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 =.