KRM₃ - RM₃ with K Modal

---

Semantics

A KRM₃ frame comprises the interpretation of sentences and predicates at a world. A KRM₃ model comprises a non-empty collection of KRM₃ frames, a world access relation \(R\), and a set of constants (the domain).

The semantics for predication, quantification, and truth-functional operators are the same as RM₃.

Modal Operators

Possibility

The value of a necessity sentence ◻A at \(w\) is the maximum value of its operand \(A\) at each world \(w'\) such that \(\langle\ w, w'\ \rangle\) is in \(R\).

Necessity

The value of a necessity sentence ◻A at \(w\) is the minimum value of its operand \(A\) at each world \(w'\) such that \(\langle\ w, w'\ \rangle\) is in \(R\).

Consequence

Logical Consequence is defined just as in KFDE.

Logical Consequence

C is a Logical Consequence of A iff all models where A has a desginated value at \(w_0\) are models where C also has a designated value at \(w_0\).

Tableaux

Nodes

Nodes for many-valued modal tableaux come in two types:

• A node consisting of a sentence, a designation marker (⊕ or ⊖), and a world.

• An access node like \(w_1Rw_2\), indicating that the pair \(\langle\ w_1, w_2\ \rangle\) is in the access relation \(R\).

Trunk

To build the trunk for an argument, add a designated node with each premise at world \(w_0\), followed by an undesignated node with the the conclusion at world \(w_0\).

To build the trunk for the argument A₁ ... A_n ∴ B write:

A₁ ⊕, w₀

⋮

A_n ⊕, w₀

B ⊖, w₀

Closure

The closure rules are the same as RM₃.

⊗LPGap Closure

A ⊖, w₀

⋮

¬A ⊖, w₀

⊗

⊗FDEDesignation Closure

A ⊕, w₀

⋮

A ⊖, w₀

⊗

Rules

Non-modal rules for KRM₃ are exactly like their RM₃ counterparts, with the addition of carrying over the world marker from the target node(s).

Modal Operator Rules

◇ Rules

◇⊕KFDEPossibility Designated

◇A ⊕, w₀

⋮

A ⊕, w₁

w₀Rw₁

◇⊖KFDEPossibility Undesignated

◇A ⊖, w₀

w₀Rw₁

⋮

A ⊖, w₁

¬◇⊕KFDEPossibility Negated Designated

¬◇A ⊕, w₀

⋮

◻¬A ⊕, w₀

¬◇⊖KFDEPossibility Negated Undesignated

¬◇A ⊖, w₀

⋮

◻¬A ⊖, w₀

◻ Rules

◻⊕KFDENecessity Designated

◻A ⊕, w₀

w₀Rw₁

⋮

A ⊕, w₁

◻⊖KFDENecessity Undesignated

◻A ⊖, w₀

⋮

A ⊖, w₁

w₀Rw₁

¬◻⊕KFDENecessity Negated Designated

¬◻A ⊕, w₀

⋮

◇¬A ⊕, w₀

¬◻⊖KFDENecessity Negated Undesignated

¬◻A ⊖, w₀

⋮

◇¬A ⊖, w₀

Operator Rules

¬ Rules

¬¬⊕FDEDouble Negation Designated

¬¬A ⊕, w₀

⋮

A ⊕, w₀

¬¬⊖FDEDouble Negation Undesignated

¬¬A ⊖, w₀

⋮

A ⊖, w₀

∧ Rules

∧⊕FDEConjunction Designated

A ∧ B ⊕, w₀

⋮

A ⊕, w₀

B ⊕, w₀

∧⊖FDEConjunction Undesignated

A ∧ B ⊖, w₀

⋮

A ⊖, w₀

B ⊖, w₀

¬∧⊕FDEConjunction Negated Designated

¬(A ∧ B) ⊕, w₀

⋮

¬A ⊕, w₀

¬B ⊕, w₀

¬∧⊖FDEConjunction Negated Undesignated

¬(A ∧ B) ⊖, w₀

⋮

¬A ⊖, w₀

¬B ⊖, w₀

∨ Rules

∨⊕FDEDisjunction Designated

A ∨ B ⊕, w₀

⋮

A ⊕, w₀

B ⊕, w₀

∨⊖FDEDisjunction Undesignated

A ∨ B ⊖, w₀

⋮

A ⊖, w₀

B ⊖, w₀

¬∨⊕FDEDisjunction Negated Designated

¬(A ∨ B) ⊕, w₀

⋮

¬A ⊕, w₀

¬B ⊕, w₀

¬∨⊖FDEDisjunction Negated Undesignated

¬(A ∨ B) ⊖, w₀

⋮

¬A ⊖, w₀

¬B ⊖, w₀

⊃ Rules

⊃⊕FDEMaterial Conditional Designated

A ⊃ B ⊕, w₀

⋮

¬A ⊕, w₀

B ⊕, w₀

⊃⊖FDEMaterial Conditional Undesignated

A ⊃ B ⊖, w₀

⋮

¬A ⊖, w₀

B ⊖, w₀

¬⊃⊕FDEMaterial Conditional Negated Designated

¬(A ⊃ B) ⊕, w₀

⋮

A ⊕, w₀

¬B ⊕, w₀

¬⊃⊖FDEMaterial Conditional Negated Undesignated

¬(A ⊃ B) ⊖, w₀

⋮

A ⊖, w₀

¬B ⊖, w₀

≡ Rules

≡⊕FDEMaterial Biconditional Designated

A ≡ B ⊕, w₀

⋮

¬A ⊕, w₀

¬B ⊕, w₀

B ⊕, w₀

A ⊕, w₀

≡⊖FDEMaterial Biconditional Undesignated

A ≡ B ⊖, w₀

⋮

A ⊖, w₀

¬B ⊖, w₀

¬A ⊖, w₀

B ⊖, w₀

¬≡⊕FDEMaterial Biconditional Negated Designated

¬(A ≡ B) ⊕, w₀

⋮

A ⊕, w₀

¬B ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

¬≡⊖FDEMaterial Biconditional Negated Undesignated

¬(A ≡ B) ⊖, w₀

⋮

¬A ⊖, w₀

¬B ⊖, w₀

B ⊖, w₀

A ⊖, w₀

→ Rules

→⊕RM₃Conditional Designated

A → B ⊕, w₀

⋮

A ⊖, w₀

¬B ⊖, w₀

A ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

¬B ⊕, w₀

→⊖RM₃Conditional Undesignated

A → B ⊖, w₀

⋮

A ⊕, w₀

B ⊖, w₀

¬A ⊖, w₀

¬B ⊕, w₀

¬→⊕FDEConditional Negated Designated

¬(A → B) ⊕, w₀

⋮

A ⊕, w₀

¬B ⊕, w₀

¬→⊖FDEConditional Negated Undesignated

¬(A → B) ⊖, w₀

⋮

A ⊖, w₀

¬B ⊖, w₀

↔ Rules

↔⊕RM₃Biconditional Designated

A ↔ B ⊕, w₀

⋮

A ⊖, w₀

B ⊖, w₀

¬A ⊖, w₀

¬B ⊖, w₀

A ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

¬B ⊕, w₀

↔⊖L₃Biconditional Undesignated

A ↔ B ⊖, w₀

⋮

A → B ⊖, w₀

B → A ⊖, w₀

¬↔⊕FDEBiconditional Negated Designated

¬(A ↔ B) ⊕, w₀

⋮

A ⊕, w₀

¬B ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

¬↔⊖RM₃Biconditional Negated Undesignated

¬(A ↔ B) ⊖, w₀

⋮

A ⊖, w₀

B ⊖, w₀

¬A ⊖, w₀

¬B ⊖, w₀

Quantifier Rules

∃ Rules

∃⊕FDEExistential Designated

∃xFx ⊕, w₀

⋮

Fa ⊕, w₀

∃⊖FDEExistential Undesignated

∃xFx ⊖, w₀

⋮

Fa ⊖, w₀

¬∃⊕FDEExistential Negated Designated

¬∃xFx ⊕, w₀

⋮

∀x¬Fx ⊕, w₀

¬∃⊖FDEExistential Negated Undesignated

¬∃xFx ⊖, w₀

⋮

∀x¬Fx ⊖, w₀

∀ Rules

∀⊕FDEUniversal Designated

∀xFx ⊕, w₀

⋮

Fa ⊕, w₀

∀⊖FDEUniversal Undesignated

∀xFx ⊖, w₀

⋮

Fa ⊖, w₀

¬∀⊕FDEUniversal Negated Designated

¬∀xFx ⊕, w₀

⋮

∃x¬Fx ⊕, w₀

¬∀⊖FDEUniversal Negated Undesignated

¬∀xFx ⊖, w₀

⋮

∃x¬Fx ⊖, w₀

Compatibility Rules

⚬ Rules

⚬⊕FDEAssertion Designated

⚬A ⊕, w₀

⋮

A ⊕, w₀

⚬⊖FDEAssertion Undesignated

⚬A ⊖, w₀

⋮

A ⊖, w₀

¬⚬⊕FDEAssertion Negated Designated

¬⚬A ⊕, w₀

⋮

¬A ⊕, w₀

¬⚬⊖FDEAssertion Negated Undesignated

¬⚬A ⊖, w₀

⋮

¬A ⊖, w₀

Notes

References

• Priest, Graham. An Introduction to Non-Classical Logic: From If to Is. Cambridge University Press, 2008.

class Rules

closure: tuple[type[ClosingRule], ...] = (<class 'pytableaux.logics.lp.Rules.GapClosure'>, <class 'pytableaux.logics.fde.Rules.DesignationClosure'>)

groups: tuple[tuple[type[Rule], ...], ...] = ((<class 'pytableaux.logics.fde.Rules.AssertionDesignated'>, <class 'pytableaux.logics.fde.Rules.AssertionUndesignated'>, <class 'pytableaux.logics.fde.Rules.AssertionNegatedDesignated'>, <class 'pytableaux.logics.fde.Rules.AssertionNegatedUndesignated'>, <class 'pytableaux.logics.fde.Rules.ConjunctionDesignated'>, <class 'pytableaux.logics.fde.Rules.ConjunctionNegatedUndesignated'>, <class 'pytableaux.logics.fde.Rules.DisjunctionNegatedDesignated'>, <class 'pytableaux.logics.fde.Rules.DisjunctionUndesignated'>, <class 'pytableaux.logics.fde.Rules.MaterialConditionalNegatedDesignated'>, <class 'pytableaux.logics.fde.Rules.MaterialConditionalUndesignated'>, <class 'pytableaux.logics.fde.Rules.ConditionalNegatedDesignated'>, <class 'pytableaux.logics.fde.Rules.ExistentialNegatedDesignated'>, <class 'pytableaux.logics.fde.Rules.ExistentialNegatedUndesignated'>, <class 'pytableaux.logics.fde.Rules.UniversalNegatedDesignated'>, <class 'pytableaux.logics.fde.Rules.UniversalNegatedUndesignated'>, <class 'pytableaux.logics.fde.Rules.DoubleNegationDesignated'>, <class 'pytableaux.logics.fde.Rules.DoubleNegationUndesignated'>, <class 'pytableaux.logics.kfde.Rules.PossibilityNegatedDesignated'>, <class 'pytableaux.logics.kfde.Rules.PossibilityNegatedUndesignated'>, <class 'pytableaux.logics.kfde.Rules.NecessityNegatedDesignated'>, <class 'pytableaux.logics.kfde.Rules.NecessityNegatedUndesignated'>), (<class 'pytableaux.logics.fde.Rules.ConjunctionNegatedDesignated'>, <class 'pytableaux.logics.fde.Rules.ConjunctionUndesignated'>, <class 'pytableaux.logics.fde.Rules.DisjunctionDesignated'>, <class 'pytableaux.logics.fde.Rules.DisjunctionNegatedUndesignated'>, <class 'pytableaux.logics.fde.Rules.MaterialConditionalDesignated'>, <class 'pytableaux.logics.fde.Rules.MaterialConditionalNegatedUndesignated'>, <class 'pytableaux.logics.fde.Rules.MaterialBiconditionalDesignated'>, <class 'pytableaux.logics.fde.Rules.MaterialBiconditionalNegatedDesignated'>, <class 'pytableaux.logics.fde.Rules.MaterialBiconditionalUndesignated'>, <class 'pytableaux.logics.fde.Rules.MaterialBiconditionalNegatedUndesignated'>, <class 'pytableaux.logics.rm3.Rules.ConditionalUndesignated'>, <class 'pytableaux.logics.fde.Rules.ConditionalNegatedUndesignated'>, <class 'pytableaux.logics.fde.Rules.BiconditionalNegatedDesignated'>, <class 'pytableaux.logics.l3.Rules.BiconditionalUndesignated'>, <class 'pytableaux.logics.rm3.Rules.BiconditionalNegatedUndesignated'>), (<class 'pytableaux.logics.rm3.Rules.ConditionalDesignated'>, <class 'pytableaux.logics.rm3.Rules.BiconditionalDesignated'>), (<class 'pytableaux.logics.kfde.Rules.NecessityDesignated'>, <class 'pytableaux.logics.kfde.Rules.PossibilityUndesignated'>), (<class 'pytableaux.logics.kfde.Rules.NecessityUndesignated'>, <class 'pytableaux.logics.kfde.Rules.PossibilityDesignated'>), (<class 'pytableaux.logics.fde.Rules.ExistentialDesignated'>, <class 'pytableaux.logics.fde.Rules.ExistentialUndesignated'>), (<class 'pytableaux.logics.fde.Rules.UniversalDesignated'>, <class 'pytableaux.logics.fde.Rules.UniversalUndesignated'>))