Page 100 - DCAP310_INTRODUCTION_TO_ARTIFICIAL_INTELLIGENCE_AND_EXPERT_SYSTEMS

Introduction to Artificial Intelligence & Expert Systems

Notes Note that we can easily check the validity, contingency, or unsatisfiability of a sentence can
easily by looking at the truth table for the propositional constants in the sentence.

Propositional Entailment

Validity, satisfiability, and unsatisfiability are properties of individual sentences. In logical
reasoning, we are not so much concerned with individual sentences as we are with the relationships
between sentences. In particular, we would like to know, given some sentences, whether other
sentences are or are not logical conclusions. This relative property is known as logical entailment.
When we are speaking about Propositional Logic, we use the phrase propositional entailment.
A set of sentences ∆ logically entails a sentence ϕ (written ∆ |= ϕ) if and only if every interpretation
that satisfies ∆ also satisfies ϕ. For example, the sentence p logically entails the sentence (p ∨ q).
Since a disjunction is true whenever one of its disjuncts is true, then (p ∨ q) must be true whenever
p is true. On the other hand, the sentence p does not logically entail (p ∧ q). A conjunction is true
if and only if both of its conjuncts are true, and q may be false. Of course, any set of sentences
containing both p and q does logically entail (p ∧ q). Note that the relationship of logical entailment
is a logical one. Even if the premises of a problem do not logically entail the conclusion, this
does not mean that the conclusion is necessarily false, even if the premises are true. It just means
that it is possible that the conclusion is false. Once again, consider the case of (p ∧ q). Although p
does not logically entail this sentence, it is possible that both p and q are true and, therefore,
(p ∧ q) is true. However, the logical entailment does not hold because it is also possible that q is
false and, therefore, (p ∧ q) is false.

Self Assessment

State whether the following statements are true or false:
3. Validity, satisfiability, and unsatisfiability are properties of individual sentences.
4. A sentence is falsifiable if and only if it is satisfied by at least one interpretation.

6.3 Well-formed Formula (WFF)

In mathematical logic, a well-formed formula, shortly WFF, often simply formula, is a word
(i.e. a finite sequence of symbols from a given alphabet) which is part of a formal language. A
formal language can be considered to be identical to the set containing all and only its formulas.
A formula is a syntactic formal object that can be informally given a semantic meaning.

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