- Logic of Paradox
is a 3-valued logic with value T, F, and B. It can be understood as without the N value.
Semantics
Truth Values
Common labels for the values include:
T |
1 |
just true |
B |
0.5 |
both true and false |
F |
0 |
just false |
Designated Values
The set of designated values for is: { T, B }
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 | ||
| B | B | ||
| F | T | ||
| Conjunction | |||
|---|---|---|---|
| ∧ | T | B | F |
| T | T | B | F |
| B | B | B | F |
| F | F | F | F |
| Disjunction | |||
|---|---|---|---|
| ∨ | T | B | F |
| T | T | T | T |
| B | T | B | B |
| F | T | B | F |
Defined Operators
The Material Conditional ⊃ is definable in terms of disjunction:
Likewise the Material Biconditional ≡ is defined in terms of ⊃ and ∧:
| Material Conditional | |||
|---|---|---|---|
| ⊃ | T | B | F |
| T | T | B | F |
| B | T | B | B |
| F | T | T | T |
| Material Biconditional | |||
|---|---|---|---|
| ≡ | T | B | F |
| T | T | B | F |
| B | B | B | B |
| F | F | B | T |
Compatibility Tables
does not have separate Assertion or Conditional operators, but we include tables and rules for them, for cross-compatibility.
| Assertion | |||
|---|---|---|---|
| ⚬ | |||
| T | T | ||
| B | B | ||
| F | F | ||
| Conditional | |||
|---|---|---|---|
| → | T | B | F |
| T | T | B | F |
| B | T | B | B |
| F | T | T | T |
| Biconditional | |||
|---|---|---|---|
| ↔ | T | B | F |
| T | T | B | F |
| B | B | B | B |
| F | F | B | T |
Predication
Like , predication defines a predicate's extenstion and anti-extension. The value of a predicated sentence is determined as follows:
A sentence with predicate P with parameters is assigned a value as follows:
- 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.
- B iff is in both the extension and anti-extension
of P.
Note, unlike , there is an exhaustion constraint on a predicate's extension/anti-extension. This means that must be in either the extension and the anti-extension of P.
Like , there is no exclusion restraint: there are permissible models where some tuple is in both the extension and anti-extension of a predicate.
Quantification
Existential
The value of an existential sentence is the maximum value of the sentences that result from replacing each constant for the quantified variable. The ordering of the values from least to greatest is: F, N, B, T.
Universal
The value of an universal sentence is the minimum value of the sentences that result from replacing each constant for the quantified variable. The ordering of the values from least to greatest is: F, N, B, T.
Consequence
Logical Consequence is defined in terms of the set of designated values { T, B }:
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
includes an additional gap closure rule. This means a branch closes when a sentence and its negation both appear as undesignated nodes on the branch.
includes the closure rule.
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
Quantifier Rules
∃ Rules
∀ Rules
Compatibility Rules
⚬ Rules
→ Rules
↔ Rules
Notes
Some notable features of include:
Like , the Law of Non-Contradiction fails ¬(A ∧ ¬A).
Unlike , has some logical truths. For example, the Law of Excluded Middle (A ∨ ¬A), and Conditional Identity (A → A).
Many classical validities fail, such as Modus Ponens, Modus Tollens, and Disjunctive Syllogism.
DeMorgan laws are valid.
References
Beall, Jc, et al. Possibilities and Paradox: An Introduction to Modal and Many-valued Logic. United Kingdom, Oxford University Press, 2003.
For futher reading see: