MH - Paracomplete Hybrid Logic
is a three-valued predicate logic with values T, F, and N. It is the gappy dual of .
Semantics
Truth Values
Common labels for the values include:
T |
1 |
just true |
N |
0.5 |
neither true nor false |
F |
0 |
just false |
Designated Values
The set of designated values for is the singleton: { T }
Truth Tables
The value of a sentence with a truth-functional operator is determined by the values of its operands according to the following tables.
| Negation | |||
|---|---|---|---|
| ¬ | |||
| T | F | ||
| N | N | ||
| F | T | ||
| Conjunction | |||
|---|---|---|---|
| ∧ | T | N | F |
| T | T | N | F |
| N | N | N | F |
| F | F | F | F |
| Disjunction | |||
|---|---|---|---|
| ∨ | T | N | F |
| T | T | T | T |
| N | T | F | N |
| F | T | N | F |
| Conditional | |||
|---|---|---|---|
| → | T | N | F |
| T | T | F | F |
| N | T | T | T |
| F | T | T | T |
Defined Operators
The Biconditional ↔, in turn, is defined in the usual way:
| Biconditional | |||
|---|---|---|---|
| ↔ | T | N | F |
| T | T | F | F |
| N | F | T | T |
| F | F | T | T |
The Material Conditional ⊃ is definable in terms of disjunction:
Likewise the Material Biconditional ≡ is defined in terms of ⊃ and ∧:
| Material Conditional | |||
|---|---|---|---|
| ⊃ | T | N | F |
| T | T | N | F |
| N | T | F | N |
| F | T | T | T |
| Material Biconditional | |||
|---|---|---|---|
| ≡ | T | N | F |
| T | T | N | F |
| N | N | F | N |
| F | F | N | T |
Compatibility Tables
does not have a separate Assertion operator, but we include a table and rules for it, for cross-compatibility.
| Assertion | |||
|---|---|---|---|
| ⚬ | |||
| T | T | ||
| N | N | ||
| F | F | ||
Predication
A sentence with n-ary predicate \(P\) over parameters has the value:
T iff is in the extension of \(P\) and not in the anti-extension of \(P\).
F iff is in the anti-extension of \(P\) and not in the extension of \(P\).
N iff is in neither the extension nor the anti-extension of \(P\).
Consequence
Logical Consequence is defined in terms of the set of designated values { T }:
Logical Consequence
C is a Logical Consequence of A iff all models where A has a desginated value are models where C also has a designated value.
Tableaux
tableaux are built similary to .
Nodes
Nodes for many-value tableaux consiste of a sentence plus a designation marker: ⊕ for designated, and ⊖ for undesignated.
Trunk
To build the trunk for an argument, add a designated node for each premise, and an undesignated node for the conclusion.
Closure
Rules
In general, rules for connectives consist of four rules per connective: a designated rule, an undesignated rule, a negated designated rule, and a negated undesignated rule. The special case of negation has a total of two rules which apply to double negation only, one designated rule, and one undesignated rule.
Operator Rules
¬ Rules
∧ Rules
∨ Rules
⊃ Rules
≡ Rules
→ Rules
↔ Rules
Compatibility Rules
⚬ Rules
Notes
References
Colin Caret. (2017). Hybridized Paracomplete and Paraconsistent Logics. The Australasian Journal of Logic, 14.