PROOF SYSTEMS FOR EXACT ENTAILMENT

2. Language

In the following, we’ll work with a fixed propositional language $\mathcal {L}$ , which has just the connectives $\neg , \land ,\lor $ and is defined over a set $\mathcal {P}=\{p_{i}:i\in I\}$ of propositional variables. The Backus–Naur Form (BNF) of $\mathcal {L}$ , correspondingly, is

$$\begin{align*}\phi::= p_{i}~|~\neg \phi~|~(\phi\land\phi)~|~(\phi\lor\phi).\end{align*}$$

We use $p,q,r,\ldots $ as meta-variables ranging over propositional variables and $\phi ,\psi ,\theta ,\ldots $ as meta-variables ranging over formulas. Unless indicated otherwise, $\Gamma ,\Delta , \Sigma ,\ldots $ range over finite sets of formulas.Footnote ⁵ We’ll follow the usual notational conventions with respect to parentheses etc.

We shall often find it convenient to rely on the following alternative syntax for $\mathcal {L}$ . Remember that a literal is either a propositional variable or its negation. We denote the set of literals by $\Lambda $ and use $\lambda $ as a meta-variable ranging over literals. We can then define the $\mathcal {L}$ -formulas using the following BNF:

$$\begin{align*}\phi::=\lambda~|~\phi\land\phi~|~\phi\lor\phi~|~\neg\neg\phi~|~\neg(\phi\land\phi)~|~\neg(\phi\lor\phi).\end{align*}$$

We shall refer to this as “the construction from literals.”

We write $\bigwedge \Gamma $ or $\bigwedge _{\phi \in \Gamma }\phi $ for the conjunction of the $\Gamma $ ’s, and $\bigvee \Gamma $ or $\bigvee _{\phi \in \Gamma }\phi $ for the disjunction of the $\Gamma $ ’s. We’ll justify this notation ex post by observing that both conjunction and disjunction are idempotent, commutative, and associative with respect to exact entailment. As a result, our notation is logically innocuous. We shall furthermore assume some background total order on $\mathcal {L}$ , which allows us to choose $\bigwedge \Gamma $ and $\bigvee \Gamma $ uniquely for given $\Gamma $ by respecting the order (relying on commutativity), ignoring repetitions (relying on idempotence), and parentheses (relying on associativity).

3. Semantics

We sketch the necessary background on exact truthmaker semantics and exact entailment.Footnote ⁶ A frame (also known as a state space in the literature) is a structure $\mathcal {F}=(S,\sqsubseteq )$ such that $S=\{s,t,u,\ldots \}$ is a non-empty set (“states”) and $\sqsubseteq $ is a partial order over S (“parthood”), such that for each $s,t\in S$ there exists a unique least upper bound $s\sqcup t\in S$ with respect to $\sqsubseteq $ (“fusion”).Footnote ⁷

A model is a structure $\mathcal {M}=(S,\sqsubseteq , v)$ , where $(S,\sqsubseteq )$ is a frame and $v=(v^+,v^-)$ is a pair of interpretation functions $v^+:\mathcal {P}\to \wp (S)$ (“exact truthmaker assignment”) and $v^-:\mathcal {P}\to \wp (S)$ (“exact falsemaker assignment”; cf. fn. 2), subject to the following condition for $\circ =+,-$ :

(Atomic Closure) $$ \begin{align} \text{If }s,t\in v^{\circ}(p)\text{, then }s\sqcup t\in v^{\circ}(p). \end{align} $$

In a model $\mathcal {M}$ , we define the exact truthmaker set $|\phi |^{+}_{\mathcal {M}}$ and the exact falsemaker set $|\phi |^{-}_{\mathcal {M}}$ by simultaneous recursion:

$$ \begin{align*} \begin{array}{c c c c} |p|^+_{\mathcal{M}}=v^+(p) & (\text{Sem-}0^{+}) & |\neg \phi|^-_{\mathcal{M}}=|\phi|^+_{\mathcal{M}}, & (\text{Sem-}{\neg}^{-}) \\ |p|^-_{\mathcal{M}}=v^-(p)& (\text{Sem-}0^{-}) & |\neg \phi|^+_{\mathcal{M}}=|\phi|^-_{\mathcal{M}}, & (\text{Sem-}{\neg}^{+}) \end{array}\\ \begin{array}{r @{\ }c @{\ } l c} |\phi\land \psi|^+_{\mathcal{M}} & =& \{s\sqcup t: s\in |\phi|^+_{\mathcal{M}},t\in |\psi|^+_{\mathcal{M}}\}, & (\text{Sem-}{\land}^{+}) \\ |\phi\land \psi|^-_{\mathcal{M}}&=&|\phi|^-_{\mathcal{M}}\cup |\psi|^-_{\mathcal{M}}\cup |\phi\lor \psi|^-_{\mathcal{M}}, & (\text{Sem-}{\land}^{-}) \\[2ex] |\phi\lor \psi|^+_{\mathcal{M}}&=&|\phi|^+_{\mathcal{M}}\cup |\psi|^+_{\mathcal{M}}\cup |\phi\land \psi|^+_{\mathcal{M}}, & (\text{Sem-}{\lor}^{+})\\ |\phi\lor \psi|^-_{\mathcal{M}}&=&\{s\sqcup t: s\in |\phi|^-_{\mathcal{M}},t\in |\psi|^-_{\mathcal{M}}\}. & (\text{Sem-}{\lor}^{-}) \end{array} \end{align*} $$

We further set $|\Gamma |^{\circ }_{\mathcal {M}}=\{|\phi |^{\circ }_{\mathcal {M}}:\phi \in \Gamma \}$ for $\circ =+,-$ .

The following canonical model is of fundamental importance to the study of exact entailment.Footnote ⁸ The canonical model $\mathfrak {M}=(\mathfrak {S}, \sqsubseteq _{\mathfrak {M}}, v_{\mathfrak {M}})$ is defined by $\mathfrak {S}=\wp (\Lambda )$ , ${\sqsubseteq _{\mathfrak {M}}}={\subseteq _{\restriction \mathfrak {S}}}$ , and $v_{\mathfrak {M}}=(v^{+}_{\mathfrak {M}},v^{-}_{\mathfrak {M}})$ , where $v^{+}_{\mathfrak {M}}(p)=\{\{p\}\}$ and $v^{-}_{\mathfrak {M}}(p)=\{\{\neg p\}\}$ .

Fine and Jago [Reference Fine and Jago17, p. 536] show that there are two natural ways of explicating guaranteed exact truthmaker preservation from premises to conclusion:

• • $\Gamma \vDash ^c\phi $ iff for all $\mathcal {M}$ , $|\bigwedge \Gamma |^{+}_{\mathcal {M}}\subseteq |\phi |^{+}_{\mathcal {M}}$ (“conjunctive exact entailment”),

• • $\Gamma \vDash ^d\phi $ iff for all $\mathcal {M}$ , $\bigcap |\Gamma |^{+}_{\mathcal {M}}\subseteq |\phi |^{+}_{\mathcal {M}}$ (“distributive exact entailment”).

As Fine and Jago [Reference Fine and Jago17] point out, $\vDash ^{c}$ and $\vDash ^{d}$ are indeed different consequence relations: while we (trivially) have $p,q\vDash ^{d}p$ , we have $p,q\nvDash ^{c}p$ . An instructive countermodel can be found at [Reference Fine and Jago17, pp. 536–537]:

What’s depicted here is the Hasse diagram of the underlying frame, where the subscripts indicate the exact truthmaking relation in the natural way. Since in this model, $|p\land q|^{+}_{\mathcal {M}}=\{s_{3}\}$ and $|p|^{+}_{\mathcal {M}}=\{s_{1}\}$ , we have a countermodel which shows that $p,q\nvDash ^{c}p$ . Note that the model shows that the familiar rule of conjunction elimination fails with respect to exact entailment—both in its conjunctive and in its disjunctive flavor. More generally, it’s easily seen that by definition, distributive exact entailment is a monotonic consequence relation: if $\Gamma \vDash ^{d}\phi $ , then $\Gamma ,\psi \vDash ^{d}\phi $ . Conjunctive exact entailment, in contrast, is not monotonic: while we trivially have $p\vDash ^{c}p$ , we don’t have $p,q\vDash ^{c}p$ .

Note, however, that conjunctive exact entailment reduces to a special case of distributive exact entailment:

Observe that this reduction is not in conflict with the different structural properties of distributive and conjunctive exact entailment. On this reduction, the failure of monotonicity for conjunctive exact entailment is preserved via the failure of conjunction elimination for distributive exact entailment.

The reduction allows us to focus our attention on distributive exact entailment for the purposes of this paper. Hence in the following, we shall simply speak of “exact entailment,” intending distributive exact entailment, and use the symbol $\vDash $ to represent the relation.Footnote ⁹ We shall write as an abbreviation for the conjunction of $\phi \vDash \psi $ and $\psi \vDash \phi $ .

We’d like to remark two facts about exact entailment that heavily influence the design of our proof systems.

First, we note that $\vDash $ is irreducibly asymmetric in the sense that we can’t reduce all premise sets to single formulas with the same consequences. To make this more precise, for $\phi (p_{1},\ldots , p_{n})$ a formula in the propositional variables $p_{1}, \ldots , p_{n}$ , let $\phi (\psi _{1}, \ldots , \psi _{n})$ denote the result of uniformly substituting $\psi _{i}$ for $p_{i}$ in $\phi $ . Let $\Gamma =\{\psi _{1},\ldots , \psi _{n}\}$ . Then a premise reduction for $\Gamma $ would then be a formula $\phi _{\Gamma }$ such that $\Gamma \vDash \theta \text { iff }\phi _{\Gamma }\vDash \theta $ . But it’s easily shown that there is no premise reduction for $\{p,q\}$ already. For suppose that there were such a $\phi _{p,q}$ . Since $p,q\vDash p$ and $p,q,\vDash q$ , we’d get $\phi _{p,q}\vDash p$ and $\phi _{p,q}\vDash q$ . Since the canonical model is a model, this entails by the definition of $\vDash $ that $|\phi _{\{p,q\}}|^{+}_{\mathfrak {M}}\subseteq |p|^{+}_{\mathfrak {M}}=\{\{p\}\}$ and $|\phi _{\{p,q\}}|^{+}_{\mathfrak {M}}\subseteq |q|^{+}_{\mathfrak {M}}=\{\{q\}\}$ . Since it’s easily checked (by induction) that for all $\psi $ , we have $|\psi |^{+}_{\mathfrak {M}}\neq \emptyset $ , it follows that $|\phi _{\{p,q\}}|^{+}_{\mathfrak {M}}=\{\{p\}\}$ and that $|\phi _{\{p,q\}}|^{+}_{\mathfrak {M}}=\{\{q\}\}$ , which is impossible since $p\neq q$ .Footnote ¹⁰

The second remark we’d like to make is that exact entailment has no theorems in the sense of formulas that are exactly made true by all states in all models (cf. [Reference Fine and Jago17, p. 539]). More precisely, if we define $\vDash \phi $ to mean that $|\phi |^{+}_{\mathcal {M}}=\mathcal {S}$ for all $\mathcal {M}$ , we get:

This means that the logic of exact entailment is “purely inferential.”

Let’s briefly talk about alternative frameworks for exact truthmaker semantics. Note that by a simple inductive argument, we can establish that Closure (p. 4) extends to all formulas, i.e., exact truthmakers and exact falsemakers are closed under (finitary) fusions:Footnote ¹¹

(Full Closure) $$ \begin{align} \text{If }s,t\in |\phi|^{\circ}_{\mathcal{M}}\text{, then }s\sqcup t\in |\phi|^{\circ}_{\mathcal{M}}\text{, for }\circ=+,-. \end{align} $$

This is why Fine [Reference Fine11, p. 206] calls the present version of exact truthmaker semantics the “inclusive” semantics. There are two prominent alternatives discussed in the literature: the non-inclusive semantics and the replete semantics.

On the non-inclusive semantics, essentially due to Fraassen [Reference van Fraassen25], we drop Closure and change ( $\text {Sem-}{\lor }^{+}$ ) and ( $\text {Sem-}{\land }^{-}$ ) to

$$\begin{align*}|\phi\lor\psi|^{+}_{\mathcal{M}}=|\phi|^{+}_{\mathcal{M}}\cup |\psi|^{+}_{\mathcal{M}} \quad \text{(Sem-}{\lor}^{+}_{\text{nc}}) \quad |\phi\land\psi|^{-}_{\mathcal{M}}=|\phi|^{-}_{\mathcal{M}}\cup |\psi|^{-}_{\mathcal{M}}. \quad \text{(Sem-}{\land}^{-}_{\text{nc}})\end{align*}$$

The most important difference between the inclusive and the non-inclusive semantics is that on the latter, the idempotence of conjunction fails. This can be seen by adjusting our countermodel for $p,q\nvDash ^{c} p$ from before (see p. 4):

We have $|p|^{+}_{\mathcal {M}}=\{s_{1},s_{2}\}$ but $|p\land p|^{+}_{\mathcal {M}}=\{s_{1},s_{2}, s_{3}\}$ , and so, since $|p\land p|^{+}_{\mathcal {M}}\nsubseteq |p|^{+}_{\mathcal {M}}$ , we have that $p\land p$ does not exactly entail p on the non-inclusive semantics. This works, of course, because on the non-inclusive semantics, the interpretation of p no longer needs to closed under fusions, and with $s_{3}=s_{1}\sqcup s_{2}$ not being an exact truthmaker of p in $\mathcal {M}$ while $s_{1},s_{2}$ are, this is exactly what we’re exploiting. Note, however, that the converse direction of idempotence—that $\phi $ exactly entails $\phi \land \phi $ —is still valid on the non-inclusive semantics.

Fine and Jago [Reference Fine and Jago17] don’t discuss the non-inclusive semantics and, for ease of exposition, we shall follow suit. The main complication is that the study of exact entailment under the non-inclusive semantics requires us to change the notion of our canonical model. To see this first note that the canonical model defined above has the (desirable) property that whenever $\phi \nvDash \psi $ , then the canonical model $\mathfrak {M}$ provides a countermodel, i.e., $|\phi |^{+}_{\mathfrak {M}}\nsubseteq |\psi |^{+}_{\mathfrak {M}}$ (we’ll prove this rigorously in Corollary 3.5). Now, while the canonical model is a model also in the sense of the non-inclusive semantics, it no longer has this important property. We can illustrate the issue with our pathological case of idempotence. When we consider $\mathfrak {M}$ as a model on the non-inclusive semantics, we have that $|p|^{+}_{\mathfrak {M}}=|p\land p|^{+}_{\mathfrak {M}}=\{\{p\}\}$ (note that the clause for conjunction are the same on the inclusive and non-inclusive semantics). In other words, $\mathfrak {M}$ doesn’t provide a countermodel to the failure of the exact entailment form $p\land p$ to p.

The point is that in order to obtain a counterexample to the inference from $p\land p$ to p, we need at least two separate states $s,t$ which individually exactly truthmake p but who’s fusion fails to be an exact truthmaker for p—just like in our modified countermodel above. Now it is possible to define a canonical model $\mathfrak {M}^{\dagger }$ , which has the desired property for the non-inclusive semantics. To achieve this, we set $\mathfrak {S}_{\dagger }=\wp (\Lambda \times \mathbb {N})$ , $\sqsubseteq _{\mathfrak {M}^{\dagger }}=\subseteq _{\restriction \mathfrak {S}_{\dagger }}$ , and $v^{+}_{\mathfrak {M}^{\dagger }}(p)=\{\{(p,i)\}:i\in \mathbb {N}\}$ as well as $v^{-}_{\mathfrak {M}^{\dagger }}(p)=\{\{(\neg p,i)\}:i\in \mathbb {N}\}$ . It’s easily verified that in this model, $|p\land p|^{+}_{\mathfrak {M}^{\dagger }}=\{\{(p,i),(p,j)\}:i,j\in \mathbb {N}\}$ , where each state $\{(p,i),(p,j)\}$ for $i\neq j$ provides an example of an exact truthmaker for $p\land p$ that fails to exactly truthmake p.Footnote ¹²

It is, in fact, possible to show that the canonical $\mathfrak {M}^{\dagger }$ can play the same role for exact entailment on the non-inclusive semantics as $\mathfrak {M}$ plays for the inclusive (and replete) semantics (see below). However, the semantic theory of non-inclusive exact truthmaking is comparatively underdeveloped in the literature. In particular, the semantic characterization results due to Fine and Jago [Reference Fine and Jago17], which play an central role in our completeness results, are not extended to the non-inclusive system. While it’s likely that we can generalize these results to the non-inclusive setting, going through the details and re-proving the relevant semantic theory is unfortunately beyond the scope of this paper. We shall therefore restrict our attention to the inclusive and replete semantics (which are covered by Fine and Jago).Footnote ¹³

On the “replete semantics,” discovered by Fine [Reference Fine11], instead, we make two additional philosophical assumptions:

• • Every statement has at least one exact truthmaker and at least one exact falsemaker (“non-vacuity”).

• • If a state lies between two exact truthmakers/falsemakers (in the sense of parthood), it is itself an exact truthmaker/falsemaker (“convexity”).Footnote ¹⁴

There are different ways in which we can formally implement the previous assumptions (cf. [Reference Fine and Jago17, pp. 547–551]). We shall follow the method described by Fine and Jago [Reference Fine and Jago17, pp 550–551]. We say that a model $\mathcal {M}=(S,\sqsubseteq , v)$ is non-vacuous iff both $v^{+}(p)\neq \emptyset $ and $v^{-}(p)\neq \emptyset $ . A straightforward induction on complexity establishes that in non-vacuous models, the non-emptiness property extends to all exact truthmaker sets and exact falsemaker sets:

(Non-Vacuity) $$ \begin{align} &\text{If }\mathcal{M}\text{ is a non-vacuous model, then }|\phi|^{\circ}_{\mathcal{M}}\neq\emptyset\text{, for }\circ=+,-. \end{align} $$

Note that the canonical model, in particular, is non-vacuous. In a frame $\mathcal {F}=(S,\sqsubseteq )$ , we define the convex closure $X_{\ast }$ of a set of states $X\subseteq S$ as $\{s\in S:\exists t,u\in X(t\sqsubseteq s\text { and }s\sqsubseteq u)\}$ . For an $\mathcal {X}\subseteq S$ , we set $(\mathcal {X})_{\ast }=\{(X)_{\ast }:X\in \mathcal {X}\}$ .

With these notions at hand, we can now define exact entailment on the replete semantics as follows (cf. [Reference Fine and Jago17, definition 7.1, p. 550]):

• • $\Gamma \vDash ^{c}_{nv\ast }\phi $ iff for all non-vacuous models $\mathcal {M}$ , $(|\bigwedge \Gamma |^{+}_{\mathcal {M}})_{\ast }\subseteq (|\phi |^{+}_{\mathcal {M}})_{\ast }$ .

• • $\Gamma \vDash ^{d}_{nv\ast }\phi $ iff for all non-vacuous models $\mathcal {M}$ , $\bigcap (|\Gamma |^{+}_{\mathcal {M}})_{\ast }\subseteq (|\phi |^{+}_{\mathcal {M}})_{\ast }$ .

Note that Propositions 3.1 and 3.2 immediately carry over to $\vDash _{nv\ast }$ (see the so-called “Convex Selection Lemma” [Reference Fine and Jago17, Lemma 6.4]), and just like in the case of $\vDash $ , there is no premise reduction for $\vDash _{nv\ast }$ either. As a result, we can focus on the distributive variant, which we’ll denote by $\vDash _{nv\ast }$ using for equivalence.

As a consequence, $\vDash _{nv\ast }$ inherits the logical laws of $\vDash $ . The main difference between $\vDash $ and $\vDash _{nv\ast }$ concerns distributivity, which we’ll discuss in the following section.

Before we move to formulating our proof systems, we would like to point out the characterization theorems provided by Fine and Jago [Reference Fine and Jago17] for both $\vDash $ and $\vDash _{nv\ast }$ , since we’ll rely heavily on them for our completeness proofs.

The theorems are stated in terms of selections. A canonical selection for a set $\Gamma $ is a function $f:\Gamma \to \wp (\Lambda )$ such that $f(\phi )\in |\phi |^{+}_{\mathfrak {M}}$ for all $\phi \in \Gamma $ . Intuitively, these selections give us a syntactic representation of the different exact truthmakers for the members of $\Gamma $ . For each $\phi \in \Gamma $ , the value $f(\phi )$ under a given selection function is a member of $|\phi |^{+}_{\mathfrak {M}}$ , so a set of literals $\{\lambda _{1},\ldots ,\lambda _{n}\}\subseteq \Lambda $ . Now a core lemma of [Reference Fine and Jago17], the so-called “Selection Lemma,” states that for any model $\mathcal {M}$ and state s therein,

$$\begin{align*}s\in |\phi|^{+}_{\mathcal{M}}\text{ iff for some selection }f\text{ for }\{\phi\}, s\in |\bigwedge f(\phi)|^{+}_{\mathcal{M}}.\end{align*}$$

In words: for a state to be an exact truthmaker for $\phi $ in some model is for the state to be an exact truthmaker for the conjunction of some exact truthmaker of $\phi $ in the canonical model (which, crucially, is just a set of literals). It is in terms of these syntactic representations that Fine and Jago characterize exact entailment:Footnote ¹⁵

The notion of a selection function can straightforwardly be generalized to the replete semantics, while preserving the underlying motivation sketched above. A convex selection for $\Gamma $ is a function $f_{\ast }:\Gamma \to \wp (\Lambda )$ , such that $f(\phi )\in (|\phi |^{\ast }_{\mathfrak {M}})_{\ast }$ for all $\phi \in \Gamma $ . Using this concept, we can prove:

4. Axiomatizing exact entailment

In this section, we directly axiomatize exact entailment as an asymmetric consequence relation. That is, we view the relation as a set of consequence pairs of the form $(\Gamma ,\phi )$ and we axiomatize membership in this set. Correspondingly, our system, $\mathsf {A}$ , operates on consequence pairs, allowing us to derive such pairs via inference rules from distinguished axiom pairs. We write $\Gamma \vdash _{\mathsf {A}}\phi $ to say that the pair $(\Gamma ,\phi )$ is derivable in our system and we write as an abbreviation for the conjunction of $\phi \vdash _{\mathsf {A}}\psi $ and $\psi \vdash _{\mathsf {A}}\phi $ .

The axioms and rules of our system are as follows:

First, a word on notation and implicit reasoning. As we noted in §2, we write $\bigwedge \Gamma $ or $\bigwedge _{\phi \in \Gamma }\phi $ for the conjunction of the $\Gamma $ ’s, and $\bigvee \Gamma $ or $\bigvee _{\phi \in \Gamma }\phi $ for the disjunction of the $\Gamma $ s. In the context of this notation, we often rely on implicit applications of $\land $ -Idempotence, $\land $ -Commutativity, and $\land $ -Associativity, as well as corresponding reasoning using $\lor $ -Introduction and $\lor $ -Elimination in treating certain expressions are “notationally equivalent.” For example, we shall treat $(\bigwedge \Gamma )\land (\bigwedge \Delta )$ and $\bigwedge (\Gamma \cup \Delta )$ as notational variants of each other, though, strictly speaking, we’re just relying on implicit reasoning in which we’re eliminating duplicates (using $\land $ -Idempotence) and re-arranging conjuncts (using $\land $ -Commutativity and $\land $ -Associativity). We also refer to this reasoning as “notational reasoning.”Footnote ¹⁷

Working towards a soundness result, we say that a pair $(\Gamma ,\phi )$ is valid iff $\Gamma \vDash \phi $ . The validity of the axioms follows directly from the corresponding laws for exact entailment observed by Fine and Jago [Reference Fine and Jago17, p. 539] and since the proofs are almost immediate by definition, we omit them here. We only cover $\land $ -Convexity, since this law has not been observed by Fine and Jago [Reference Fine and Jago17].

It turns out that $\land $ -Convexity is of central importance to the logic of exact entailment and closely related to the Fine–Jago characterization theorems. Note that $\land $ -Convexity is the only genuine multi-premise axiom of our system, and other multi-premise laws, like $\phi ,\psi \vdash _{\mathsf {A}}\phi \land \psi $ , are derived (cf. Proposition 4.7). We shall find it convenient to prove its validity in the following more general form:

The corresponding generalized axiom for our system, $\bigwedge \Gamma ,\bigwedge \Sigma \vdash _{\mathsf {A}}\bigwedge \Delta $ where $\Gamma \subseteq \Delta \subseteq \Sigma $ , is easily derived from the official axiom of $\land $ -Convexity and repeated applications of $\land $ -Monotonicity (see Proposition 4.7).

In a rule, we call the expressions on top of the inference line the “premises” of the rule and the expression below the line its “conclusion.” We say that a rule is sound iff its conclusion is valid whenever its premises are. The structural rules of Weakening and Cut are straightforwardly seen to be valid from the definition of $\vDash $ . Fine and Jago [Reference Fine and Jago17, p. 554, proof of theorem 9.2] effectively prove the soundness of $\land $ -Monotonicity and $\lor $ -Elimination by proving the soundness of corresponding rules for their sequent calculus. Since the corresponding proofs for our system are essentially notational variants of the proofs for the Fine–Jago system, we shall only provide the argument for $\land $ -Monotonicity and leave the case of $\lor $ -Elimination for the reader to work out.

Next, we extend our system to a system for exact entailment on the replete semantics. By Proposition 3.3, all axioms that are valid with respect to $\vDash $ are also valid with respect to $\vDash _{nv\ast }$ . The main difference concerns the distributivity of $\lor $ over $\land $ . As the reader will easily confirm (perhaps using the derived rules given in Proposition 4.7), we can derive

$$\begin{align*}\phi\lor(\psi\land\theta)\vdash_{\mathsf{A}}(\phi\lor\psi)\land(\phi\lor\theta).\end{align*}$$

The inverse direction, however, is not derivable—in fact, it is invalid on the inclusive semantics. Just observe that

$$\begin{align*}|p\lor (q\land r)|_{\mathfrak{M}}^{+}=\{\{p\}, \{q,r\}, \{p,q,r\}\},\end{align*}$$

$$\begin{align*}|(p\lor q)\land (p\lor r)|_{\mathfrak{M}}^{+}=\{\{p\},\{p,r\}, \{p,q\}, \{q,r\},\{p,q,r\}\}.\end{align*}$$

Since $|(p\lor q)\land (p\lor r)|_{\mathfrak {M}}^{+}\nsubseteq |p\lor (q\land r)|_{\mathfrak {M}}^{+}$ , we get that $(p\lor q)\land (p\lor r)\nvDash p\lor (q\land r)$ .

On the replete semantics, instead, the inference becomes valid. Just observe that

$$\begin{align*}(|p\lor (q\land r)|_{\mathfrak{M}}^{+})_{\ast}=\{\{p\},\{p,r\}, \{p,q\}, \{q,r\},\{p,q,r\}\},\end{align*}$$

$$\begin{align*}(|(p\lor q)\land (p\lor r)|_{\mathfrak{M}}^{+})_{\ast}=\{\{p\},\{p,r\}, \{p,q\}, \{q,r\},\{p,q,r\}\}.\end{align*}$$

Using Corollary 3.7 from Theorem 3.6, we can infer that . Indeed, the law holds in its general form, i.e., $(\phi \lor \psi )\land (\phi \lor \theta )\vDash _{nv\ast }\phi \lor (\psi \land \theta )$ , as is easily (but perhaps tediously) seen via Theorem 3.4 (left to the reader).

We denote derivability in our system for $\vDash _{nv\ast }$ correspondingly by $\vdash _{\mathsf {A}^{nv\ast }}$ . The rules and axioms for this system are the same as for $\vdash _{\mathsf {A}}$ except that we add the missing distributivity law as an axiom:

$$\begin{align*}(\phi\lor\psi)\land(\phi\lor\theta)\vdash_{\mathsf{A}^{nv\ast}}\phi\lor(\psi\land\theta).\end{align*}$$

We note that the arguments for the soundness of the “structural” rules, Weakening and Cut, are straightforward. The arguments for the soundness of $\land $ -Monotonicity $_{\mathsf {A}}$ and $\lor $ -Elimination $_{\mathsf {A}}$ use Theorem 3.6 in a similar way as the proof of Lemma 4.2 uses Theorem 3.4. We provide the proof for the soundness of $\lor $ -Elimination $_{\mathsf {A}}$ to illustrate the idea and leave the (easier) case of $\land $ -Monotonicity $_{\mathsf {A}}$ to the interested reader:

Proof. Assume that $\Gamma ,\phi _{1}\vDash _{nv\ast }\psi $ and $\Gamma ,\phi _{2}\vDash _{nv\ast }\psi $ . We wish to derive that $\Gamma ,\phi _{1}\lor \phi _{2}\vDash _{nv\ast }\psi $ . First, note that by $\land $ -Monotonicity and $\land $ -Idempotence, we can infer that $\Gamma ,\phi _{1}\land \phi _{2}\vDash _{nv\ast }\psi $ . We proceed using the Fine–Jago characterization theorem for the replete semantics (Theorem 3.6).

We want to show that for each convex selection $f_{\ast }$ for $\Gamma \cup \{\phi _{1}\lor \phi _{2}\}$ , we find a $\Delta \in (|\theta |^{+}_{\mathfrak {M}})_{\ast }$ such that the conditions of Theorem 3.6.(a and b) are satisfied, i.e., there’s a $\theta \in \Gamma \cup \{\phi _{1}\lor \phi _{2}\}$ with $f_{\ast }(\theta )\subseteq \Delta \subseteq \bigcup _{\theta \in (\Gamma \cup \{\phi _{1}\lor \phi _{2}\})}f_{\ast }(\theta )$ . So take an arbitrary convex selection $f_{\ast }$ for $\Gamma \cup \{\phi _{1}\lor \phi _{2}\}$ . Consider $f_{\ast }(\phi _{1}\lor \phi _{2})=\Sigma $ . Note that $\Sigma \in (|\phi _{1}\lor \phi _{2}|^{+}_{\mathfrak {M}})_{\ast }$ iff there exists $\Sigma _{\uparrow }\in |\phi _{1}|^{+}_{\mathfrak {M}}\cup |\phi _{2}|^{+}_{\mathfrak {M}}$ and a $\Sigma _{\downarrow }\in |\phi _{1}\land \phi _{2}|^{+}_{\mathfrak {M}}$ such that $\Sigma _{\uparrow }\subseteq \Sigma \subseteq \Sigma _{\downarrow }$ . Without loss of generality, we can assume that for our $\Sigma $ , we have $\Sigma _{\uparrow }\in |\phi _{1}|^{+}_{\mathfrak {M}}$ since the case for $\Sigma _{\uparrow }\in |\phi _{2}|^{+}_{\mathfrak {M}}$ is analogous.

We define a convex selection $f_{\ast }^{\uparrow }$ for $\Gamma \cup \{\phi _{1}\}$ by setting $(f^{\uparrow }_{\ast })_{\restriction _{\Gamma \setminus \{\phi _{1}\}}}=f_{\ast }$ and $f_{\ast }^{\uparrow }(\phi _{1})=\Sigma ^{\uparrow }$ . By our assumption that $\Gamma ,\phi _{1}\vDash _{nv\ast }\psi $ and Theorem 3.6, we get that there exists a $\theta ^{\uparrow }\in \Gamma \cup \{\phi _{1}\}$ and a $\Delta ^{\uparrow }\in |\psi |^{+}_{\mathfrak {M}}$ with $f_{\ast }^{\uparrow }(\gamma )\subseteq \Delta ^{\uparrow }\subseteq \bigcup _{\xi \in \Gamma \cup \{\phi _{1}\}}f_{\ast }^{\uparrow }(\xi )$ . We distinguish two cases: (a) $\theta ^{\uparrow }\in \Gamma \setminus \{\phi _{1}\}$ and (b) $\theta ^{\uparrow }=\phi _{1}$ . In case (a), we can simply let $\Delta ^{\uparrow }$ be our $\Delta $ and $\theta ^{\uparrow }$ our $\theta $ since by definition of $f^{\uparrow }_{\ast }$ we have

$$\begin{align*}f_{\ast}^{\uparrow}(\theta^{\uparrow})=f_{\ast}(\theta^{\uparrow})\subseteq \Delta^{\uparrow}\subseteq\bigcup_{\xi\in\Gamma\cup\{\phi_{1}\}}f^{\downarrow}_{\ast}(\xi)\subseteq \bigcup_{\xi\in\Gamma\cup\{\phi_{1}\lor\phi_{2}\}}f_{\ast}(\xi).\end{align*}$$

In case (b), we have $f^{\uparrow }_{\ast }(\phi _{1})\subseteq \Delta ^{\uparrow }\subseteq \bigcup _{\xi \in \Gamma \cup \{\phi _{1}\}}f^{\uparrow }_{\ast }(\xi )$ . We define another convex selection $f^{\downarrow }_{\ast }$ , this time for $\Gamma \cup \{\phi _{1}\land \phi _{2}\}$ by setting $(f^{\downarrow }_{\ast })_{\restriction _{\Gamma \setminus \{\phi _{1}\land \phi _{2}\}}}=f_{\ast }$ and $f_{\ast }^{\downarrow }(\phi _{1}\land \phi _{2})=\Sigma ^{\downarrow }$ . Since we’ve already seen that $\Gamma ,\phi _{1}\land \phi _{2}\vDash \theta $ , we can apply Theorem 3.6 to infer that there exists a $\theta ^{\downarrow }\in \Gamma \cup \{\phi _{1}\land \phi _{2}\}$ and a $\Delta ^{\downarrow }\in (|\psi |^{+}_{\mathfrak {M}})$ where $f_{\ast }^{\downarrow }(\theta ^{\downarrow })\subseteq \Delta ^{\downarrow }\subseteq \bigcup _{\xi \in \Gamma \cup \{\phi _{1}\land \phi _{2}\}}f^{\downarrow }_{\ast }(\xi )$ . We again distinguish two cases: (a.1) $\theta ^{\downarrow }\notin \Gamma \setminus \{\phi _{1}\land \phi _{2}\}$ and (a.2) $\theta ^{\downarrow }=\phi _{1}\land \phi _{2}$ .

In case (a.1), we can set $\theta =\theta ^{\downarrow }$ and $\Delta =\Delta ^{\uparrow }\cup f_{\ast }(\theta ^{\downarrow })$ . To establish this, we first note that since $\theta ^{\downarrow }\in \Gamma $ , we have $f^{\downarrow }_{\ast }(\theta ^{\downarrow })=f^{\uparrow }_{\ast }(\theta ^{\downarrow })=f_{\ast }(\theta ^{\downarrow })$ by the definition of $f^{\downarrow }_{\ast }$ . We then infer using basic set-theory that $f^{\downarrow }_{\ast }(\theta ^{\downarrow })\subseteq \Delta ^{\uparrow }\cup f^{\downarrow }_{\ast }(\theta ^{\downarrow })$ . Furthermore, since $\Delta ^{\uparrow }\subseteq \bigcup _{\xi \in \Gamma \cup \{\phi _{1}\}}f^{\uparrow }_{\ast }(\xi )$ , we can infer that $\Delta ^{\uparrow }\cup f^{\downarrow }_{\ast }(\theta ^{\downarrow })\subseteq \bigcup _{\xi \in \Gamma \cup \{\phi _{1}\}}f^{\uparrow }_{\ast }(\xi )\cup f^{\downarrow }_{\ast }(\theta ^{\downarrow })=\bigcup _{\xi \in \Gamma \cup \{\phi _{1}\}}f^{\uparrow }_{\ast }(\xi )$ . We now have that

$$\begin{align*}f_{\ast}(\theta^{\downarrow})\subseteq \Delta^{\uparrow}\cup f_{\ast}(\theta^{\downarrow})\subseteq \Delta^{\uparrow}=\Delta\subseteq\bigcup_{\xi\in\Gamma\cup\{\phi_{1}\}}f^{\downarrow}_{\ast}(\xi)\subseteq \bigcup_{\xi\in\Gamma\cup\{\phi_{1}\lor\phi_{2}\}}f_{\ast}(\xi).\end{align*}$$

What remains to be shown is that $\Delta ^{\uparrow }\cup f_{\ast }(\theta ^{\downarrow })\in (|\psi|^{+}_{\mathfrak{M}})_{\ast}$ . To see this note that since $\Delta ^{\uparrow },\Delta ^{\downarrow }\in (|\psi |^{+}_{\mathfrak {M}})_{\ast }$ , by Full Closure we have $\Delta ^{\uparrow }\cup \Delta ^{\downarrow }\in (|\psi |^{+}_{\mathfrak {M}})_{\ast }$ . We already know that $f_{\ast }(\theta ^{\downarrow })=f^{\downarrow }_{\ast }(\theta ^{\downarrow })\subseteq \Delta ^{\downarrow }$ , so it follows that $\Delta ^{\uparrow }\subseteq \Delta ^{\uparrow }\cup f_{\ast }(\theta ^{\downarrow })\subseteq \Delta ^{\uparrow }\cup \Delta ^{\downarrow }$ . By the convexity of $(|\psi |^{+}_{\mathfrak {M}})_{\ast }$ , we can infer that $\Delta ^{\uparrow }\cup f_{\ast }(\theta ^{\downarrow })\in (|\psi |^{+}_{\mathfrak {M}})_{\ast }$ as desired.

We’ve arrived at our final case, viz. $\theta ^{\uparrow }=\phi _{1}$ and $\theta ^{\downarrow }=\phi _{1}\land \phi _{2}$ . We summarize

$$ \begin{align*} \begin{array}{r @{ \ensuremath{\subseteq} } c @{ \ensuremath{\subseteq}} l} f^{\downarrow}_{\ast}(\phi_{1}\land\phi_{2}) & \Delta^{\downarrow}& \displaystyle\bigcup_{\xi\in\Gamma\cup\{\phi_{1}\land\phi_{2}\}}f_{\ast}^{\downarrow}(\xi),\\ f^{\uparrow}_{\ast}(\phi_{1}) & \Delta^{\uparrow} & \displaystyle\bigcup_{\xi\in\Gamma\cup\{\phi_{1}\}}f_{\ast}^{\uparrow}(\xi)=\displaystyle\bigcup_{\xi\in\Gamma}f_{\ast}(\xi)\cup f^{\uparrow}_{\ast}(\phi_{1}). \end{array} \end{align*} $$

Set $\Sigma ^{\ast }=\Sigma \setminus \Sigma ^{\uparrow }$ . Since $f_{\ast }^{\uparrow }(\phi _{1})=\Sigma ^{\uparrow }$ , we get that

$$\begin{align*}\underbrace{\Sigma^{\uparrow}\cup \Sigma^{\ast}}_{=\Sigma=f_{\ast}(\phi_{1}\lor\phi_{2})}\subseteq \Delta^{\uparrow}\cup\Sigma^{\ast}\subseteq \bigcup_{\xi\in\Gamma}f_{\ast}(\xi)\cup \underbrace{\Sigma^{\uparrow}\cup \Sigma^{\ast}}_{=f_{\ast}(\phi_{1}\lor\phi_{2})}=\bigcup_{\xi\in\Gamma\cup\{\phi_{1}\lor\phi_{2}\}}f_{\ast}(\xi).\end{align*}$$

Since $f^{\downarrow }_{\ast }(\phi _{1}\land \phi _{2})=\Sigma _{2}$ and $\Sigma ^{\ast }\subseteq \Sigma _{2}$ , we get that $\Delta ^{\uparrow }\subseteq \Delta ^{\uparrow }\cup \Sigma ^{\ast }\subseteq \Delta ^{\uparrow }\cup \Delta ^{\downarrow }$ . And just like in case (a.1), $\theta ^{\downarrow }\in \Gamma \setminus \{\phi _{1}\land \phi _{2}\}$ , we can infer that $\Delta ^{\uparrow }\cup \Sigma ^{\ast }\in (|\psi |^{+}_{\mathfrak {M}})_{\ast }$ . So in our final case, we can set $\theta =\phi _{1}\lor \phi _{2}$ and $\Delta =\Delta ^{\uparrow }\cup \Sigma ^{\ast }$ , establishing our claim.

Summarily, we get:

Before turning to completeness, we observe some facts about the logic of exact entailment via our proof system. As Odintsov and Wansing [Reference Odintsov and Wansing23, p. 51] point out, thinking about Cresswell’s definition of hyperintensional contexts as ones that do not respect logical equivalence, naturally leads us to the notion of self-extensionality of a logic. A logic is called fully self-extensional or congruential iff all the operators respect logical equivalence in that logic. In our case, that means that exact entailment is congruential (on the inclusive semantics) iff

We observe that exact entailment on the inclusive semantics is not fully congruential since it is not $\neg $ -congruential. To see this, note first that by ${\land }/{\lor }$ -Distributivity, we have that . But observe that

$$\begin{align*}|\neg(\neg p\lor (\neg q\land \neg r))|_{\mathfrak{M}}^{+}=\{\{\neg p\}, \{\neg q,\neg r\}, \{\neg p,\neg q,\neg r\}\},\end{align*}$$

$$\begin{align*}|\neg((\neg p\lor \neg q)\land (\neg p\lor \neg r))|_{\mathfrak{M}}^{+}=\{\{\neg p\},\{\neg p,\neg r\}, \{\neg p,\neg q\}, \{\neg q,\neg r\},\{\neg p,\neg q,\neg r\}\}.\end{align*}$$

It follows that $\neg ((p\land q)\lor (p\lor r))\nvDash \neg (p\land (q\lor r))$ and so $\neg ((p\land q)\lor (p\lor r))\nvdash _{\mathsf {A}} \neg (p\land (q\lor r))$ by Soundness.Footnote ¹⁸

An immediate consequence of this observation is that the following pair of rules is not sound:

(Replacement)

where $\theta (p)$ is any formula in the propositional variable p. That is, on the inclusive semantics, we cannot replace exactly equivalent formulas in all contexts while preserving exact entailment—in contexts involving negation, things might break down.

It is, however, easy to establish that our logic is what we might call positively congruential:

We can use this observation to show that positive replacement rules are admissible in our system. By the “admissibility” of a rule we mean that whenever premises of the rule are derivable, so is the conclusion. Remember that an occurrence of a subformula within a formula is positive iff the occurrence is not within the scope of an odd number of negations. The rules we shall prove admissible, then, are

(Positive Replacement)

where p occurs only positively in $\theta (p)$ . We have:

This means that we can freely use Positive Replacement in our system. It shall be convenient to abbreviate a certain pattern of reasoning involving Positive Replacement. It’s easily seen that using Positive Replacement and Cut, we can replace exactly equivalent formulas for one another anywhere within a derivation given that they occur positively. To see this, consider the following example:

To see that the application of Positive Replacement is indeed valid, just note that p occurs positively in $p\lor \psi $ , as well as $(p\lor \psi )(\phi \land \phi )=(\phi \land \phi )\lor \theta $ and $(p\lor \psi )(\phi )=\phi \lor \theta $ . For conciseness’ sake, we shall abbreviate the reasoning pattern as in the following example:

The fact that the logic of exact entailment on the inclusive semantics is not $\neg $ -congruential means that the logic is not only hyperintensional in the usual sense of distinguishing classically equivalent formulas,Footnote ¹⁹ but also hyperintensional in the sense of Odintsov and Wansing [Reference Odintsov and Wansing23]: it is hyperintensional by its own logical standards. Odintsov and Wansing [Reference Odintsov and Wansing23, p. 53] argue for a conception of hyperintensionality where (a) a logic is only hyperintensional if it is not congruential and (b) a connective is hyperintensional within a logic only if it is not congruential in the logic. So, one way to summarize our observations so far is that exact entailment on the inclusive semantics is hyperintensional in the sense of Odintsov and Wansing because negation is hyperintensional in the logic. This result has the potential for philosophical application when one tries to determine the most appropriate system for hyperintensional logic.

There is a debate in the literature on which concept(s) to build a basic system of hyperintensional logic. Fine [Reference Fine, Hale, Wright and Miller14, p. 565], for example, argues in favor of using exact truthmaking over an alternative inexact notion, which doesn’t require complete relevance but only partial relevance. Deigan [Reference Deigan6], instead, argues for taking the inexact notion as our starting point and Leitgeb [Reference Leitgeb, Giodani and Milanowski22] provides further arguments to support this position. Odintsov and Wansing [Reference Odintsov and Wansing23] show that Leitgeb’s HYPE, a system of hyperintensional logic based on an inexact conception of truthmaking, does not qualify as hyperintensional by their standards. Together with our previous observation, we can use this result to argue in favor of exact truthmaker semantics over HYPE: if Odintsov–Wansing hyperintensionality is what you’re after (for all the reasons given by them), you should go with exact truthmaking rather than HYPE. We leave further philosophical exploration of the result, e.g., of how the behavior of negation can be used to capture certain philosophical phenomena, for future work.

If, instead, a fully congruential is what we’re after, there are some options. One way to go would be to impose additional constraints on exact entailment that ensure its self-extensionality, such as exact falsemaker anti-inclusion, or formally, $|\phi |^{-}\subseteq \bigcup |\Gamma |^{-}$ .Footnote ²⁰ As it turns out, however, we’ve already described a fully congruential logic for exact entailment, viz. the logic on the replete semantics. We shall prove this next.

First, we note for later use:

Proof. Most arguments are standard and/or straightforward. Since the general arguments are somewhat opaque, we give derivations of simplified cases that are easily seen to generalize. 1. is derived using $\land $ -Monotonicity and $\land $ -Idempotence as the following simplified example:

The derivation for 2. is a generalization of the following:

We call 2. “Closure,” since it’s another syntactic expression of the semantic fact that truthmakers are closed under fusions next to $\land $ -Idempotence. The derivation for 3. is simply repeated applications of $\land $ -Monotonicity to $\land $ -Convexity. Finally, the derivation of 4. is a generalization of the following sketch:

The proof that exact entailment on the replete semantics is fully congruential relies on one of the core theorems for exact entailment, viz. its Disjunctive Normal Form (DNF) theorem. We shall now state and prove this theorem, which plays a central role in our completeness argument.

A conjunctive clause is a formula of the form $\bigwedge \Gamma $ for $\Gamma \subseteq \Lambda $ . A formula $\phi $ is in DNF iff it is of the form $\bigvee \phi _{i}$ , where the $\phi _{i}$ ’s are conjunctive clauses.

Proof. The proof of 1. is by induction on $\phi $ following the construction from literals. The proof is more or less the same as the standard proof of the DNF theorem for classical logic with an additional use of the fact that , which is quickly derived using Proposition 4.7. We cover only one case besides the base case to illustrate the relevant reasoning.

The base cases are straightforward, since $|p|^{+}_{\mathfrak {M}}=\{\{p\}\}$ and $|\neg p|^{+}_{\mathfrak {M}}=|p|^{-}_{\mathfrak {M}}=\{\{\neg p\}\}$ . Next, we cover the case for $\phi _{1}\lor \phi _{2}$ . By the induction hypothesis, we have

and

. By $\lor $ -Elimination and $\lor $ -Introduction, we get

The right-hand side of this equivalence is easily seen to be notationally equivalent to $\bigvee _{\Gamma \in |\phi _{1}|^{+}_{\mathfrak {M}}\cup |\phi _{2}|^{+}_{\mathfrak {M}}}\bigwedge \Gamma $ . Using

repeatedly, we get

Since $\bigwedge \Gamma _{1}\land \bigwedge \Gamma _{2}$ is notationally equivalent to $\bigwedge (\Gamma _{1}\cup \Gamma _{2})$ and $|\phi _{1}\land \phi _{2}|^{+}_{\mathfrak {M}}=\{\Gamma _{1}\cup \Gamma _{2}:\Gamma _{i}\in |\phi _{i}|^{+}_{\mathfrak {M}}\}$ (by Sem- ${\land }^{+}$ ), we get that $\bigvee _{\Gamma _{i}\in |\phi _{i}|^{+}_{\mathfrak {M}}}\left (\bigwedge \Gamma _{1}\land \bigwedge \Gamma _{2}\right )$ is notationally equivalent to $\bigvee _{\Gamma \in |\phi _{1}\land \phi _{2}|^{+}_{\mathfrak {M}}}\bigwedge \Gamma $ . This gives us, again via notational equivalence, that

Since $|\phi _{1}\lor \phi _{2}|^{+}_{\mathfrak {M}}=|\phi _{1}|^{+}_{\mathfrak {M}}\cup |\phi _{2}|^{+}_{\mathfrak {M}}\cup |\phi _{1}\land \phi _{2}|^{+}_{\mathfrak {M}}$ by Sem- ${\lor }^{+}$ the case is complete.

We leave the remaining cases to the interested reader and turn our attention to 2. First, we establish (a). Since $(|\phi |^{+}_{\mathfrak {M}})_{\ast }=|\phi |^{+}_{\mathfrak {M}}\cup \{\Delta :\exists \Gamma _{1},\Gamma _{2}\in |\phi |^{+}_{\mathfrak {M}}\text { with }\Gamma _{1}\subseteq \Delta \subseteq \Gamma _{2}\}$ , we know that $\bigvee _{\Gamma \in (|\phi |^{+}_{\mathfrak {M}})_{\ast }}\bigwedge \Gamma $ is notationally equivalent to

$$\begin{align*}\left(\bigvee_{\Gamma\in|\phi|^{+}_{\mathfrak{M}}}\bigwedge\Gamma\right)\lor\left( \bigwedge_{ \exists\Gamma_{1},\Gamma_{2}\in |\phi|^{+}_{\mathfrak{M}},\Gamma_{1}\subseteq\Delta\subseteq\Gamma_{2}}\bigwedge\Delta \right).\end{align*}$$

By 1., we know that

. Repeatedly using $\lor $ -Convexity (Proposition 4.7), we can derive

From this, our claim quickly follows via Cut and notational reasoning.

Finally, we establish 2.(b) by induction on $\phi $ following the construction from literals. We only sketch the argument since it’s essentially a dual version of the proof for 1. The base cases are again trivially since $(|p|^{\circ }_{\mathfrak {M}})_{\ast }=|p|^{\circ }_{\mathfrak {M}}$ for $\circ =+,-$ . The most interesting case is for $\phi _{1}\lor \phi _{2}$ , since it involves an argument via ${\lor }/{\land }$ -Distribution, which is not available for $\vdash _{\mathsf {A}}$ . Hence this case shows why we can’t prove a comparable theorem for $\vdash _{\mathsf {A}}$ .

By the induction hypothesis, we have

and

. Using $\lor $ -Congruence, we can derive

Using ${\lor }/{\land }$ -Distribution repeatedly as well as notational reasoning, we can derive

From here, we can reason as in 1. using

, as in 2.(a) using $\lor $ -Convexity, and using the (rather convoluted) semantic fact that $(|\phi _{1}\lor \phi _{2}|^{+}_{\mathfrak {M}})_{\ast }=(|\phi _{1}|^{+}_{\mathfrak {M}})_{\ast }\cup (|\phi _{2}|^{+}_{\mathfrak {M}})_{\ast }\cup (|\phi _{1}\land \phi _{2}|^{+}_{\mathfrak {M}})_{\ast }\cup \{\Delta :\exists \Gamma _{1},\Gamma _{2}\in (|\phi _{1}|^{+}_{\mathfrak {M}})_{\ast }\cup (|\phi _{2}|^{+}_{\mathfrak {M}})_{\ast }\cup (|\phi _{1}\land \phi _{2}|^{+}_{\mathfrak {M}})_{\ast }, \Gamma _{1}\subseteq \Delta \subseteq \Gamma _{2}\}$ , to derive

From this our claim follows via a series of Cut $_{\mathsf {A}}$ ’s.

Note that the DNFs in our theorem are indeed canonical DNFs: $\bigvee _{\Gamma \in |\phi |^{+}_{\mathfrak {M}}}\bigwedge \Gamma $ is what’s known as the “standard” DNF of $\phi $ , and $\bigvee _{\Gamma \in (|\phi |^{+}_{\mathfrak {M}})_{\ast }}\bigwedge \Gamma $ is what Fine [Reference Fine11, p. 215] calls “maximally standard” DNFs. Note also that these DNFs are unique up to logical equivalence since by Soundness (Theorem 4.4), we get: if , then $|\phi |^{+}_{\mathfrak {M}}=|\psi |^{+}_{\mathfrak {M}}$ ; and if , then $(|\phi |^{+}_{\mathfrak {M}})_{\ast }=(|\psi |^{+}_{\mathfrak {M}})_{\ast }$ .

We get now as straightforward corollaries:

Proof. It suffices to show that the logic is $\neg $ -Congruential since the arguments for $\land $ -Congruentiality and $\lor $ -Congruentiality go through as for $\vdash _{\mathsf {A}}$ . So, suppose that

. By Theorem 4.8, we have

But since DNFs are unique up to logical equivalence, we get that ${\bigwedge _{\Gamma \in (|\phi |^{+}_{\mathfrak {M}})_{\ast }}\bigvee _{\theta \in \Gamma }\neg \theta }$ and ${\bigwedge _{\Gamma \in (|\psi |^{+}_{\mathfrak {M}})_{\ast }}\bigvee _{\theta \in \Gamma }\neg \theta }$ are identical. So we can derive

by a single application of Cut $_{\mathsf {A}}$ .

We conclude the section by proving completeness.

Proof. For 1. take $\Gamma =\{\psi _{1},\ldots ,\psi _{n}\}$ with $\Gamma \vdash _{\mathsf {A}}\phi $ . By the Fine–Jago theorem for the inclusive semantics (Theorem 3.4), we get that for each selection function f for $|\Gamma |^{+}_{\mathfrak {M}}$ , there exists a $\Delta \in |\phi |^{+}_{\mathfrak {M}}$ , such that for some $\psi _{i}\in \Gamma , f(|\psi _{i}|^{+}_{\mathfrak {M}})\subseteq \Delta \subseteq \bigcup _{\psi _{i}\in \Gamma }f(|\psi _{i}|^{+}_{\mathfrak {M}})$ . For $\psi _{i}\in \Gamma $ , we can write $|\psi _{i}|^{+}_{\mathfrak {M}}=\{\Gamma _{i}^{1},\ldots , \Gamma _{i}^{j(i)}\}$ , where j maps i to the number of elements in $|\psi _{i}|^{+}_{\mathfrak {M}}$ . Now pick a selection function such that $f(|\psi _{i}|^{+}_{\mathfrak {M}})=\Gamma _{i}^{1}$ for $1\leq i\leq n$ . We get that there exists a $\Delta \in |\phi |^{+}_{\mathfrak {M}}$ such that for some $\Gamma _{i}^{1}$ , $\Gamma _{i}^{1}\subseteq \Delta \subseteq \bigcup _{1\leq i \leq n}\Gamma _{i}^{1}$ . Using $\land $ -Convexity, Proposition 4.7, we get that $\bigwedge \Gamma _{i}^{1},\bigwedge _{1\leq i \leq n}\Gamma _{i}^{1}\vdash _{\mathsf {A}}\bigwedge \Delta $ . Using $\land $ -Introduction, Proposition 4.7, together with Cut, we can infer that $\bigwedge \Gamma _{1}^{1},\ldots ,\bigwedge \Gamma _{n}^{1}\vdash _{\mathsf {A}}\bigwedge \Delta $ . Using Lemma 4.11 and Cut, we get that $\bigwedge \Gamma _{1}^{1},\ldots ,\bigwedge \Gamma _{n}^{1}\vdash _{\mathsf {A}} \phi $ . Completely analogously, just by choosing $f(|\psi _{1}|^{+}_{\mathfrak {M}})=\Gamma _{1}^{2}$ and $f(|\psi _{i}|^{+}_{\mathfrak {M}})=\Gamma _{i}^{1}$ for $1<i\leq n$ , we get $\bigwedge \Gamma _{1}^{2},\bigwedge _{2}^{1},\ldots ,\bigwedge \Gamma _{n}^{1}\vdash _{\mathsf {A}} \phi $ . And so on, giving us

$$ \begin{align*} \begin{array}{c @{} c @{} c } \bigwedge\Gamma_{1}^{1} &, \Gamma_{2}^{1}, \ldots,\Gamma_{n}^{1} &\ \vdash_{\mathsf{A}}\phi\\ \vdots & \vdots & \vdots\\ \bigwedge\Gamma_{1}^{j(1)} &, \Gamma_{2}^{1}, \ldots,\Gamma_{n}^{1} &\ \vdash_{\mathsf{A}}\phi.\\ \end{array} \end{align*} $$

Repeated application of $\lor $ -Elimination gives us $\bigvee _{\Gamma \in |\psi _{1}|^{+}_{\mathfrak {M}}}\bigwedge \Gamma , \Gamma _{2}^{1},\ldots ,\Gamma _{n}^{1}\vdash _{\mathsf {A}}\phi $ . By the DNF theorem (Theorem 4.8), we have , so by Cut, we get $\psi _{1},\Gamma _{2}^{1},\ldots ,\Gamma _{n}^{1}\vdash _{\mathsf {A}}\phi $ . We repeat this reasoning with suitable selection functions to obtain

$$ \begin{align*} \begin{array}{c @{} c @{} c @{} c } \psi_{1} &, \Gamma_{2}^{1}, & \Gamma_{3}^{1},\ldots,\Gamma_{n}^{1} &\ \vdash_{\mathsf{A}}\phi\\ & \vdots & \vdots & \vdots\\ \psi_{1} &, \Gamma_{2}^{j(2)} & , \Gamma_{3}^{1},\ldots,\Gamma_{n}^{1} &\ \vdash_{\mathsf{A}}\phi.\\ \end{array} \end{align*} $$

From this we get $\psi _{1},\psi _{2},\Gamma _{3}^{1},\ldots ,\Gamma _{n}^{1}\vdash _{\mathsf {A}}\phi $ using $\lor $ -Elimination, the DNF theorem and Cut. By repeating this reasoning, we finally obtain $\psi _{1},\ldots ,\psi _{n}\vdash _{\mathsf {A}}\phi $ .

The proof for 2. proceeds exactly analogously just that it relies on Theorem 3.6, Lemma 4.11 and Theorem 4.8.

Note that while the proof of completeness is direct (i.e., not via contrapositive reasoning), it is not constructive (i.e., it doesn’t generate a proof, it just shows that one exists). This is because of the non-constructive application of the Fine–Jago characterization theorem in our proof.

5. Hilbert calculus

In this section, we present two Hilbert calculi for exact entailment, one for the inclusive semantics and one for the replete semantics. The calculi are inspired by the Hilbert calculus for FDE described by Font [Reference Font18], which in turn relies on ideas used by Rebagliato and Ventura [Reference Rebagliato and Ventura24] to obtain a calculus for the implicationless fragment of intuitionistic logic. In these calculi, certain logical inferences are “nested” within disjunctive contexts, as for example in the inference from $\neg \neg \phi \lor \psi $ to $\phi \lor \psi $ , where $\psi $ provides a “disjunctive context” for the logical inference from $\neg \neg \phi $ to $\phi $ . The use of disjunctive contexts essentially allows us to absorb disjunction-elimination-style reasoning—inferences from $\Gamma ,\phi \vdash \theta $ and $\Gamma ,\psi \vdash \theta $ to $\Gamma ,\phi \lor \psi \vdash \theta $ —as a meta-rule (see Proposition 5.5). Without the disjunctive contexts, this meta-rule would need to become an explicit rule of our calculus. This would change the nature of our calculus from a Hilbert calculus for formula-to-formula inferences to something more akin to the direct axiomatization from the previous section.

It turns out that in order to accommodate exact entailment on the inclusive semantics, in particular in light of the failure of ${\land }$ -Elimination, we need an additional conjunctive context, nested within the disjunctive context as in the inference from $(\neg \neg \phi \land \psi )\lor \xi $ to $(\phi \land \psi )\lor \xi $ . Just like the disjunctive contexts allow us to absorb the disjunction elimination as a meta-rule, the conjunction ultimately allow us to prove $\land $ -Monotonicity as a meta-rule (see Lemma 5.3 and Proposition 5.5). The use of disjunctive and conjunctive contexts together is what allows us to formulate a proper formula-to-formula Hilbert calculus for exact entailment.

Since exact entailment has no theorems (Proposition 3.2), there are no axioms. The calculus, $\mathsf {H}$ , consists entirely of the following rules:

First, a quick remark on notation. Observe that we’ve absorbed the idempotence, associativity, and commutativity of conjunction in the single rule R $_{3}$ .Footnote ²¹ The rule R $_{2}$ is still necessary since in §2, we’ve decided on a canonical background ordering which the $\bigwedge $ and $\bigvee $ notation respects.

We write $\Gamma \vdash _{\mathsf {H}}\phi $ to say that there is a derivation of $\phi $ using the above rules from assumptions exclusively in $\Gamma $ . Just like before, we write as an abbreviation for both $\phi \vdash _{\mathsf {H}}\psi $ and $\psi \vdash _{\mathsf {H}}\phi $ .

Rather than proving soundness and completeness of the system directly, we shall show that the system is deductively equivalent to $\mathsf {A}$ .

Proof. The arguments are all analogous: in each case, the idea is to use R $_{5}$ to introduce the desired conclusion as a disjunctive context, apply the relevant rule, and then use R $_{6}$ to infer the conclusion. We provide the derivation for 1. as an example:

We leave verifying the remaining cases to the interested reader.

Next, we establish that the rules of our previous calculus hold as meta-theorems for our Hilbert calculus. First, note that Reflexivity, Weakening, and Cut are covered using standard structural reasoning about Hilbert calculi.

The following lemma is where the additional conjunctive contexts are really put to work.

Proof. The arguments for 1. are all analogous in that they essentially rely on permuting the relevant rules with ${\lor }/{\land }$ -Distribution $_{\mathsf {H}}$ . We shall show $(\phi \lor \xi ), (\psi \lor \xi )\land \gamma \vdash _{\mathsf {H}}((\phi \land \psi )\lor \xi )\land \gamma $ as an example:

For 1.(b), we show R $_{3}$ as an example:

The cases 2. are all straightforward given the disjunctive contexts in the premises.

Note the crucial role played by the conditional contexts in the derivation for R $_{3}$ in 1.(b).

Using the previous lemma, we prove:

We’re now in a position to prove that the rules of our axiomatic system from §4 hold as meta-theorems for our Hilbert calculus.

Putting Propositions 5.1–5.5 together in a straightforward induction on the length of derivations, we get:

In order to establish the converse of the previous theorem, thereby giving us that $\vdash _{\mathsf {A}}$ and $\vdash _{\mathsf {H}}$ are deductively equivalent, we first establish:

Proof. For 1., we sketch the derivation for R $_{1}$ and leave the analogous derivation for R $_{2}$ to the interested reader:

For 2., first note that R $_{5-8}$ are standard using $\lor $ -Introduction and $\lor $ -Introduction. The arguments for the remaining rules are all analogous applications of Positive Replacement with the formula $\bigwedge (\Gamma \cup \{p\})\lor \xi $ and the corresponding axioms for $\vdash _{\mathsf {A}}$ as in the following example:

A straightforward inductive argument using the previous lemma then gives us:

So, we’ve seen that $\mathsf {H}$ is deductively equivalent to $\mathsf {A}$ . To obtain a calculus for exact entailment on the replete semantics, we simply add the following two rules to $\mathsf {H}$ :

We write $\Gamma \vdash _{\mathsf {H}^{nv\ast }}\phi $ for derivability in the resulting calculus and

as an abbreviation for $\phi \vdash _{\mathsf {H}^{nv\ast }}\psi $ and $\psi \vdash _{\mathsf {H}^{nv\ast }}\phi $ .

For soundness and completeness, we can be quick: Proposition 5.2 carries over to $\vdash _{\mathsf {H}^{nv\ast }}$ without any adjustments:

For Proposition 5.5, just note that R $_{17,18}$ are of the same form as R $_{9,10}$ and thus we can carry over all the arguments building up to the relevant proof:

And finally, again using the fact that R $_{17,18}$ are of the same form as R $_{9,10}$ , we can easily adjust the proof of Lemma 5.7 to include R $_{17,18}$ :

We summarily conclude:

We conclude the section by noting that since $\mathsf {A}$ and $\mathsf {H}$ are deductively equivalent, we shall simply write