Semantics
Interpretations
The mapping of elements of logical language (i.e., atoms) with elements of world (i.e., propositions) is known as interpretation.
If an atom β has value True iff it is associated with proposition P. Otherwise it has value False. Since every proposition of the world is either true or false, the atoms associated with propositions are assigned with values directly. We store these values in knowledge bas, of agent's memory structure.
The Propositional Truth Table
The truth tables are tabular forms to establish semantics of propositional connectives. We can compute any wff from truth table.
Table: Truth Table
The wff P v Q is false only if P and Q are false. The wff P ∧ Q is true only if P and Q are true. The wff P⊃ Q is false only if P is true and Q is false.
If the world is described by agent in n features, and each feature value is associated with an atom, then the total number of ways its world can be is 2n. We can derive values for any wff from atoms of agent, but converse may not be possible. Because the derived values of wff contain more than one true's or false's.
Satisfiability and Models
A statement is satisfiable if there is some interpretation for which it is true. Here the interpretation is known as model. By using truth table we can tell whether the interpretation is satisfiable or not.
An wff is said to be inconsistent or unsatisfiable if there is no interpretation satisfies. the wff.
Validity
A sentence (wff) is valid if it is true for every interpretation. For example, the sentence P v ¬P is valid. Valid sentences are also called tautologies.
Some of the tautologies are,
1) Q ⊃ Q
2) T
3) P v T
4) P ⊃ (R ⊃ P)
Equivalence
Two sentences are logically equivalent if they have the same truth value under every, interpretation, written as P ∧ Q. For example, we can show (using truth tables) that P ∧ Q and Q ∧ P are logically equivalent.
Table: Standard Logical Equivalence
Entailment
We say a set Δ is logically entails a sentence P iff all the interpretations of P are true for which each of the wffs (sentences) in set Δ has value true. And also ,we can say, P logically follows from Δ. Here P is logical consequence of Δ. The entailment is represented with symbol ⊨.
Example:
1) T ⊨ P
2) {Q} ⊨ Q
3) P ∧ R ⊨ P
