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I've been studying entailment in classical logic and am struggling with how to interpret $X \vDash$, where the conclusion is empty. I'm looking for clarification on, and reasoning behind, whether $X \vDash$ means that $X$ is a tautology or that $X$ is unsatisfiable. Any insights or formal explanations would be greatly appreciated!

According to the definition of entailment (as given in Logic: An Introduction by Restall),

$X \vDash A$ if and only if any evaluation satisfying everything in $X$ also satisfies $A$.

I understand that $\vDash A$ (where there is no assumption) means that $A$ is a tautology, because there is nothing in the empty set to make an evaluation false.

However, does $X \vDash$ (where $A$ is "nothing" or empty) mean that $X$ is a tautology or that $X$ is unsatisfiable? My text says the latter, but I think that $X$ can be a tautology for the same reason that $\vDash A$ implies that $A$ is a tautology: $X$ being a tautology and $A$ being empty does satisfy the above definition: every interpretation that satisfies a tautology also satisfies "nothing".

ryang
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numq
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    As far as I understand, this gives us absolutely no information about $X$. – Smiley1000 Oct 29 '24 at 00:17
  • @Smiley1000 This is the case if you understand X⊨ as Reading 2 instead of Reading 1 in my answer below. On a separate note, I've just expanded its first bullet point to include an example. – ryang Oct 30 '24 at 04:38

2 Answers2

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The book is correct, an empty conclusion makes the set of premises unsatisfiable.

More typically we deal with a single (at least one and no more than one) conclusion, but multiple-conclusion logic gives rise to a natural extension to answer this question.

The interpretation of a multiple-conclusion sequent is that the conjunction of the premises entails the disjunction of the conclusions, i.o.w., under any given evaluation either all of the premises are false or at least one of the conclusions is true:

$X \vDash Y$
$\Longleftrightarrow$ For each $v$: If $\models_v x$ for all $x \in X$ then $\models_v y$ for at least one $y \in Y$
$\Longleftrightarrow$ For each $v$: Either $\not \models_v x$ for some $x \in X$ or $\models_v y$ for at least one $y \in Y$

Now when $Y$ is empty, there is nothing to satisfy on the right-hand side. So in order for the disjunction to hold, the first clause must be the case, meaning that not all of the $X$ can be satisfied under any given evaluation $v$. This means that the set is unsatisfiable.

You can now also see why an empty set of premises renders a single conclusion a tautology: We have

$\vDash \{y\}$
$\Longleftrightarrow$ For each $v$: Either ⊭vx for some x∈X or $\models_v y$ for at least one y∈Y the one $y$ with $\{y\} = Y$
$\Longleftrightarrow$ For each $v$: $\models_v y$
- i.e., $y$ is a tautology.

Natalie Clarius
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    Thank you. I understand how X being unsatisfiable would work for an empty Y. But my question is, wouldn't X being a tautology also work for an empty Y? Since is empty, the entailment is trivially satisfied because there's no way for to fail to have a true conclusion. So it seems to me that X could either be unsatisfiable or a tautology here. – numq Oct 29 '24 at 02:19
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    No, if $X$ is a tautology (as in, all formulas in it are tautologies) then we can't have $\vDash$ with an empty $Y$. The entailment does not hold then because we don't only need to make sure to satisfy one of the conclusions (none in this case), we have the condition for the left-hand side to take care of as well. And $X$ not being unsatisfiable is incompatible with the requirement established above that $\not \models_v x$ for some $x \in X$. If the conclusion is empty then the premises must be unsatisfiable, that's what I've been trying to show. – Natalie Clarius Oct 29 '24 at 02:46
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    "there's no way for to fail to have a true conclusion" Yes, there is - because there is no way to have a true conclusion at all. So the only way to maintain that every evaluation makes either one of the conclusions true or not all of the premises (or iow, that there is no evaluation that makes all of the premises true but none of the conclusions), we need to make sure that not all of the premises can become true under any evaluation either. – Natalie Clarius Oct 29 '24 at 03:09
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    I see what you're saying now, and I appreciate the detailed explanation. However, I’m still a bit confused about why vacuous truth wouldn’t apply here. In other logical contexts, like universal statements with empty domains or conditionals with false antecedents, vacuous truth is accepted because there's nothing to contradict the statement. So, I wonder why we can't apply a similar idea here — given that Y is empty, is there a specific reason we can’t consider the entailment trivially true? I'm a novice, so I apologize if the answer is obvious. – numq Oct 29 '24 at 03:27
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    In vacuous truth it goes the other way round - when the antecedent is empty the statement holds trivially because there is nothing to consider in the restriction. But here are failing to satisfy the conclusion because it is empty, which is something else. – Natalie Clarius Oct 29 '24 at 03:47
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    That's why an empty conclusion works differently from an empty premise: We're looking at the other side of the "if ... then ..." being empty now. – Natalie Clarius Oct 29 '24 at 03:48
  • Thank you for your help. I want to check my understanding. In the case of ∅ ⊨ {}, y has to be a tautology because ⊨⟺ For each : If ⊨ for all ∈, then ⊨ for at least one ∈. The antecedent here is vacuously true because it is a universal statement with an empty set. This means the conditional has the antecedent true in every interpretation. That means all that is required is for the consequent to be true in every interpretation, therefore y must be a tautology – numq Oct 29 '24 at 06:26
  • Then for the case of ⊨ ∅. X has to be unsatisfiable because here, the consequent cannot be satisfied because it is an existential statement (which cannot be vacuously true) with an empty domain. So the only way for the conditional to still hold is if the antecedent is false. Which means X must be unsatisfiable – numq Oct 29 '24 at 06:37
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    Yes, you got it. – Natalie Clarius Oct 29 '24 at 12:21
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According to the definition of entailment (as given in Logic: An Introduction by Restall),

⊨ iff any evaluation satisfying everything in also satisfies .

To resolve your dilemma, just know that the above definition applies only when is a single formula, and that the general definition of the entailment $$X\vDash Y,\tag1$$ where each of $X$ and $Y$ is a possibly empty list of formulae, is

  1. every interpretation that satisfies every formula in $X$ satisfies some formula in $Y$ ✅

    (when $X$ and $Y$ are nonempty, this means that the conjunction of $X$ logically entails the disjunction of $Y),$

rather than

  1. every interpretation that satisfies every formula in $X$ satisfies every formula in $Y$ ❌

  2. every interpretation that satisfies $X$ satisfies $Y$ ❌

    (this is the definition that you quoted).

Observe that when $Y$ comprises multiple formulae, Reading 1 is obviously equivalent to neither Reading 2 nor Reading 3; and that when $Y$ is empty, Reading 2 gives absolutely no information about $X$ while Reading 3 does not even fully make sense.

  • By definition (1), the statements \begin{align}&\vDash\phi_1,\ldots,\phi_n\\\top&\vDash\phi_1,\ldots,\phi_n\end{align} synonymously mean that in each interpretation, some formula in the succedent is true; in other words, they mean that $$\text{the }\textbf{disjunction }(\phi_1\lor\ldots\lor\phi_n) \text{ is }\textbf{valid};$$ note that when $n\ge2,$ it isn't necessary that at least one of $\boldsymbol{\phi_1,\ldots,\phi_n}$ is valid! For example, $$\top\vDash (\pi→\psi),(\psi→\pi)\quad,$$ yet neither $(\pi→\psi)$ nor $(\psi→\pi)$ is a valid formula.
  • It is vacuously true that every formula in the empty set is not satisfied, so in each interpretation it is false that some formula in the empty set is satisfied, so, by definition (1), the statements \begin{align}\phi_1,\ldots,\phi_n&\vDash\\\phi_1,\ldots,\phi_n&\vDash\bot\end{align} synonymously mean that in each interpretation, some formula in the antecedent is false; in other words, they mean that $$\text{the }\textbf{conjunction }(\phi_1\land\ldots\land\phi_n) \text{ is }\textbf{unsatisfiable},\\\text{i.e., the assumptions }\phi_1,\ldots,\phi_n\text{ are inconsistent};$$ note that when $n\ge2,$ the formulae $\phi_1,\ldots,\phi_n$ can nonetheless be individually satisfiable.
ryang
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