Page 115 - DCAP310_INTRODUCTION_TO_ARTIFICIAL_INTELLIGENCE_AND_EXPERT_SYSTEMS

Unit 6: Formalized Symbolic Logics

Notes
Example: In an axiomatic system proof of soundness amounts to verifying the validity
of the axioms and that the rules of inference preserve validity (or the weaker property, truth).
Most axiomatic systems have only the rule of modus ponens (and sometimes substitution), so it
requires only verifying the validity of the axioms and one rule of inference.
Soundness properties come in two main varieties: weak and strong soundness, of which the
former is a restricted form of the latter.
Soundness

Soundness of a deductive system is the property that any sentence that is provable in that
deductive system is also true on all interpretations or structures of the semantic theory for the
language upon which that theory is based. In symbols, where S is the deductive system, L the
language together with its semantic theory, and P a sentence of L: if P, then also P.
S L
Strong Soundness

Strong soundness of a deductive system is the property that any sentence P of the language upon
which the deductive system is based that is derivable from a set Γ of sentences of that language
is also a logical consequence of that set, in the sense that any model that makes all members of
Γ true will also make P true. In symbols, where Γ is a set of sentences of L: if Γ P, then also
S
Γ P. Notice that in the statement of strong soundness, when Γ is empty, we have the statement
L
of weak soundness.
Arithmetic Soundness

If T is a theory whose objects of discourse can be interpreted as natural numbers, we say T is
arithmetically sound if all theorems of T are actually true about the standard mathematical
integers.

6.7.4 Representations Using Rules Dealing with Inconsistencies and
Uncertainties

AI research is highly technical and specialized, deeply divided into subfields that often fail to
communicate with each other. Some of the division is due to social and cultural factors: subfields
have grown up around particular institutions and the work of individual researchers. AI research
is also divided by several technical issues. There are subfields which are focused on the solution
of specific problems, on one of several possible approaches, on the use of widely differing tools
and towards the accomplishment of particular applications. The central problems (or goals) of
AI research include reasoning, knowledge, planning, learning, communication, perception and
the ability to move and manipulate objects. General intelligence (or “strong AI”) is still among
the field’s long term goals. Currently popular approaches include statistical methods,
computational intelligence and traditional symbolic AI. There are an enormous number of tools
used in AI, including versions of search and mathematical optimization, logic, methods based
on probability and economics, and many others. The field was founded on the claim that a
central property of humans, intelligence—the sapience of Homo sapiens—can be so precisely
described that it can be simulated by a machine. This raises philosophical issues about the nature
of the mind and the ethics of creating artificial beings, issues which have been addressed by
myth, fiction and philosophy since antiquity. Artificial intelligence has been the subject of
tremendous optimism but has also suffered stunning setbacks. Today it has become an essential
part of the technology industry, providing the heavy lifting for many of the most difficult
problems in computer science.

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