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5.5.3
A Glimpse of the History of Formal Logic
Formal Logic originates with Aristotle and concerns the human activity of drawing
inferences. Of course, since the time when language developed, man has deduced
conclusions from premises, but Aristotle inaugurated the systematic study of the
rules involved in the construction of valid reasoning. The first important discovery
of this approach was that the logical structure of sentences and deductions is given
by relations between signs in abstraction from their meaning. This aspect motivates
the attribute
formal
. Modern logic, as it has been developed since the middle of 19th
century, emphasizes this aspect by developing notational systems able to represent
and analyze the logical form of sentences and deductions. In this sense formal logic
is also referred as
Symbolic Logic
,oralso
Mathematical Logic
, because the emer-
gence of the symbolic perspective was stimulated by certain trends within math-
ematics, namely, the generalization of algebra, the development of the axiomatic
method, especially in geometry, and the
reductionism
, that is, the tendency, espe-
cially in analysis, to find basic concepts for a foundation of the whole mathematics.
The elaboration of the formal method in modern logic was chiefly founded by the
works of: Gottfried Wilhelm von Leibniz (1646-1716), De Morgan (1806-1871),
Boole (1815-1864), Peirce (1839-1914), Schr oeder (1841-1902), Frege (1848-
1925), Peano (1858-1932), Hilbert (1862-1943), Russell (1872-1970), Lowenheim
(1878-1957), Skolem (1887-1963), Tarski (1901-1983), Godel (1906-1978), Her-
brand (1908-1931), Gentzen (1909-1945), and Turing (1912-1954). The essential
aspect of formal methods, in the representation of sentences and deductions, con-
sists of a clear distinction between
syntax
and
semantics
. This is an intrinsic feature
of any
formal language
, as opposed to natural language. Syntax establishes which
(linear) arrangements of symbols of a specified alphabet have to be considered well-
formed expressions, the categories in which they are classified, and symbolic rules
according to which some relations between expressions are defined. Semantics es-
tablishes how to define the general concepts of interpretation, satisfiability, truth,
compatibility, independence, and entailment. This distinction does not mean that
syntax and semantics are opposed, but rather, complementary. Syntactic categories,
and syntactic rules, are given in such a way that important semantic concepts can
be adequately expressed in syntactic terms. When this is possible those concepts
can be
calculated
or
mechanized
. One of the most important questions investigated
in modern logic is that of the completeness of given logical systems. A way of ex-
pressing completeness, as in celebrated Godel's completeness theorem (1930), is
that any sentence that is provable, by means of the syntactic rules of a logical sys-
tem (e.g. Russell's system of
Principia Mathematica
), is universally valid, that is,
true in all of its possible interpretations. Therefore,
form
and
content
of sentences
are distinguished in order to achieve
effective
, general methods in elaborating and
understanding sentences. Syntax defined apart from semantics, can individuate pro-
cesses which elaborate formulae by using only symbolic structures, which by virtue
of their finite nature, can be encoded as physical states of a machine (the famous
Godel's incompleteness theorem was based on a suitable encoding of FOL syn-
tax in arithmetical terms). Semantics, which deal with no particular interpretation
