Bilateralism, Trilateralism, Multilateralism and Poly-Sequents

Abstract

The paper introduces the formula structure of poly-sequents, allowing the expression of poly-positions: positions with any number of stances, of which bilateralism and trilateralism are special cases. The paper also puts forward the view that s-coherence (strong coherence) of such poly-positions can be defined inferentially, without appealing to their validity under interpretations of the object language.

Appendix A: Multi-valued Logics Based on Poly-sequents

In this Appendix, I very briefly review parts of [2], viewing ${\mathcal C}^{n}$ as as having a counterpart ${\mathcal N}^{n}$, multi-valued (families of) logics. Here, instead o f n stances in a poly-position, we have a collection ${\mathcal V}=\{v_{1},\cdots ,v_{n}\},\ n \ge 2$ truth-values.

Suppose we are given a (propositional) object language L_n, where the connectives of which are are given by truth-tables over ${\mathcal V}$.

Definition A.7 (truth-value assignment)

A truth-value assignment σ is a mapping of the formulas in the object language to ${\mathcal V}$ assigning to each formula φ a truth-value $\sigma {\textbf {[}\! [} \varphi {\textbf {]\!]}} \in {\mathcal V}$. The value of σ for atomic sentences is arbitrary, and the extension to arbitrary formulas respects the truth-tables of the connectives.

Definition A.8 (satisfaction, validity, consequence)

satisfaction: :

A truth-value assignment σ satisfies a poly-sequent π, denoted ⊧_σπ, iff the following holds: If for every 1 ≤ i ≤ n and every φ ∈Γ_i, σ[ [φ]] = v_i, then for some j, 1 ≤ j ≤ n, and some ψ ∈Δ_j, σ[ [ψ]] = v_j. If σ[ [ψ]]≠v_j for every 1 ≤ j ≤ n and every ψ ∈Δ_j we say that σ satisfies π vacuously.

σ satisfies π, denoted ⊧_σπ, iff ⊧_σπ for every π ∈π.

validity: :

π is valid, denoted ⊧π, iff ⊧_σπ for every truth-value assignment σ.

consequence: :

π is a consequence of a set of poly-sequents π, denoted π⊧π, iff for every assignment σ, if ⊧_σπ then ⊧_σπ.

Note that unlike consequence relations in multi-valued logics defined over formulas, the consequence relation over poly-sequents does not depend on any designation of some of the truth-values. All the ruth-values take part in the definition of consequence. Designation is considered when retrievability of specific formula-logics within poly-sequent logics is considered. In [2], some well-known 3-valued logics are shown to be retrievable in ${\mathcal N}^{3}$ (the logical counterpart of ${\mathcal C}^{3}$).

The logics ${\mathcal C}^{n}$ are defined by adding to the structural rules presented above the following logical rules, systematically and uniformly constructed out of the given truth-table for the connectives.

Logical rules:

The guiding lines for the construction are the following, expressed in terms of a generic p-ary operator, say ‘∗’.

(∗ I)::

Such rules introduce a conclusion $\overline {{\Gamma }} : \overline {{\Delta }}_{\overline {k}} | {\Delta }_{k},*(\varphi _{1},\cdots ,\varphi _{p})$.

• In general, if in the truth-table for ‘∗’ the values $v_{i_{j}}$ for φ_j, 1 ≤ j ≤ p, yield the value v_k for ∗ (φ₁,⋯ ,φ_p), then there is a rule

  $$ \frac{\{\overline{{\Gamma}} : \overline{{\Delta}}_{\overline{i_{j}}} | {\Delta}_{i_{j}},\varphi_{j}\},\ 1 \le j \le p}{\overline{{\Gamma}} : \overline{{\Delta}}_{\overline{k}} |{\Delta}_{k},*(\varphi_{1},\cdots,\varphi_{p})}~ (* I_{i_{1},\cdots,i_{p},k}) $$

  (A.68)

  The rule $(* I_{i_{1},\cdots ,i_{p},k})$ has, thus, p premises.

(∗ E)::

Such rules have a major premise $\overline {{\Gamma }} : \overline {{\Delta }}_{\overline {k}} | {\Delta }_{k},*(\varphi _{1},\cdots ,\varphi _{p})$.

• Let I_k be the collection of all $\{{i_{1},\cdots ,i_{p}\}} \subseteq \hat {n}^{p}$ such that in the truth-table for ‘∗’ the values $v_{i_{j}}$ for φ_j yield the value v_k for ∗ (φ₁,⋯ ,φ_p). Then, there is a rule

  $$ \frac{ \overline{{\Gamma}} \!:\! \overline{{\Delta}}_{\overline{k}} | {\Delta}_{k},*(\varphi_{1},\!\cdots,\varphi_{p}) \{ \overline{{\Gamma}}_{\overline{ \{i_{1},\cdots,i_{p}\} } }| {\Gamma}_{i_{1}},\varphi_{1} \!|\! {\cdots} \!| {\Gamma}_{i_{p}},\varphi_{p} : \overline{{\Delta}}\ \textbf{s.t.}\ {\langle} i_{1},\!\cdots,i_{p}{\rangle\!} \in I_{k} \} }{\overline{{\Gamma}} : \overline{{\Delta}}}~(* E_{I_{k}}) $$

  (A.69)

  Thus, the rule $(* E_{I_{k}})$ has |I_k| + 1 premises.

See an important Corollary (A) regarding the E-rules.

Theorem A.1 (soundness and strong completeness)

For everyπand π:

$$ \boldsymbol{{\Pi}} {\models} {\Pi}\ \text{iff}\ \boldsymbol{{\Pi}} {\vdash}_{{\mathcal N}^{n}} {\Pi} $$

(A.70)

The proof is in [2] and not repeated here.

Corollary A (admissibility of the E-rules)

The E-rules of ${\mathcal N}^{n}$areadmissible.

The corollary follows from the fact that the E-rules are not used in the completeness proof. Actually, as shown for the special case of bilateralism above, they are derivable.