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.


2 Answers 2


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.

  • Using what you said in the second paragraph. Since a ∨ b is true if either a or b is true or both. Then would this prove that a ∨ b entails c?
    – Amous
    Oct 18, 2017 at 14:40
  • @Amous - Correct. Oct 18, 2017 at 14:47

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.

  • Did you look at Mauro's answer? I think you're misreading the symbols.
    – virmaior
    Oct 24, 2017 at 2:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.