CPL - Classical Predicate Logic
is the standard bivalent logic with values T and F.
Semantics
Truth Values
Common labels for the values include:
T |
1 |
true |
F |
0 |
false |
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 | |
| F | T | |
| Conjunction | ||
|---|---|---|
| ∧ | T | F |
| T | T | F |
| F | F | F |
| Disjunction | ||
|---|---|---|
| ∨ | T | F |
| T | T | T |
| F | T | 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 | F |
| T | T | F |
| F | T | T |
| Material Biconditional | ||
|---|---|---|
| ≡ | T | F |
| T | T | F |
| F | F | 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 | |
| F | F | |
| Conditional | ||
|---|---|---|
| → | T | F |
| T | T | F |
| F | T | T |
| Biconditional | ||
|---|---|---|
| ↔ | T | F |
| T | T | F |
| F | F | T |
Predication
The value of predicated sentences are handled in terms of a predicate's extension.
A sentence with n-ary predicate \(P\) over parameters has the value T iff is in the extension of \(P\).
Consequence
Logical Consequence is defined as follows:
Logical Consequence
C is a Logical Consequence of A iff all models where the value of A is T are models where C also has the value T.
Tableaux
Nodes
Each node consists of a sentence.
Trunk
To build the trunk for an argument, add a node for each premise, and a node with the negation of the conclusion.
Closure
Rules
In general, rules for connectives consist of two rules per connective: a plain rule, and a negated rule. The special case of negation only one rule for double negation.
There is also a special rule for the Identity predicate.