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 $\Gamma \vdash \phi $ and (and similarly for the $nv\ast $ variants).

6. Sequent calculus

In this section, we present a sequent calculus for exact entailment. We begin by reviewing the calculus presented by Fine and Jago [Reference Fine and Jago17, pp. 551–556].

What sets the calculus apart from ordinary sequent calculi is that it operates on sequents of the form $\{\Gamma _1, \ldots , \Gamma _n\}\Rightarrow \Delta $ . That is, a sequent in the Fine–Jago calculus has a (finite) set of sets of formulas on the left and a single set of formulas on the right. The intended reading of a sequent is $\bigwedge \Gamma _1, \ldots , \bigwedge \Gamma _n\vDash \bigwedge \Delta $ . That is, the $\Gamma _{i}$ ’s and $\Delta $ are read conjunctively, while $\{\Gamma _{1},\ldots ,\Gamma _{n}\}$ is read distributively. This makes the Fine–Jago calculus akin to a single-conclusion sequent calculus. In the following, we’ll use $\mathcal {X},\mathcal {Y}, \mathcal {Z}, \ldots $ as variables for finite sets of sets of formulas. So, we can represent the form of a sequent as $\mathcal {X}\Rightarrow \Delta $ .Footnote ²² To cut down on set-braces, we use “;” to separate the members of $\mathcal {X}$ and we use “,” to separate the members of $\Delta $ and of the $\Gamma \in \mathcal {X}$ . So, for example, $\phi ,\psi ;\theta \Rightarrow \gamma ,\delta $ is shorthand for $\{\{\phi ,\psi \}, \{\theta \}\}\Rightarrow \{\gamma ,\delta \}$ .

Note that the following structural rules are absorbed in the notation:

From a proof-theoretic perspective, this is slightly unsatisfactory since it gives us less control over the structural aspects of the calculus. Semantically, however, the issue is immaterial: the validity of $(W,L), (W,R), (Ex, L)$ , and $(Ex,R)$ follows immediately from the idempotence and commutativity of conjunction and the validity of $(W;L)$ and $(Ex;L)$ follows from the definition of exact entailment.

The Fine–Jago calculus, $\mathsf {G}_{FJ}$ , has the following axioms and rules:

Logical Rules

We write $\mathcal {X}\vdash _{\mathsf {G}_{FJ}}\Delta $ to say that the sequent $\mathcal {X}\Rightarrow \Delta $ is derivable in the calculus.

We say that a sequent $\mathcal {X}\Rightarrow \Delta $ is valid, symbolically $\mathcal {X}\vDash \Delta $ , iff $\{\bigwedge \Gamma :\Gamma \in \mathcal {X}\}\vDash \bigwedge \Delta $ . In a rule, we call the sequents above the inference line “upper sequents” and the one below the “lower sequent.” A rule is sound iff its lower sequent is valid whenever its upper sequents are. Fine and Jago [Reference Fine and Jago17, theorem 9.2, p. 554, and theorem 9.6, p. 555] establish:

Since their proof doesn’t make use of $Cut_{\mathsf {G}_{FJ}}$ , Fine and Jago [Reference Fine and Jago17, theorem 9.7, p. 556] infer as a corollary that their calculus has the Cut-elimination property, i.e., if $\mathcal {X}\vdash _{\mathsf {G}_{FJ}}\Delta $ , then the sequent is derivable without any applications of Cut. We shall now investigate the calculus in more detail from a proof-theoretic perspective.

First, note that the rule Weak cannot be eliminated from the calculus: without the rule we already couldn’t derive $p;q\Rightarrow p$ .Footnote ²³ Having a weakening rule around in a sequent calculus is not ideal since it complicates proof searches (though it is, of course, strictly speaking not problematic). But this is an easy fix: just take as axioms all sequents of the form $\mathcal {X},\Gamma \Rightarrow \Gamma $ . Call the resulting calculus $\mathsf {G}_{FJ'}$ . Then it’s straightforward to see:

Since all axioms of $\mathsf {G}_{FJ}$ are also axioms of $\mathsf {G}_{FJ'}$ (just let $\mathcal {X}=\emptyset $ ), the two calculi are clearly deductively equivalent.

It turns out that once we’ve eliminated Weak, we can straightforwardly eliminate $(,1)$ as well!

Proof. We show that for every derivation with exactly one application of $(,1)$ , there exists a derivation without $(,1)$ . The theorem then follows by a simple inductive argument.

Without loss of generality, we can focus on derivations without Weak where the application of $(,1)$ is the last step in the derivation:

We prove the claim by an induction on the height of this derivation, that is on $max(|\mathcal {D}_1|, |\mathcal {D}_2|)$ .

For the base case, note that if $max(|\mathcal {D}_1|, |\mathcal {D}_2|)=0$ , then both premises of $(,1)$ are axioms, that is, the derivation looks like this:

But then also the conclusion is an axiom.

We need to go through the possible last rules of $\mathcal {D}_{1}$ and of $\mathcal {D}_{2}$ . Though there are many such possibilities, they reduce to a manageable amount of cases by relying on the duality of the $\neg $ -rules and the positive rules. Here we only show the case where the left upper sequent was derived via $(,2)$ and the right upper sequent via $(\neg \neg R)$ to illustrate the idea. Consider:

This derivation can be transformed into:

Since $max(|\mathcal {D}_1^1|,|\mathcal {D}^{\prime }_2|), max(|\mathcal {D}_1^2|,|\mathcal {D}^{\prime }_2|)< max(|\mathcal {D}_1|, |\mathcal {D}_2|)$ , we can derive the conclusions of the $(,1)$ -applications without $(,1)$ by the induction hypothesis, giving us a $(,1)$ -free derivation of the final sequent. We leave verifying the remaining cases to the interested reader.

Note that if we hadn’t absorbed $(Weak)$ , the argument wouldn’t straightforwardly go through, since $(,1)$ requires shared contexts in the premises and $(Weak)$ applied backwards might delete formulas from the context.

This leaves us with $(,2)$ as the last remaining structural rule. Unfortunately, however, it turns out that $(,2)$ cannot be eliminated from $\mathsf {G}_{FJ'}$ . To see this, consider the following derivation:

A very simple inductive argument on the height of derivations without $(,2)$ establishes that $p;p,q,r\Rightarrow p,q$ is not derivable without the rule. Given Propositions 6.2 and 6.3, we can focus our attention on derivations without $(Weak)$ and $(,1)$ . Now, clearly $p;p,q,r\Rightarrow p,q$ is not a logical axiom. And since all the remaining rules of $\mathsf {G}_{FJ'}$ introduce connectives and $p;p,q,r\Rightarrow p,q$ is connective free, the rules couldn’t have been used to derive the sequent. Since $\mathsf {G}_{FJ}$ and $\mathsf {G}_{FJ'}$ are deductively equivalent, the sequent is not derivable in the original calculus without $(,2)$ either. In short, $(,2)$ is not eliminable $\mathsf {G}_{FJ}$ .

In fact, we can use the above derivation to show that $\mathsf {G}_{FJ}$ doesn’t enjoy the subformula property, i.e., it’s not the case that every formula that occurs in a derivation is a subformula of a formula in the derived sequent. Consider:

The formula $r\lor s$ , which we first introduce via $(\lor R_1)$ only to immediately delete it via $(,2)$ , does not occur as a subformula in the derived sequent $p;p,q,r\Rightarrow p,q$ . Hence, the subformula property fails.

An unfortunate consequence of this observation is that the Fine–Jago calculus doesn’t allow for proof searches. In general, the idea of a proof search algorithm is to consider all the possible ways in which a given sequent could have been derived. In a calculus that absorbs all the structural rules and enjoys the subformula property, the search-space is finite. To see this, note that by the subformula property, we can restrict our attention to derivations involving only subformulas of formulas in the sequent. Since the structural rules are absorbed, the only way in which a sequent can be derived is either as an axiom or by means of a left or right rule applied to the immediate subformulas of a formula in the sequent. Putting these two observations together, it follows that the search space is finite. In fact, the height of a derivation of a sequent is bounded by the maximum complexity of the formulas in the sequent.

In the presence of $(,2)$ , however, this observation fails. Since $(,2)$ deletes formulas from a derivation, a derivation may involve formulas which don’t occur in the final sequent. In fact, since we can always first introduce some formula and then delete it via a (superfluous) application of $(,2)$ , the height of derivations of a sequent is not finitely bound. Since there are sequents that can only be derived using $(,2)$ , this means that a search algorithm may fail to terminate.

It’s worth thinking about the role that $(,2)$ plays in the Fine–Jago calculus. Essentially what it allows us to do is to derive instances of $\land $ -Convexity. To see this, suppose that $\Gamma \subseteq \Delta \subseteq \Sigma $ . We get the following derivation:

Note especially that $\Gamma ;\Sigma \Rightarrow \Gamma $ and $\Gamma ;\Sigma \Rightarrow \Delta \cup (\Sigma \setminus \Delta )$ are axioms. Note further that no rules other than $(,2)$ and $(\land L)$ and $(\land R)$ are used in this derivation. Since the use of $(\land L)$ and $(\land R)$ in the derivation is simply to make the reading of the sequent $\Gamma ,\Sigma \Rightarrow \Delta $ explicit, this indicates that the role of $(,2)$ is precisely to derive $\land $ -Convexity.

At this point, we have a couple of options if we wish to develop a sequent calculus with the subformula property. The first would be to rely on a formulation of $(,2)$ that doesn’t eliminate formulas. An example of such a rule would be

The idea is that rather than deleting formulas on the right, we can just introduce formulas on the left. It’s straightforward to see that $(,2)$ and $(,2')$ are inter-derivable:

• • $(,2)\Rightarrow (,2')$

  

• • $(,2')\Rightarrow (,2)$

  

Since $(,2')$ doesn’t eliminate formulas, there would no longer be formula-deleting rules were we to replace $(,2)$ with $(,2')$ . Consequently, the resulting calculus would indeed have the subformula property—as desired.

While we do obtain a calculus with the subformula property in this way, from a proof-theoretic perspective, the resulting calculus is still not ideal. It’s easily seen, via an analogous argument as for $(,2)$ , that the rule $(,2')$ would not be eliminable. Just consider the derivation

For the same reasons why this sequent wasn’t derivable without $(,2)$ , it is not derivable without $(,2')$ . Though $(,2')$ is not problematic in itself, just like $(Weak)$ , the rule is not ideal for proof-searches. We shall now develop a calculus that absorbs all structural rules. It turns out that we can use the same idea as in the case of $(Weak)$ : we can absorb $(,2)$ on the axiomatic level.

To further facilitate proof searches, we shall move to a multi-conclusion sequent calculus, the benefits of which we shall reap in our completeness proof. Correspondingly, our sequents shall now be of the form $\mathcal {X}\Rightarrow \mathcal {Y}$ . The notational conventions for sequents carry over to these sequents in a straightforward way. The intended reading of a sequent of the form $\Gamma _1;\ldots ;\Gamma _n\Rightarrow \Delta _1;\ldots ;\Delta _m$ is that $\bigwedge \Gamma _1,\ldots ,\bigwedge \Gamma _n\vDash \bigwedge \Delta _1\lor \cdots \lor \bigwedge \Delta _m$ . That is, we read sets of formulas conjunctively as before, but the reading of $\mathcal {X}$ and $\mathcal {Y}$ in $\mathcal {X}\Rightarrow \mathcal {Y}$ is different: while the former is read distributively, the latter is read disjunctively.Footnote ²⁴

In our new setting, the following structural rules are absorbed in notation:

The validity of the new right rules is straightforwardly justified using the idempotence, associativity, and commutativity of disjunction.

In line with the intended reading of our sequents, we say that $\mathcal {X}\Rightarrow \mathcal {Y}$ is valid iff $\{\bigwedge \Gamma :\Gamma \in \mathcal {X}\}\vDash \bigvee \{\bigwedge \Delta :\Delta \in \mathcal {Y}\}$ . What our calculus will do is to derive all valid sequences from valid sequences only involving literals, while essentially following the construction of the formulas from literals.

To motivate our choice of axioms, we shall begin with a couple of semantic observations. First, note that if $\Gamma \subseteq \Lambda $ , then $|\bigwedge \Gamma |^{+}_{\mathfrak {M}}=\{\Gamma \}$ , as is easily seen via (Sem- ${\land }^{+}$ ). Let’s further define $cl(\mathcal {X})$ , for $\mathcal {X}\subseteq \wp (\mathcal {L})$ , to be the closure of $\mathcal {X}$ under $\cup $ , i.e., $cl(\mathcal {X})=\bigcap \{\mathcal {Z}:\mathcal {X}\subseteq \mathcal {Z}\text { and whenever }X,Y\in \mathcal {Z}\text {, then }X\cup Y\in \mathcal {Z}\}$ . Relying on the previous observation and (Sem- ${\lor }^{+}$ ), we can show that if $\mathcal {Y}\subseteq \wp (\Lambda )$ , then $|\bigvee _{\Delta \in \mathcal {Y}}\bigwedge \Delta |^{+}_{\mathfrak {M}}=cl(\mathcal {Y})$ using a straightforward induction on cardinality of $\mathcal {Y}$ . Using Theorem 3.4, we’re now in a position to prove:

Note that the condition that one might expect in light of the Fine–Jago characterization theorem (Theorem 3.4), viz. that there be $\Gamma \in \mathcal {X}$ and $\Delta \in \mathcal {Y}$ with $\Gamma \subseteq \Delta \subseteq \bigcup \mathcal {X}$ , will not suffice to characterize all valid sequents. Take, for example, $\{p,q\}\Rightarrow \{p\}, \{q\}$ . This sequent is clearly valid since $p\land q\vDash p\lor q$ (cf. Proposition 4.7), but it doesn’t satisfy the above constraint. The issue is, of course, that for an application of Theorem 3.4, we need to recruit our $\Delta $ from $|\bigvee \{\bigwedge \Delta :\Delta \in \mathcal {Y}\}|_{\mathfrak {M}}^{+}$ , which is identical to $cl(\mathcal {Y})$ and not $\mathcal {Y}$ . In our example, we can indeed pick $\{p,q\}\in cl(\{\{p\},\{q\}\})$ as our $\Delta $ , which witnesses the validity of $\{p,q\}\Rightarrow \{p\},\{q\}$ since trivially $\{p,q\}\subseteq \{p,q\}\subseteq \{p,q\}$ .

Now it’s precisely the sequents that satisfy the constraint of Lemma 6.4 that will be the axioms of our calculus:

Logical Axioms

$$\begin{align*}\mathcal{X}\Rightarrow\mathcal{Y},\end{align*}$$

where $\mathcal {X},\mathcal {Y}\subseteq \wp (\Lambda )$ and $\mathcal {X}\vDash \mathcal {Y}$ .

Note that by Lemma 6.4, the condition that characterizes our axioms, $\mathcal {X}\vDash \mathcal {Y}$ for $\mathcal {X},\mathcal {Y}\subseteq \wp (\Lambda )$ , is really just shorthand for the syntactic condition there are $\Gamma \in \mathcal {X}$ and $\Delta \in cl(\mathcal {Y})$ with $\Gamma \subseteq \Delta \subseteq \bigcup \mathcal {X}$ .

Our calculus has no structural rules and the following logical rules:

Logical Rules

We shall refer to this calculus as $\mathsf {G}$ and correspondingly denote derivability by $\mathcal {X}\vdash _{\mathsf {G}}\mathcal {Y}$ .

First, soundness:

There are several routes to completeness, but our calculus allows for a particularly pleasing proof, which relies on the following lemma:

Using this insight, we are now in the position to show:

Proof. The proof makes use of a measure of complexity tracking the construction from literals:

• • $c_\Lambda (\lambda )=0$ .

• • $c_\Lambda (\phi \circ \psi )=c_\Lambda (\phi )+c_\Lambda (\psi )+1$ for $\circ =\lor ,\land $ .

• • $c_\Lambda (\neg \neg \phi )=c_\Lambda (\phi )+1$ .

• • $c_\Lambda (\neg (\phi \circ \psi ))=c_\Lambda (\neg \phi )+c_\Lambda (\neg \psi )+1$ for $\circ =\lor ,\land $ .

We proceed by induction on $c_{\Lambda }(\mathcal {X}\Rightarrow \mathcal {Y}):=\sum _{\phi \in \bigcup (\mathcal {X}\cup \mathcal {Y})}c_\Lambda (\phi )$ .

For the base case, note that $\sum _{\phi \in \bigcup (\mathcal {X}\cup \mathcal {Y})}c_\Lambda (\phi )$ iff $c_\Lambda (\phi )=0$ for all $\phi \in \bigcup (\mathcal {X}\cup \mathcal {Y})$ , i.e., all these $\phi $ ’s are literals. But this just means that $\mathcal {X},\mathcal {Y}\subseteq \Lambda $ and so, by Lemma 6.4, $\mathcal {X}\vDash \mathcal {Y}$ iff it’s an initial sequent, and thus provable.

For the induction step, assume that $\sum _{\phi \in \bigcup (\mathcal {X}\cup \mathcal {Y})}c_\Lambda (\phi )=n$ . Now pick any formula $\phi \in \cup (\mathcal {X}\cup \mathcal {Y})$ with $c_\Lambda (\phi )\neq 0$ (such a $\phi $ will exist by the assumption that $c_{\Lambda }(\mathcal {X}\Rightarrow \mathcal {Y})=n>0$ ). What we’ll do is to backwards apply a suitable rule of our calculus to get valid (Lemma 6.6) sequents of a lower measure which by the induction hypothesis will be provable. Then we simply apply the rule forwards to get a prove of our sequent.

To illustrate, assume that $\phi =\neg (\phi _{1}\land \phi _{2})$ and $\Gamma \cup \{\phi \}\in \mathcal {X}$ , i.e., our sequent is of the form $\mathcal {X}';\Gamma ,\neg (\phi _{1}\lor \phi _{2})\Rightarrow \mathcal {Y}$ . Since $\mathcal {X}';\Gamma ,\neg (\phi _{1}\lor \phi _{2})\vDash \mathcal {Y}$ , applying Lemma 6.6 with $(\neg {\lor }L_{\mathsf {G}})$ , we get that $\mathcal {X}';\Gamma ,\neg \phi _{1}\vDash \mathcal {Y}$ and $\mathcal {X}';\Gamma ,\neg \phi _{2}\vDash \mathcal {Y}$ . Since $c_{\Lambda }(\mathcal {X}';\Gamma ,\neg \phi _{i}\vDash \mathcal {Y})=n-1$ , our induction hypothesis gives us $\mathcal {X}';\Gamma ,\neg \phi _{i}\vdash _{\mathsf {G}}\mathcal {Y}$ with witnessing derivations $\mathcal {D}_{i}$ . We get

We leave verifying the remaining cases to the interested reader.

Note that our completeness proof is essentially just a convenient proof-search method, based on invertability. Indeed, we can look at the method described in the proof as a concrete implementation (and slight simplification) of the decision procedure sketched by Fine and Jago [Reference Fine and Jago17, p. 547]: each leaf in our proof-search corresponds to a selection function for our initial sequent, and the leaves are axioms just in case they satisfy the condition of the Fine–Jago theorem.

Note also that our calculus absorbs the relevant structural rules:

Theorem 6.9 (Admissibility of the structural rules)

The following rules are admissible in the sense that if their upper sequents are derivable, then their lower sequent is derivable, too:

It is, in fact, also possible to prove the admissibility of the structural rules using genuinely proof-theoretic methods. The idea is, in each case, that applications of the structural rules can be pushed to the axioms (by means of an induction on the height of derivations) where they are very easily absorbed. Note, in particular, that this also works for $(,2_{\mathsf {G}})$ , which wasn’t the case with $(,2)$ in the original Fine–Jago calculus. Since the arguments are more or less standard (modulo our new notation), we shall leave verifying these claims to the interested reader.

We shall conclude this section by extending $\mathsf {G}$ to the replete semantics. Luckily, this is rather simple: we just need to add certain logical axioms. First, we straightforwardly extend the notion of validity to the replete semantics by setting $\mathcal {X}\vDash _{nv\ast }\mathcal {Y}$ iff $\{\bigwedge \Gamma :\Gamma \in \mathcal {X}\}\vDash _{nv\ast }\bigvee \{\bigwedge \Delta :\Delta \in \mathcal {Y}\}$ .

Proof. The proof is via the Fine–Jago theorem for the replete semantics (Theorem 3.6) using some semantic facts. Remember from the proof of Lemma 6.4, that if $\Gamma \subseteq \Lambda $ , then $|\bigwedge \Gamma |^{+}_{\mathfrak {M}}=\{\Gamma \}$ . Since $|\bigwedge \Gamma |^{+}_{\mathfrak {M}}$ is a singleton set, it follows that $(|\bigwedge \Gamma |^{+}_{\mathfrak {M}})_{\ast }=|\bigwedge \Gamma |^{+}_{\mathfrak {M}}$ . It follows that there is only one selection function for $\{(|\bigwedge \Gamma |^{+}_{\mathfrak {M}})_{\ast }:\Gamma \in \mathcal {X}\}$ , the one with $f((|\Gamma |^{+}_{\mathfrak {M}})_{\ast })=\Gamma $ . The main difference to before is that we need to consider the convex closure $(|\bigvee _{\Delta \in \mathcal {Y}}\bigwedge \Delta |^{+}_{\mathfrak {M}})_{\ast }$ of $|\bigvee _{\Delta \in \mathcal {Y}}\bigwedge \Delta |^{+}_{\mathfrak {M}}=\{\Delta _{1}\cup \Delta _{2}:\Delta _{i}\in \mathcal {Y}\}$ . But it’s easily checked that

$$\begin{align*}\left(|\bigvee_{\Delta\in\mathcal{Y}}\bigwedge\Delta|^{+}_{\mathfrak{M}}\right)_{\ast}=\{\Delta:\exists\Sigma\in\mathcal{Y}\text{ with }\Sigma\subseteq\Delta\subseteq\bigcup\mathcal{Y}\}.\end{align*}$$

From these observations, our claim follows by Theorem 3.6.

We define the calculus $\mathsf {G}_{nv\ast }$ to be the calculus where the axioms are all the valid sequents $\mathcal {X}\Rightarrow \mathcal {Y}$ with $\mathcal {X},\mathcal {Y}\subseteq \wp (\Lambda )$ such that $\mathcal {X}\vDash _{nv\ast }\mathcal {Y}$ . The rules remain the same as in $\mathsf {G}$ .

To obtain soundness and completeness, we observe that our arguments for the validity and invertability of our rules go through on the replete semantics, as well.

This means virtually the same completeness proof is available for $\mathsf {G}_{nv\ast }$ as for $\mathsf {G}$ :

We conclude with an example derivation of the crucial direction of ${\lor }/{\land }$ -Distribution in $\mathsf {G}_{nv\ast }$ since it’s both an instructive example for our proof-search method and for the axiom choice in $\mathsf {G}_{nv\ast }$

To see that the sequents at the top are all axioms of $\mathsf {G}_{nv\ast }$ note from left to right that: $\{p\}\subseteq \{p\}\subseteq \{p,q,r\}$ , $\{p\}\subseteq \{p,r\}\subseteq \{p,q,r\}$ , $\{p\}\subseteq \{q,p\}\subseteq \{p,q,r\}$ , and $\{q,r\}\subseteq \{q,r\}\subseteq \{p,q,r\}$ .