S5LP - LP with S5 Modal


Semantics

S5LP semantics behave just like KLP semantics.

S5LP 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

S5LP 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 ... AnB write:
A1 , w0
An , w0
B , w0

Closure

S5LP includes the LP and FDE closure rules.

LPGap Closure[source]
A , w0
¬A , w0
FDEDesignation Closure[source]
A , w0
A , w0

Rules

S5LP contains all the KFDE 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

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
FDEConditional Designated[source]
A → B , w0
¬A , w0
B , w0
FDEConditional Undesignated[source]
A → 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
FDEBiconditional Designated[source]
A ↔ B , w0
¬A , w0
¬B , w0
B , w0
A , w0
FDEBiconditional Undesignated[source]
A ↔ B , w0
A , w0
¬B , w0
¬A , w0
B , w0
¬FDEBiconditional Negated Designated[source]
¬(A ↔ B) , w0
A , w0
¬B , w0
¬A , w0
B , w0
¬FDEBiconditional Negated Undesignated[source]
¬(A ↔ B) , w0
¬A , w0
¬B , w0
B , w0
A , w0

Notes

References