S5RM₃ - RM₃ with S5 Modal

Semantics 

S5RM₃ semantics behave just like KRM₃ semantics.

S5RM₃ 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 

S5RM₃ 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 

S5RM₃ includes the LP and FDE closure rules.

⊗LPGap Closure[source]: A ⊖, w₀

⋮

¬A ⊖, w₀

⊗

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

⋮

A ⊖, w₀

⊗

Rules 

S5RM₃ contains all the KRM₃ 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

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

⋮

¬A ⊕, w₀

¬B ⊕, w₀

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

⋮

¬A ⊖, w₀

¬B ⊖, w₀

∨ Rules

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

⋮

A ⊕, w₀

B ⊕, w₀

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

⋮

A ⊖, w₀

B ⊖, w₀

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

⋮

¬A ⊕, w₀

¬B ⊕, w₀

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

⋮

¬A ⊖, w₀

¬B ⊖, w₀

⊃ Rules

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

⋮

¬A ⊕, w₀

B ⊕, w₀

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

⋮

¬A ⊖, w₀

B ⊖, w₀

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

⋮

A ⊕, w₀

¬B ⊕, w₀

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

⋮

A ⊖, w₀

¬B ⊖, w₀

≡ Rules

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

⋮

¬A ⊕, w₀

¬B ⊕, w₀

B ⊕, w₀

A ⊕, w₀

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

⋮

A ⊖, w₀

¬B ⊖, w₀

¬A ⊖, w₀

B ⊖, w₀

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

⋮

A ⊕, w₀

¬B ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

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

⋮

¬A ⊖, w₀

¬B ⊖, w₀

B ⊖, w₀

A ⊖, w₀

→ Rules

→⊕RM₃Conditional Designated[source]: A → B ⊕, w₀

⋮

A ⊖, w₀

¬B ⊖, w₀

A ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

¬B ⊕, w₀

→⊖RM₃Conditional Undesignated[source]: A → B ⊖, w₀

⋮

A ⊕, w₀

B ⊖, w₀

¬A ⊖, w₀

¬B ⊕, w₀

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

⋮

A ⊕, w₀

¬B ⊕, w₀

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

⋮

A ⊖, w₀

¬B ⊖, w₀

↔ Rules

↔⊕RM₃Biconditional Designated[source]: A ↔ B ⊕, w₀

⋮

A ⊖, w₀

B ⊖, w₀

¬A ⊖, w₀

¬B ⊖, w₀

A ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

¬B ⊕, w₀

↔⊖L₃Biconditional Undesignated[source]: A ↔ B ⊖, w₀

⋮

A → B ⊖, w₀

B → A ⊖, w₀

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

⋮

A ⊕, w₀

¬B ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

¬↔⊖RM₃Biconditional Negated Undesignated[source]: ¬(A ↔ B) ⊖, w₀

⋮

A ⊖, w₀

B ⊖, w₀

¬A ⊖, w₀

¬B ⊖, 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

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.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.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.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.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'>,))

S5RM3 - RM3 with S5 Modal