TFDE - FDE with T Modal

---

Semantics

TFDE semantics behave just like KFDE semantics.

TFDE includes the T reflexive restriction on the access relation for models.

Reflexivity

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

Tableaux

TFDE 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

⊗FDEDesignation Closure

A ⊕, w₀

⋮

A ⊖, w₀

⊗

Rules

TFDE contains all the KFDE rules plus the T Reflexive rule.

The Reflexive rule applies to an open branch b when there is a node n on b with a world w but there is not a node where w accesses w (itself).

Access Rules

R≤TReflexive

A, w₀

⋮

w₀Rw₀

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

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

→ Rules

→⊕FDEConditional Designated

A → B ⊕, w₀

⋮

¬A ⊕, w₀

B ⊕, w₀

→⊖FDEConditional Undesignated

A → 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

↔⊕FDEBiconditional Designated

A ↔ B ⊕, w₀

⋮

¬A ⊕, w₀

¬B ⊕, w₀

B ⊕, w₀

A ⊕, w₀

↔⊖FDEBiconditional Undesignated

A ↔ B ⊖, w₀

⋮

A ⊖, w₀

¬B ⊖, w₀

¬A ⊖, w₀

B ⊖, w₀

¬↔⊕FDEBiconditional Negated Designated

¬(A ↔ B) ⊕, w₀

⋮

A ⊕, w₀

¬B ⊕, w₀

¬A ⊕, w₀

B ⊕, w₀

¬↔⊖FDEBiconditional Negated Undesignated

¬(A ↔ B) ⊖, w₀

⋮

¬A ⊖, w₀

¬B ⊖, w₀

B ⊖, 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.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.ConditionalUndesignated'>, <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.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.fde.Rules.ConditionalDesignated'>, <class 'pytableaux.logics.fde.Rules.ConditionalNegatedUndesignated'>, <class 'pytableaux.logics.fde.Rules.BiconditionalDesignated'>, <class 'pytableaux.logics.fde.Rules.BiconditionalNegatedDesignated'>, <class 'pytableaux.logics.fde.Rules.BiconditionalUndesignated'>, <class 'pytableaux.logics.fde.Rules.BiconditionalNegatedUndesignated'>), (<class 'pytableaux.logics.fde.Rules.ExistentialDesignated'>, <class 'pytableaux.logics.fde.Rules.ExistentialUndesignated'>), (<class 'pytableaux.logics.fde.Rules.UniversalDesignated'>, <class 'pytableaux.logics.fde.Rules.UniversalUndesignated'>))