Logical Implication: Evaluating Truth of Claims

[Yankel]

Hello all,

I am trying to find if the following two claims are true or false:

1) If
\[\alpha \models \left ( \beta \rightarrow \gamma \right )\]
then
\[\alpha ,\beta \models \gamma\]

2) If
\[\alpha \models \left ( \beta \rightarrow \gamma \right )\]
then
\[\alpha \vee \beta \models \gamma\]

where \[\models\] is logical implication, meaning that if everything on the left side of the operator is T, then whatever on the right side is also T.

I cannot build truth tables, because \[\alpha ,\beta ,\gamma\] are not necessarily atoms.

Thank you in advance.

 

[Evgeny.Makarov]

1) If
\[\alpha \models \left ( \beta \rightarrow \gamma \right )\]
then
\[\alpha ,\beta \models \gamma\]

This holds by definition of $\models$ and $\to$. Suppose $\alpha\models \beta\to\gamma$. This means that for every interpretation $I$, if $I\models\alpha$, then $I\models\beta\to\gamma$. But the latter statement means that if $I\models\beta$, then $I\models\gamma$. Thus, if $I\models\alpha$ and $I\models\beta$, then $I\models\gamma$, or $\alpha,\beta\models\gamma$.

In general, $\alpha\models\beta$ iff $\models\alpha\to\beta$ (i.e., $\alpha\to\beta$ is true in every interpretation). In this sense, the connective $\to$ formalizes (i.e., is an analog on the level of the formal language being studied) the concept of logical implication $\models$. Also, $\alpha\land\beta\to\gamma$ is equivalent to $\alpha\to(\beta\to\gamma)$, i.e., these formulas are true in the same interpretations.

2) If
\[\alpha \models \left ( \beta \rightarrow \gamma \right )\]
then
\[\alpha \vee \beta \models \gamma\]

According to the above, $\alpha\models\beta\to\gamma$ is equivalent to $\alpha\land\beta\models\gamma$. Can you decide if this implies $\alpha\lor\beta\models\gamma$ and vice versa?

 

[Yankel]

I am not sure. Is there a way to see it using a truth table?

 

[Evgeny.Makarov]

Consider when $\gamma$ is $\alpha\land\beta$.

 

[Yankel]

I see what you mean, so it ain't true, is it ? I need both $$\alpha$$ and $$\beta$$ to be true, while $$\lor$$ requires that at least one is true, not necessarily both.

 

[Evgeny.Makarov]

I see what you mean, so it ain't true, is it ? I need both $$\alpha$$ and $$\beta$$ to be true, while $$\lor$$ requires that at least one is true, not necessarily both.

You are right, \(\displaystyle \alpha\lor\beta\not\models\alpha\land\beta\) in general, in particular, when $\alpha$ and $\beta$ are propositional variables. If $\alpha$ and $\beta$ are formulas, this implication may sometimes be true, for example, when $\alpha$ and $\beta$ are tautologies.