Modal System L5: A Unified Logical Framework

• Modal System L5 is a canonical logic that combines intuitionistic, classical, and modal reasoning with positive and negative introspection for unified expressivity.
• It employs a single-sorted propositional modal language and integrates axioms from IPC, CPC, and modal postulates to extend traditional logical systems.
• L5 offers robust algebraic semantics via Heyting algebras with the disjunction property and Kripke semantics using partially ordered frames, impacting logic programming and theoretical computer science.

The modal system L5 is a canonical example of a logic combining intuitionistic, classical, and modal reasoning, enhanced with positive and negative introspection principles to achieve both algebraic and Kripke completeness. Introduced as an extension of a base system called L, L5 is designed to address limitations in the expressivity of Kripke semantics by the addition of key modal postulates. L5 serves as a powerful tool for the analysis of truth in intermediate propositional logics and provides a unified approach to both algebraic and relational semantics for a large class of logics, including logics of interest in theoretical computer science.

1. Syntax and Language

L5 employs a single-sorted propositional modal language. The signature consists of a countable set of propositional variables $x_0, x_1, x_2, \ldots$; binary connectives $\wedge, \vee, \to$; a nullary constant $\bot$ (with $\top$ defined as $\bot \to \bot$); and a unary modal operator $\Box$. The set of formulas $\mathrm{Fm}$ is defined in the usual way, with $\mathrm{Fm}_0$ denoting the purely propositional (non-modal) fragment.

The system introduces standard abbreviations: $\neg\varphi := \varphi \to \bot$ and $$\varphi \leftrightarrow \psi := (\varphi \to \psi)\wedge(\psi \to \varphi)$$. A key extension is the propositional identity connective $$\varphi = \psi := \Box(\varphi \to \psi) \wedge \Box(\psi \to \varphi),$$ which functions as a syntactic surrogate for identity between propositions.

2. Axiom Schemas and Inference Rules

L5 is axiomatized by the following schemas (for all formulas $\varphi$ and $\psi$):

1. Intuitionistic Propositional Calculus (IPC): All theorems of IPC.
2. Classical Propositional Calculus (CPC): All theorems of CPC. Alternatively, this is equivalent to adding tertium non datur, $\varphi \vee \neg \varphi$.
3. Distribution of Strict Implication (K): $(\Box\varphi \to \psi)\to \Box(\varphi \to \psi)$.
4. Disjunction Property (DP): $$\Box(\varphi \vee \psi) \to (\Box\varphi \vee \Box\psi)$$.
5. Positive Introspection: $\Box\varphi \to \Box\Box\varphi$.
6. Negative Introspection: $\neg\Box\varphi \to \Box\neg\Box\varphi$.
7. Propositional Identity Substitution (SP): For every formula $\chi(x)$,

$$(\varphi = \psi) \to \bigl(\chi(x)[x:=\varphi] = \chi(x)[x:=\psi]\bigr).$$

The inference rules are:

• Modus Ponens (MP): From $\varphi$ and $\varphi\to\psi$ infer $\psi$.
• Axiom Necessitation (AN): From an axiom instance infer its boxed form $\Box$.

L5 strictly includes both IPC and CPC and extends them uniformly with the modal, introspection, and disjunction postulates (Lewitzka, 2015).

3. Algebraic Semantics

L5 admits a robust algebraic semantics. An L5-model is a tuple $$\mathcal{M} = (M, \mathit{TRUE}, \wedge, \vee, \to, \bot, \top, f_\Box)$$, where:

1. $(M, \wedge, \vee, \to, \bot, \top)$ is a Heyting algebra.
2. $\mathit{TRUE} \subseteq M$ is a designated ultrafilter, i.e., nonempty, proper, closed under meets, and prime.
3. Disjunction Property (DP): For all $m, n\in M$, if $m\vee n\in \mathit{TRUE}$, then $m\in \mathit{TRUE}$ or $n\in \mathit{TRUE}$.
4. The modal operator $f_\Box: M\to M$ satisfies:
   • (i) $f_\Box(m)\leq m$;
   • (ii) $f_\Box(m\to n) \leq (f_\Box m \to f_\Box n)$;
   • (iii) $f_\Box(m\vee n) \leq f_\Box m \vee f_\Box n$;
   • (iv) $f_\Box(m)=\top$ iff $m=\top$.

A valuation $v:V\to M$ extends recursively, with $\Box\varphi$ interpreted as $f_\Box(v(\varphi))$. Truth for $(\mathcal{M}, v)$ is $v(\varphi) \in \mathit{TRUE}$. Consequence $\Gamma\models_{\mathbf{L5}} \varphi$ holds if, in every L5-model, every valuation assigning all formulas in $\Gamma$ to $\mathit{TRUE}$ also assigns $\varphi$ to $\mathit{TRUE}$.

Algebraic completeness: For all $\Gamma \cup \{\varphi\} \subseteq \mathrm{Fm}$,

$$\Gamma \vdash_{\mathbf{L5}} \varphi \iff \Gamma \models_{\mathbf{L5}} \varphi.$$

(Lewitzka, 2015)

4. Kripke Semantics

The Kripke-style semantics for L5 is based on L5-frames $(W, R)$, in which $W$ is a non-empty set of worlds partially ordered by $R$ with a least element $w_\bot$, and every $R$-chain has an upper bound. A valuation $g:V \to \mathcal{P}(W)$ is monotonic: $wRw'$ and $w\in g(x)$ imply $w'\in g(x)$.

Truth relations are defined by induction:

• $w \models x$ iff $w \in g(x)$.
• $w \models \varphi \wedge \psi$ iff $w \models \varphi$ and $w \models \psi$.
• $w \models \varphi \vee \psi$ iff $w \models \varphi$ or $w \models \psi$.
• $w \models \varphi \to \psi$ iff $$\forall w' \succeq w: w' \models \varphi \implies w' \models \psi$$.
• $w \models \Box\varphi$ iff $w_\bot \models \varphi$.

A local consequence relation $\Gamma \models^{\mathrm{Kr}}_{\mathbf{L5}} \varphi$ holds if, in every L5-frame, for every maximal world $w_{\max}$, whenever $w_{\max} \models \psi$ for all $\psi \in \Gamma$, then $w_{\max} \models \varphi$.

Kripke completeness: For all $\Gamma \cup \{\varphi\} \subseteq \mathrm{Fm}$,

$$\Gamma \vdash_{\mathbf{L5}} \varphi \iff \Gamma \models^{\mathrm{Kr}}_{\mathbf{L5}} \varphi.$$

Positive and negative introspection are both valid, as $\Box\varphi$ refers to the bottom world, ensuring $\Box\varphi \to \Box\Box\varphi$ and $\neg\Box\varphi \to \Box\neg\Box\varphi$ hold (Lewitzka, 2015).

5. Parametrized Logics: $L5(I)$ and Intermediate Logics

Given any intermediate propositional logic $I$ ($\mathrm{IPC} \subseteq I \subseteq \mathrm{CPC}$), axiomatized by $\mathrm{IPC}$ plus a set of propositional axioms $\Delta$, the logic $\mathbf{L5}(I)$ is defined replacing all IPC-theorems in L5 with all $I$-theorems.

Principal properties include:

• $\mathbf{L5}(I)$ is a conservative extension of CPC.
• There is an embedding $p\mapsto \Box p$ of $I$ into $\mathbf{L5}(I)$, giving: $I \vdash \varphi$ iff $\mathbf{L5}(I) \vdash \Box\varphi$.
• The algebraic semantics: Algebras for $\mathbf{L5}(I)$ are non-trivial Heyting algebras with DP validating $I$.
• The Kripke semantics: Frames for $\mathbf{L5}(I)$ are those where $\Delta$ holds at the bottom world.

This framework yields completeness for $I$ by focusing on Heyting algebras with the disjunction property, and the method generalizes to a range of intermediate logics (Lewitzka, 2015).

6. Notable Examples and Computer Science Applications

Logic of Here-and-There (HT)

• Axiomatized as $\mathrm{IPC}+\{p \vee (p\to q) \vee \neg q\}$.
• L5(HT)-semantics restricts reduct Heyting algebras to at most three elements, corresponding to the “here-and-there” interpretation.
• Completeness w.r.t. finite Kripke frames of size $\leq 2$ (here below there).
• Application: In logic programming, HT-equivalence characterizes strong equivalence of programs.

Gödel–Dummett Logic (G)

• Axiomatized as $\mathrm{IPC}+\{(p\to q)\vee(q\to p)\}$.
• Reduct algebras are the non-trivial linearly ordered Heyting algebras.
• Frames are linearly ordered Kripke frames.
• Provides a concise algebraic proof of Dummett's completeness theorem.

Jankov Logic (KC)

• Defined via $\mathrm{IPC}+\{\neg p \vee \neg\neg p\}$.
• Algebras are “KC-algebras”: non-trivial Heyting algebras with DP in which elements above $\bot$ have infimum above $\bot$, equivalent to having a unique ultrafilter.
• Kripke frames: finite rooted frames with a single maximal world.

For each, completeness is established by restricting attention to Heyting algebras with DP, demonstrating the uniformity and power of the approach via $\mathbf{L5}(I)$ (Lewitzka, 2015).

7. Structural and Methodological Contributions

The central significance of L5 lies in the unification and simplification of completeness proofs for a wide class of intermediate logics. By reducing completeness (both algebraic and Kripke) for various propositional logics to the existence of suitable Heyting algebras with the disjunction property and suitably constructed Kripke frames, L5 provides a modal framework in which strong theorems of logic—including results due to Hosoi, Dummett/Horn, and Jankov—are obtained transparently.

A plausible implication is that the L5 methodology offers a systematic modal pathway for analyzing the algebraic and semantic structure of non-classical logics encountered in computer science, particularly those relevant for logic programming and stable semantics. The explicit substitution principle for propositional identity further facilitates formal manipulation and comparison of formulas in these settings.

Reference: "Combining intermediate propositional logics with classical logic" (Lewitzka, 2015).