Finite Tree-Countermodels via Refutation Systems in Extensions of Positive Logic with Strong Negation

A sufficient condition for an extension of positive logic with strong negation to be characterized by a class of finite trees is given.


Introduction
In this paper, extensions of positive logic with strong negation are studied.We give a sufficient condition (called the reduction property) for the completeness of such a logic with respect to the class of finite trees.The condition (implicitly) involves a certain refutation system employing Mints-style normal forms, and it generalizes some results in [5,6].Our completeness proof is constructive.As an example, the method is applied to the connexive logic C (introduced in [7]), an important non-classical logic.
A frame is a pair W = (W, R), where W is a non-empty set of points (worlds) and R is a reflexive, transitive relation on W .A model is a triple M = (W, V + , V − ), where W is a frame and V + (V − ) is a verification (falsification) valuation (that is, a function assigning to a propositional variable a the set of points at which a is true (false)) that satisfies the persistency condition: ).The verification and falsification relations (|= + and |= − ) between M worlds and formulas are defined as follows. M, We say that a formula ϕ is valid in a frame W (in symbols ϕ ∈ VAL(W)) iff for every model M = (W, V + , V − ) and w ∈ W , we have: M, w |= + ϕ.
The condition for "M, w |= − ϕ → ψ" is that for the logic C.However, it can be replaced with some other standard condition because it is not really used in our general result.
For convenience, we say that a set Φ of formulas is (not) true at w iff so is every ϕ ∈ Φ.
The one-point frame • = ({x 0 }, (x 0 , x 0 )) will be especially important.We also write "m = ( We say that a logic L is complete with respect to a class W of frames (or We assume that every logic defined by an axiom system has a general characterization by a class W L of (possibly infinite) frames obtained by using canonical models (see e.g.[3]).The frames in W L are also referred to as Lframes.
By a generated frame we mean a frame W with a least point x (that is, xRy for all y ∈ W ).
Let W i be an x i -generated frame (1 ≤ i ≤ k).Then x 0 (W 1 , . . ., W k ) is the x 0 -generated frame shown below.(It is obtained from W 1 , . . ., W k by adding a new root, and it resembles Jaśkowski's construction from S. McCall, Polish Logic: 1920-1939.) x 0 In this paper, we focus on the logics satisfying the following condition.
If generated frames W 1 , . . ., W k are L-frames, then so is x 0 (W 1 , . . ., W k ).In that case, we also say that W L is root-closed.The rank r(α) of α is k, which is the size of Δ. (If Δ = ∅ then k = 0.)

L Refutation System
The refutation system R L is defined as follows.
• Refutation axioms: All L special normal forms of rank 0.
• Refutation rules: Normalization rules: Normal-form rules: where α(= Δ; Γ −→ Θ) is an L special normal form of rank k > 0, and We say that a formula ϕ is refutable (in symbols ϕ) iff ϕ is derivable from refutation axioms by refutation rules.And we say that a logic L is R L -complete iff for every normal form α, we have: Either α ∈ L or α.
Remark 4.1.Since Lp ⊆ L, we have: (i) The rule R i and all normalization rules have the property that ϕ → ψ ∈ L, where ψ is the premise and ϕ is the conclusion.

Theorem 5.2. If a logic L has the reduction property, then
Proof.Assume that L has the reduction property.We show, by induction on the rank of a normal form α, that either α ∈ L or α.
Consider the general forms α i , β i (1 ≤ i ≤ k).They are of rank < k.By Definition 5.1(2), each of them is L-reducible to some normal forms of rank < k, which (by the induction hypothesis) are in L or refutable.So, all α i , β i are also in L or refutable (by Definition 4.2).
If some β i is refutable, then so is α (by R i ), so we assume that every β i ∈ L.Then, the formula Δ; Γ; (a i → b i ) −→ Θ is in L as well (by Remark 4.1(iii)).Now, if some α i ∈ L, then the formula Δ; Γ −→ (a i → b i ) is in L, so it follows that α ∈ L, and so we may assume that each α i is refutable.Also, either m(α) Therefore, either α ∈ L or α, as required.

R L Refutation Trees
The derivations in R L can be presented as refutation trees.By an R L refutation tree for a formula ϕ we mean a finite immediate-successor tree RT whose nodes are labelled with formulas and which satisfies the following conditions.(For any node x in RT , ϕ(x) is the label of x).
• ϕ is the label of the origin x 0 .
• If x is an end node, then ϕ(x) is an R L axiom.
• If x 1 , . . ., x n are the immediate successors of a node x, then ϕ(x) is obtained from ϕ(x 1 ), . . ., ϕ(x n ) by an R L rule.

Finite Tree-Countermodels
Recall that a logic is an extension of Lp such that its general characterization W L is root-closed.By modifying some results in [4], we now transform refutation trees into countermodels.
Let RT be an R L refutation tree for a normal form α. We construct a finite tree-countermodel (T , V + , V − ) as follows.
• First, we define the finite, reflexive, transitive tree RT ↑ • by taking the reflexive, transitive closure of the irreflexive, intransitive relation in RT .
• Second, we delete all nodes in RT ↑ • that are obtained by R i or a normalization rule, getting the subtree T of RT ↑ • .Note that every node in T is either an end node or a node obtained by R L , so the label of each node in T is an L special normal form.
x ∈ V + (ϕ) iff ϕ ∈ Γ, and By inspecting the refutation rules, we can see that if y is a successor of x in T , then a literal is in Φ whenever it is in Φ, where ϕ(x) = Φ −→ Ψ and ϕ(y) = Φ −→ Ψ .So, the persitency condition is satisfied.Let M = (T , V + , V − ).Note that for any literal a, we have: M, x 0 |= + a iff m(α) |= + a. • Finally, for any node x in RT , we define its world x in T as follows.If x is an end node or obtained by R L , then x = x; and if x is obtained from x 1 by R i or a normalization rule, then x = x 1 .
Lemma 7.1.If x is a node in RT , then ϕ(x) is not true at x .
Proof.(by induction on the number n x of nodes in the subtree of RT generated by x).
(2.1) ϕ(x) is obtained from ϕ(x 1 ) by R i or a normalization rule.Then x = x 1 .Since n x1 < n x , by the induction hypothesis, ϕ(x 1 ) is not true at x 1 .Hence ϕ(x 1 ) is not true at x .So, by Remark 4.1(i), ϕ(x) is not true at x (because ϕ(x) → ϕ(x 1 ) is true everywhere).(2.2) ϕ(x) is obtained from ϕ(x 1 ), . . ., ϕ(x k ) by R L .Then ϕ(x) = Δ; Γ −→ Θ is an L special normal form of rank k > 0, and ϕ( and n xi < n x for all i.By the induction hypothesis, ϕ(x i ) is not true at x i (1 ≤ i ≤ k), so Δ; Γ −→ (a i → b i ) is not true at x i (1 ≤ i ≤ k) (by Remark 4.1(ii)).Hence, every a i → b i is not true at x i and Δ; Γ is true at each x i .So, every a i → b i is not true at x (because x precedes every x i ).Also, Δ; Γ is true at all x i , so Δ is true at x and every a → γ ∈ Γ is true at x (see (1) above).Of course, every literal in Γ is true and Θ is not true at x .Therefore ϕ(x) is not true at x , as required.α ∈ L for some normal form α. So, by Theorem 5.2, α is refutable.Hence, by Lemma 7.1, α is not true at some point in some model based on a finite, reflexive, transitive tree.Thus, ϕ is not true there either (because, by Corollary 4.3, ϕ → α is true everywhere).Therefore, ϕ is not valid in some finite, reflexive, transitive tree, which gives the result.
are literals, Γ is a finite set of formulas of the kind: a or a → b or a → (b → c) or a → b ∨ c, where a, b, c are literals, and Θ is a finite, nonempty set of literals.

Theorem 7 . 2 .
If a logic L has the reduction property, then L is characterized by the class of finite, reflexive, transitive trees.Proof.Assume that L has the reduction property, and assume that ϕ ∈ L.Then, (by Definition 5.1(1)) its general form α ϕ ∈ L, and (by Definition 5.1(2))