GO - Gappy Object Logic
GO is a 3-valued logic with values T, F, and N. It has non-standard readings of disjunction and conjunction, as well as different behavior of the quantifiers.
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 | F | F |
| N | F | F | F |
| F | F | F | F |
| Disjunction | |||
|---|---|---|---|
| ∨ | T | N | F |
| T | T | T | T |
| N | T | F | F |
| F | T | F | F |
Defined Operators
An Assertion ⚬ operator is definable in terms of ∧:
| Assertion | |||
|---|---|---|---|
| ⚬ | |||
| T | T | ||
| N | F | ||
| F | F | ||
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 | F | F |
| N | T | F | F |
| F | T | T | T |
| Material Biconditional | |||
|---|---|---|---|
| ≡ | T | N | F |
| T | T | F | F |
| N | F | F | F |
| F | F | F | T |
The Conditional → is definable as follows:
This can be read as: either A ⊃ B or both A and B are gappy (i.e. have the value T{N}).
The Biconditional ↔, in turn, is defined in the usual way:
| Conditional | |||
|---|---|---|---|
| → | T | N | F |
| T | T | F | F |
| N | T | T | F |
| F | T | T | T |
| Biconditional | |||
|---|---|---|---|
| ↔ | T | N | F |
| T | T | F | F |
| N | F | T | F |
| F | F | F | T |
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 \(P\).
Quantification
For quantification, we introduce a crunch function:
Crunched Value
The crunched value of v is 1 (T) if v is 1, else 0 (F).
Existential
The value of an existential sentence is the maximum of the crunched values of the sentences that result from replacing each constant for the quantified variable.
This is in accord with interpreting the existential quantifier in terms of generalized disjunction.
Universal
The value of an universal sentence is the minimum of the crunched values of the sentences that result from replacing each constant for the quantified variable.
This is in accord with interpreting the universal quantifier in terms of generalized conjunction.
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
↔ Rules
Quantifier Rules
∃ Rules
∀ Rules
Notes
GO has some similarities to . Material Identity A ⊃ A and the Law of Excluded Middle A ∨ ¬A fail.
Unlike , there are logical truths, e.g. the Law of Non-Contradiction ¬(A ∧ ¬A).
GO contains an additional conditional operator besides the material conditional, which is similar to . However, this conditional is non-primitive, unlike , and it obeys contraction (A → (A → B) ⊢ A → B).
Conjunctions and Disjunctions always have a classical value (T or F). This means that only atomic sentences (with zero or more negations) can have the non-classical N value.
This property of "classical containment" means that we can define a conditional operator that satisfies Conditional Identity A → A. It also allows us to give a formal description of a subset of sentences that obey all principles of classical logic. For example, although the Law of Excluded Middle fails for atomic sentences A ∨ ¬A, complex sentences -- those with at least one binary connective -- do obey the law: ⊢ (A ∨ A) ∨ ¬(A ∨ A).
References
Doug Owings (2012). Indeterminacy and Logical Atoms. Ph.D. Thesis, University of Connecticut.
Further Reading
Colin Caret. (2017). Hybridized Paracomplete and Paraconsistent Logics. The Australasian Journal of Logic, 14.