I have these two statements:

False $\models$ True

Reads as : False logicially entails True if all models that evaluate False to True also evaluate True to True.

True $\models$ False

Reads as : True logically entails False if all models that evaluate to True also evaluate False to True.

If my understanding of the concept of "Entails" is correct, then both of these are incorrect because False can never be evaluated to True. Am I correct in my thinking, or am I missing something? This seems like a trick question, so I'm second guessing myself.


You are partly correct.

"True $\models$ False" is indeed false: every model makes "true" true, but no model makes "false" true, so every model provides a counterexample.

However, since no model makes "false" true, "False $\models$ True" is actually true! It's an instance of vacuous implication: think of it as being true for the same reason the statement $$\mbox{"If $0=1$, then I'm the president"}$$ is true.

  • $\begingroup$ I'm gonna have to spend some time wrapping my head around this concept. Cheers! $\endgroup$ – Talen Kylon Mar 31 '16 at 1:44
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    $\begingroup$ It's important to realize that in logic, unlike semantics, "if" doesn't have any causality. It's simply a statement about the combined truth values of two statements. If p then q just means q is true when p is. If p isn't true, the statements not false so it's true. $\endgroup$ – fleablood Mar 31 '16 at 8:10
  • $\begingroup$ If T |= T is true, T |= F is F, F |= T is T, and F |= F is T, then wouldn't entailment and implication have the same truth tables? $\endgroup$ – Eric Wiener Oct 17 at 0:43
  • $\begingroup$ @EricWiener Yes - entailment is essentially "implication, one level higher" (although one needs to be careful about this). This can be made precise: for example, we have $$\varphi\models\psi$$ iff $$\emptyset\models\varphi\rightarrow \psi,$$ so we have a kind of translation between the two. The differences between $\models$ and $\rightarrow$ aren't in terms of how they behave truth-tably but rather what type of thing they are in the first place: "$\varphi\rightarrow\psi$" is a statement which is either true or false in any given model but "$\varphi\models\psi$" is a statement about all models. $\endgroup$ – Noah Schweber Oct 17 at 2:02
  • $\begingroup$ Saying "$\varphi\rightarrow\psi$ is true" is grammatically incorrect: we need to specify a model in which we're evaluating truth (so we should instead say something like "$\varphi\rightarrow\psi$ is true in $\mathcal{M}$"). Saying "$\varphi\models\psi$," by contrast, is saying that every model in which $\varphi$ is true also makes $\psi$ true - so we don't need to specify a model for this to make sense. $\endgroup$ – Noah Schweber Oct 17 at 2:04

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