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As far as I understand the notion of semantic consequence (denoted by $\models$), $ \models A$ means $A$ is a semantic consequence of the empty set. So the "empty space" on the left side of the double turnstile means "empty set".

However, when we take a look at $A \models $, now that means $A$ is a contradiction, i.e., everything is a semantic consequence of $A$. Now the empty space means "everything".

Why is that? Is there any explanation for that difference?

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  • $\begingroup$ What was not clear about the two highly upvoted answers to your previous identical post ? $\endgroup$ – Mauro ALLEGRANZA Apr 28 '20 at 7:04
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    $\begingroup$ It's not an identical post, I invite you to read it thoroughly. The older post asks what these notations mean, this post asks why that is. $\endgroup$ – SlowerPhoton Apr 28 '20 at 11:48
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To understand this difference, it is better to see these notations in a more general context. Actually, there is a uniform explanation for them.

Let $\Gamma$ and $\Delta$ be sets of formulas. The notation $\Gamma \models \Delta$ means that every structure (in the language $\mathcal{L}$ for the formulas in $\Gamma$ and $\Delta$) that satisfies all the formulas in $\Gamma$, satisfies also at least one formula in $\Delta$. This is the key semantic notion of logical consequence.

Now, what happen if $\Gamma = \emptyset$ and $\Delta = \{A\}$? It is vacuously true that every structure satisfies all the formulas in $\Gamma$, since $\Gamma$ is empty. Therefore, the notation $\Gamma \models \Delta$, i.e. $\models A$, says that every structure satisfies $A$ (the only formula in $\Delta$), i.e. $A$ is universally valid (a tautology in propositional logic).

What happen if $\Gamma = \{A\}$ and $\Delta = \emptyset$? It is necessarily false that a structure satisfies at least one the formulas in $\Delta$, since $\Delta$ is empty. Therefore, the notation $\Gamma \models \Delta$, i.e. $A \models \,$, says that there is no structure that satisfies $A$ (otherwise it would satisfy at least one formula in $\Delta = \emptyset$), i.e. $A$ is a contradiction (or unsatisfiable). Moreover, since there is no structure that satisfies $A$, it is also vacuously true that every structure that satisfies $A$ satisfies also a formula $B$ whatsoever. This is the reason why if $A \models \, $ (i.e. if $A$ is a contradiction) then everything is a semantic consequence of $A$: this is the so-called principle of explosion, aka ex falso quodlibet.


In model theory, the notation $\Gamma \models \Delta$ is often used in the case $\Delta$ is a singleton, i.e. in the form $\Gamma \models A$. The idea is that $\Gamma$ represents the set of hypotheses and $A$ is the thesis. So, in the literature is not quite common the notation $A \models \,$, but it is not impossible to find it. For instance, if I remember well, Epstein’s textbook Classical Mathematical Logic: the Semantic Foundations of Logic uses the notation $A \models \, $.

What I want to stress is that the notations $\models A$ and $A \models \,$ are perfectly consistent and have the general and uniform explanation I sketched above.

Moreover, in proof theory, especially in the sequent calculus for classical logic, it is very natural to deal with objects, called sequents, of the form $\Gamma \vdash \Delta$ where $\Gamma$ and $\Delta$ are finite sets (or multisets or sequences), possibly empty, of formulas. A sequent $\Gamma \vdash \Delta$ intuitively means that there is a derivation from the conjunction of the formulas in $\Gamma$ to the disjunction of the formulas in $\Delta$, in a derivation system with precise inference rules. Apparently, this syntactical notion of sequent has nothing to do with the semantic notion of logical consequence, but according to the completeness and soundness theorems (which holds for many logics, in particular for propositional and first-order classical logic), these two notions coincide, so it is equivalent to say $\Gamma \vdash \Delta$ and $\Gamma \models \Delta$, when $\Gamma$ and $\Delta$ are finite (possibly empty) sets of formulas.

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It helps to think of semantic consequence as a disjunction, where the premises are negated and the conclusion is positive:

$A_1, ..., A_n \vDash B$

means

"[For all valuations, ] if $A_1$ and ... and $A_n$ are all true [under that valuation], then $B$ is true [under that same valuation], too"

which can, using the fact that "If X then Y" is in mathematical use equivalent to "either X is not the case or Y is the case", be reformulated as

"either not all of $A_1$ and ... and $A_n$ are true, or $B$ is true"

which is in turn, using the equivalence between "not both X and Y" and "not X or not Y", equivalent to

"not $A_1$ or not ... or not $A_n$, or $B$".


Now

$\vDash A$

where there are no (negated) premises and only the (unnegated) conclusion

means

"[For all valuations, ] (nothing) or $A$"

so $A$ is the only option to make the "or"-statement happen -- $A$ must be true under all circumstances, i.e., $A$ is a tautology.

However, if $A$ occurs on the left side of the sequent

$A \vDash$

it is negated, so we have

"[For all valuations, ] not $A$ or (nothing)"

so now "not $A$" is the only option to satisfy the disjunction, meaning that $A$ can only ever be false -- i.o.w., $A$ is a contradiction.

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