S5K³_W - K³_W with S5 Modal

Semantics 

S5K³_W semantics behave just like KK³_W semantics.

S5K³_W includes the access relation restrictions on models of S5:

Reflexivity

In every model, for each world \(w\), \(\langle\ w,w\ \rangle\) is in the access relation.

Transitivity

In every model, for each world \(w\), and each world \(w'\), for any world \(w''\) such that \(\langle\ w,w'\ \rangle\) and \(\langle\ w',w''\ \rangle\) are in the access relation, then \(\langle\ w,w''\ \rangle\) is in the access relation.

Symmetry

In every model, for each world \(w\) and each world \(w'\), if \(\langle\ w,w'\ \rangle\) is in the access relation, then so is \(\langle\ w',w\ \rangle\).

Tableaux 

S5K³_W tableaux are constructed just like KFDE system 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 

S5K³_W includes the K₃ and FDE closure rules.

⊗K₃Glut Closure[source]: A ⊕, w₀

⋮

¬A ⊕, w₀

⊗

⊗FDEDesignation Closure[source]: A ⊕, w₀

⋮

A ⊖, w₀

⊗

Rules 

S5K³_W contains all the KK³_W rules plus the S5 access rules.

Access Rules

R+S4Transitive[source]: w₀Rw₁

w₁Rw₂

⋮

w₀Rw₂

R≤TReflexive[source]: A, w₀

⋮

w₀Rw₀

R⌯S5Symmetric[source]: w₀Rw₁

⋮

w₁Rw₀

Operator Rules

¬ Rules

¬¬⊕FDEDouble Negation Designated[source]: ¬¬A ⊕, w₀

⋮

A ⊕, w₀

¬¬⊖FDEDouble Negation Undesignated[source]: ¬¬A ⊖, w₀

⋮

A ⊖, w₀

∧ Rules

∧⊕FDEConjunction Designated[source]: A ∧ B ⊕, w₀

⋮

A ⊕, w₀

B ⊕, w₀

∧⊖FDEConjunction Undesignated[source]: A ∧ B ⊖, w₀

⋮

A ⊖, w₀

B ⊖, w₀

¬∧⊕K³_WConjunction Negated Designated[source]: ¬(A ∧ B) ⊕, w₀

⋮

A ⊕, w₀

¬B ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

¬A ⊕, w₀

¬B ⊕, w₀

¬∧⊖K³_WConjunction Negated Undesignated[source]: ¬(A ∧ B) ⊖, w₀

⋮

A ⊖, w₀

¬A ⊖, w₀

B ⊖, w₀

¬B ⊖, w₀

A ⊕, w₀

B ⊕, w₀

∨ Rules

∨⊕K³_WDisjunction Designated[source]: A ∨ B ⊕, w₀

⋮

A ⊕, w₀

¬B ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

A ⊕, w₀

B ⊕, w₀

∨⊖K³_WDisjunction Undesignated[source]: A ∨ B ⊖, w₀

⋮

A ⊖, w₀

¬A ⊖, w₀

B ⊖, w₀

¬B ⊖, w₀

¬A ⊕, w₀

¬B ⊕, w₀

¬∨⊕FDEDisjunction Negated Designated[source]: ¬(A ∨ B) ⊕, w₀

⋮

¬A ⊕, w₀

¬B ⊕, w₀

¬∨⊖K³_WDisjunction Negated Undesignated[source]: ¬(A ∨ B) ⊖, w₀

⋮

A ∨ B ⊕, w₀

A ⊖, w₀

¬A ⊖, w₀

B ⊖, w₀

¬B ⊖, w₀

⊃ Rules

⊃⊕K³_WMaterial Conditional Designated[source]: A ⊃ B ⊕, w₀

⋮

¬A ∨ B ⊕, w₀

⊃⊖K³_WMaterial Conditional Undesignated[source]: A ⊃ B ⊖, w₀

⋮

¬A ∨ B ⊖, w₀

¬⊃⊕K³_WMaterial Conditional Negated Designated[source]: ¬(A ⊃ B) ⊕, w₀

⋮

¬(¬A ∨ B) ⊕, w₀

¬⊃⊖K³_WMaterial Conditional Negated Undesignated[source]: ¬(A ⊃ B) ⊖, w₀

⋮

¬(¬A ∨ B) ⊖, w₀

≡ Rules

≡⊕K³_WMaterial Biconditional Designated[source]: A ≡ B ⊕, w₀

⋮

(A ⊃ B) ∧ (B ⊃ A) ⊕, w₀

≡⊖K³_WMaterial Biconditional Undesignated[source]: A ≡ B ⊖, w₀

⋮

(A ⊃ B) ∧ (B ⊃ A) ⊖, w₀

¬≡⊕K³_WMaterial Biconditional Negated Designated[source]: ¬(A ≡ B) ⊕, w₀

⋮

¬((A ⊃ B) ∧ (B ⊃ A)) ⊕, w₀

¬≡⊖K³_WMaterial Biconditional Negated Undesignated[source]: ¬(A ≡ B) ⊖, w₀

⋮

¬((A ⊃ B) ∧ (B ⊃ A)) ⊖, w₀

Quantifier Rules

∃ Rules

∃⊕FDEExistential Designated[source]: ∃xFx ⊕, w₀

⋮

Fa ⊕, w₀

∃⊖FDEExistential Undesignated[source]: ∃xFx ⊖, w₀

⋮

Fa ⊖, w₀

¬∃⊕FDEExistential Negated Designated[source]: ¬∃xFx ⊕, w₀

⋮

∀x¬Fx ⊕, w₀

¬∃⊖FDEExistential Negated Undesignated[source]: ¬∃xFx ⊖, w₀

⋮

∀x¬Fx ⊖, w₀

∀ Rules

∀⊕FDEUniversal Designated[source]: ∀xFx ⊕, w₀

⋮

Fa ⊕, w₀

∀⊖FDEUniversal Undesignated[source]: ∀xFx ⊖, w₀

⋮

Fa ⊖, w₀

¬∀⊕FDEUniversal Negated Designated[source]: ¬∀xFx ⊕, w₀

⋮

∃x¬Fx ⊕, w₀

¬∀⊖FDEUniversal Negated Undesignated[source]: ¬∀xFx ⊖, w₀

⋮

∃x¬Fx ⊖, w₀

Compatibility Rules

⚬ Rules

⚬⊕FDEAssertion Designated[source]: ⚬A ⊕, w₀

⋮

A ⊕, w₀

⚬⊖FDEAssertion Undesignated[source]: ⚬A ⊖, w₀

⋮

A ⊖, w₀

¬⚬⊕FDEAssertion Negated Designated[source]: ¬⚬A ⊕, w₀

⋮

¬A ⊕, w₀

¬⚬⊖FDEAssertion Negated Undesignated[source]: ¬⚬A ⊖, w₀

⋮

¬A ⊖, w₀

→ Rules

→⊕K³_WConditional Designated[source]: A → B ⊕, w₀

⋮

¬A ∨ B ⊕, w₀

→⊖K³_WConditional Undesignated[source]: A → B ⊖, w₀

⋮

¬A ∨ B ⊖, w₀

¬→⊕K³_WConditional Negated Designated[source]: ¬(A → B) ⊕, w₀

⋮

¬(¬A ∨ B) ⊕, w₀

¬→⊖K³_WConditional Negated Undesignated[source]: ¬(A → B) ⊖, w₀

⋮

¬(¬A ∨ B) ⊖, w₀

↔ Rules

↔⊕K³_WBiconditional Designated[source]: A ↔ B ⊕, w₀

⋮

(A ⊃ B) ∧ (B ⊃ A) ⊕, w₀

↔⊖K³_WBiconditional Undesignated[source]: A ↔ B ⊖, w₀

⋮

(A ⊃ B) ∧ (B ⊃ A) ⊖, w₀

¬↔⊕K³_WBiconditional Negated Designated[source]: ¬(A ↔ B) ⊕, w₀

⋮

¬((A ⊃ B) ∧ (B ⊃ A)) ⊕, w₀

¬↔⊖K³_WBiconditional Negated Undesignated[source]: ¬(A ↔ B) ⊖, w₀

⋮

¬((A ⊃ B) ∧ (B ⊃ A)) ⊖, w₀

Notes 

References 

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

class Rules[source]

closure: tuple[type[ClosingRule], ...] = (<class 'pytableaux.logics.k3.Rules.GlutClosure'>, <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.DisjunctionNegatedDesignated'>, <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.k3w.Rules.MaterialConditionalDesignated'>, <class 'pytableaux.logics.k3w.Rules.MaterialConditionalUndesignated'>, <class 'pytableaux.logics.k3w.Rules.MaterialConditionalNegatedDesignated'>, <class 'pytableaux.logics.k3w.Rules.MaterialConditionalNegatedUndesignated'>, <class 'pytableaux.logics.k3w.Rules.ConditionalDesignated'>, <class 'pytableaux.logics.k3w.Rules.ConditionalUndesignated'>, <class 'pytableaux.logics.k3w.Rules.ConditionalNegatedDesignated'>, <class 'pytableaux.logics.k3w.Rules.ConditionalNegatedUndesignated'>, <class 'pytableaux.logics.k3w.Rules.MaterialBiconditionalDesignated'>, <class 'pytableaux.logics.k3w.Rules.MaterialBiconditionalUndesignated'>, <class 'pytableaux.logics.k3w.Rules.MaterialBiconditionalNegatedDesignated'>, <class 'pytableaux.logics.k3w.Rules.MaterialBiconditionalNegatedUndesignated'>, <class 'pytableaux.logics.k3w.Rules.BiconditionalDesignated'>, <class 'pytableaux.logics.k3w.Rules.BiconditionalUndesignated'>, <class 'pytableaux.logics.k3w.Rules.BiconditionalNegatedDesignated'>, <class 'pytableaux.logics.k3w.Rules.BiconditionalNegatedUndesignated'>, <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.s4.Rules.Transitive'>,), (<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.t.Rules.Reflexive'>,), (<class 'pytableaux.logics.fde.Rules.ConjunctionUndesignated'>,), (<class 'pytableaux.logics.k3w.Rules.DisjunctionDesignated'>, <class 'pytableaux.logics.k3w.Rules.DisjunctionUndesignated'>, <class 'pytableaux.logics.k3w.Rules.ConjunctionNegatedDesignated'>, <class 'pytableaux.logics.k3w.Rules.ConjunctionNegatedUndesignated'>, <class 'pytableaux.logics.k3w.Rules.DisjunctionNegatedUndesignated'>), (<class 'pytableaux.logics.fde.Rules.ExistentialDesignated'>, <class 'pytableaux.logics.fde.Rules.ExistentialUndesignated'>), (<class 'pytableaux.logics.fde.Rules.UniversalDesignated'>, <class 'pytableaux.logics.fde.Rules.UniversalUndesignated'>), (<class 'pytableaux.logics.s5.Rules.Symmetric'>,))

S5K3W - K3W with S5 Modal