can construct the proposition $ \forall x[Bx \implies Cx]$. This is read as "For every $x$, if $x$ has the property $B$, then $x$ has the property $C$." The universal quantifier $\forall x$ generalizes the idea so that it can be applied to any specific circumstance. In this case, the specific circumstance is $Ba$, that is, $a$ has the property $b$.

This leads to the following proof:

\begin{array}{l} & \{1\} & 1. & \forall x[Bx \implies Cx] & \text{ Premise }\\ & \{2\} & 2. & Ba & \text{ Premise }\\ & \{1\} & 3. & Ba \implies Ca & \text{ 1 Universal elimination }\\ & \{1,2\} & 4. & Ca & \text{ 2,3 Modus ponens }\\ \end{array}

Therefore, claiming that $a$ has the property $c$ can be proven by predicate logic, but the ideas of predication and quantification are a little too involved to be captured by truth tables alone.