Laws of Thought) and Frege (1884, Die Grundlagen der
Arithmetik). Further developments were due to Russel
(1903, Principia mathematical Peano (1895, Formulario
di matematica) and more recently, to Carnap, (1934,
Logische Syntax der Sprache), Tarski (1950, Logic,
Semantics, Mathematics) and Quine (1960, World and
Object).
2. Mathematical Logic
Boole took as basic feature of human thought, the ability
to isolate classes of objects and to refer them, as names or
symbols, able to identify them. Boole analyzed the art of
thought and decomposed it as follows:
■ act of election, to identify a certain class
■ the result of the said act
■ the universe, made with objects and classes of
objects, subject to the act of election.
Boole said:” The objects of logic are the relation among
classes and the ways the human mind contemplates such
relations. Before one can perceive the existence of
propositions, there are prime laws to which the
conception of a class hands upon. The laws depend upon
the structure of intellect and determine the character and
shape of reasoning process. They can be expressed
mathematically, thus being the basis of a comprehensible
calculus.
According to Boole, it is also possible to define, although
not so strictly, algebraic simple operations. So, operation
x+y shows the whole of elements belonging to classe x or
y, whenever they lack common elements.
allx arey:x(l-y) = 0
no x is y : xy = 0
s ont e x is y : v = xy
s ont e x is not y : v = x(l - y)
Boole also tried to express, in algebraic form, the four
basic propositions:
Furthermore, he used all available devices in order to
express the traditional syllogistic forms, thus establishing
a symbolic treatment of Logic A syllogism includes, as it
is widely known, three propositions (two premises plus
one conclusion, the premises having a common term):
Boole changed all propositions regarding two classes into
one equation: the switch to conclusion becomes a process
of elimination for the common symbol.
X 2 = X
In the Laws of Thought (1854), Boole clearly stated his
view: “the laws of logical symbols arise from laws ruling
the human mind”. So, he even tried to prove that the
principle of contradiction ensues from the basic rule of
thought, that is a second degree law, like:
As an immediate consequence, human mind is supposed
to operate analysis and classification by a division into
pairs of opposites, that is on the basis of dicotomy.
The principle of opposites has been quoted by the
founders of modem Linguistics (de Saussure and
Trubeckoj) as the concept of phoneme which will be
discussed later.
Also speaks Boole:” What makes Logic possible, is the
existence, in our mind, of general concepts: our ability to
conceive a class and to denominate, all together, all
individuals being part of that”. Thus, logic is intimately
linked to the theory of language from the very beginning.
Logic is the philosophy of the whole thinking which is to
be expressed by signs. Even the natural language is
reduced to a set of symbols and rules of combination for
the same. Boole appeared close to define a science, above
all the others, whose object is the study of symbols and
their combinations.
Many years later, the huge complex of theories and
problems of mathematical logic, also called “Algebra of
logic” (of which Boolean Algebra is the core), has given
a first rank support to developing of “informatic
revolution”.
Human intellect has obviously enough some remarkable
features in common with computers, being the latter
logical machines, based upon reconnaisance, evolution,
judgement and inferences.
However human intellect also beholds illogical features
(like intuition,...) impossible to express in binary logic
(like Hadamard said).
Most natural languages include ambiguity and multiple
meanings. Above all, adjective are not quite precise,
specially as long as, the amplitude of meaning is
involved. Words are often qualitative: however, term like
“old” or “high” are basically quantitative. In this case
Fuzzy Logic tries to take into account differences
between merely logical though and human way of
though. Fuzzy set theory works with the quantification of
the meaning. The function that gives the degree to which
an element x of a set is included in a subset is called
membership function.
If £ is a crisp set (non fuzzy set), the equivalent function
is called characteristic function and the grade is two
valued: if x is included in £ it is 7, if not 0; the grade in a
fuzzy set can be anything from 0 to 7.
3. Symbolic Logic versus Artificial Intelligence
Logic has two main issues:
■ what can be said
■ what can be deduced
logic is a formal effort: it deals with the forms (or syntax)
of statements and with assessment of truth by formulas or
syntactic operation.
The expressive capacity of a representative system, based
upon logic, comes down from this architecture: one starts
with a plain statement (true or false) and by inclusion of
additional information (as conjunction or predication)
develops a more expressive logic, able to expose more
subtle ideas.
A statement has two possible states: true or false
(statements are all assertions that one may find true or
false). The simplest statement are not very meaningful:
any sentential connective are thus necessary, as:
and a, or v, not —i, implies —», equivalent =.