Page 114 - DCAP310_INTRODUCTION_TO_ARTIFICIAL_INTELLIGENCE_AND_EXPERT_SYSTEMS

Introduction to Artificial Intelligence & Expert Systems

Notes ponens goes back to antiquity. While modus ponens is one of the most commonly used concepts
in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for
the construction of deductive proofs that includes the “rule of definition” and the “rule of
substitution”. Modus ponens allows one to eliminate a conditional statement from a logical
proof or argument (the antecedents) and thereby not carry these antecedents forward in an
ever-lengthening string of symbols; for this reason modus ponens is sometimes called the rule
of detachment. For example, modus ponens can produce shorter formulas from longer ones and
Russell observes that “the process of the inference cannot be reduced to symbols. Its sole record
is the occurrence of q [the consequent] . . . an inference is the dropping of a true premise; it is the
dissolution of an implication”.
A justification for the “trust in inference is the belief that if the two former assertions [the
antecedents] are not in error, the final assertion [the consequent] is not in error”. In other words,
if one statement or proposition implies a second one, and the first statement or proposition is
true, then the second one is also true. If P implies Q and P is true, then Q is true. An example is:
If it’s raining, I’ll meet you at the movie theater.
It’s raining.

Therefore, I’ll meet you at the movie theater.
P → Q , P
Modus ponens can be stated formally as:
∴ Q
where the rule is that whenever an instance of “P → Q” and “P” appear by themselves on lines
of a logical proof, Q can validly be placed on a subsequent line; furthermore, the premise P and
the implication “dissolves”, their only trace being the symbol Q that is retained for use later e.g.
in a more complex deduction.
6.7.2 Law of Syllogism

A relation of inference is monotonic if the addition of premises does not undermine previously
reached conclusions; otherwise the relation is no monotonic. Deductive inference, is monotonic:
if a conclusion is reached on the basis of a certain set of premises, then that conclusion still holds
if more premises are added. By contrast, everyday reasoning is mostly non-monotonic because
it involves risk: we jump to conclusions from deductively insufficient premises. We know when
it is worth or even necessary (e.g. in medical diagnosis) to take the risk. Yet we are also aware
that such inference is defensible—that new information may undermine old conclusions. Various
kinds of defensible but remarkably successful inference have traditionally captured the attention
of philosophers (theories of induction, Peirce’s theory of abduction, inference to the best
explanation, etc.). More recently logicians have begun to approach the phenomenon from a
formal point of view. The result is a large body of theories at the interface of philosophy, logic
and artificial intelligence.

6.7.3 Deductive Logic: Validity and Soundness

In mathematical logic, a logical system has the soundness property if and only if its inference
rules prove only formulas that are valid with respect to its semantics. In most cases, this comes
down to its rules having the property of preserving truth, but this is not the case in general.
Soundness is among the most fundamental properties of mathematical logic. A soundness
property provides the initial reason for counting a logical system as desirable. The completeness
property means that every validity (truth) is provable. Together they imply that all and only
validities are provable. Most proofs of soundness are trivial.

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