Prove whether statement is true or false

Question

Consider the following:

If a |= c or b |= c, then a ∨ b |= c. Prove whether this statement is true or false.

My gut instinct is to compare truth tables, but I don't think a truth table is possible with double turnstiles. Can someone offer me a better way to tackle this problem. I'm more interested in advice and hints.

Answer 1

The question deals with ⊨ i.e. with Logical consequence:

A formula A is a logical consequence within some formal system of a set of statements S if and only if there is no model in which all members of S are true and A is false.

We can apply the definition above: by assumption, we have that every truth valuation that satisfy a will also satisfy c and every valuation that satisfy b will also satisfy c.

But a valuation that satisfy a ∨ b must satisfy either a or b.

Answer 2

This is false in general, as in the following example.

Let a = c = some false statement, and let b be some true statement. Then:

• (a |= c) is true, therefore ( a|=c or b|=c ) is also true.
• b is true, therefore "a or b" is true.
• c is false.
• By the previous two points, ((a or b) |= c ) is false.

You can get the same result by analysing the formula "(( a -> c ) v ( b -> c )) -> ( a v b -> c )".

But if you assume that both a|=c AND b|=c hold, then you can conclude that (a or b)|=c also holds.