MHB Logical Implication: Evaluating Truth of Claims

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The discussion evaluates the truth of two logical claims regarding implications. The first claim asserts that if \(\alpha \models (\beta \rightarrow \gamma)\), then \(\alpha, \beta \models \gamma\) is true, supported by the definitions of logical implication and the connective \(\to\). The second claim suggests that if \(\alpha \models (\beta \rightarrow \gamma)\), then \(\alpha \vee \beta \models \gamma\), which is not generally true since \(\alpha \vee \beta\) requires only one of the statements to be true, while \(\alpha\) and \(\beta\) both need to be true for \(\gamma\) to hold. The discussion concludes that the second claim does not hold in general, although it may be true under specific conditions, such as when both \(\alpha\) and \(\beta\) are tautologies. Understanding these implications is crucial for logical reasoning.
Yankel
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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.
 
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Yankel said:
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.

Yankel said:
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?
 
I am not sure. Is there a way to see it using a truth table?
 
Consider when $\gamma$ is $\alpha\land\beta$.
 
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.
 
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Yankel said:
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, $$\alpha\lor\beta\not\models\alpha\land\beta$$ 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.
 
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