Formal systems and the notion of syntactic consequence were developed through specific historical milestones in logic
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Historical references and encyclopedia entries document how the development of formal logic systems and deductive calculi evolved through specific milestones involving figures like Frege and Russell.
al one.
The works of Frege and Russell introduced a different perspective on
the way to approach logic. In those works a logic system was given by
a formal language and a deductive calculus, namely a set of axioms and
a set of inference rules. Let us call such a pair a logical
deduction system , and the formulas derivable in the calculus the
theorems of the system (nowadays it is common practice to
call this kind of calculi Hilbert style calculi). In Frege and
Russell's approach a formal (mathematical) semantics of whatever kind
(algebraic, model-theoretic, etc.) for the formal languages they used
was lacking. The only semantics present was of an intuitive informal
kind.
The systems introduced by Frege and Russell were systems of classical
logic, but soon after systems of non-classical logics were considered
by other logicians. The first influential attempts to introduce logics
different from classical logic remained within the Frege-Russell
tradition of presenting a logical deduction system without any formal
semantics. They include the first modal systems of C.I. Lewis (1918)
and the axiomatization of intuitionistic logic by Heyting (1930).
The idea underlying the design of Frege and Russell's logical
deduction systems is that the theorems should be the formulas that
correspond (intuitively) to the logical truths or logical
validities. The concept of logical consequence was not central to the
development and this was also the case in the many systems of
non-classical logics that were to be designed following in the
footsteps of the first modal systems of C.I. Lewis. This situation
influenced the way in which the research on some non-classical logics
has usually been presented and sometimes also its real
evolution. However the concept of logical consequence has been the one
that logic has traditionally dealt with. Tarski put it once again into
the center of modern logic, both semantically and
syntactically. Nowadays, a general theory of the algebraization of
logics
The Emergence of First-Order Logic (Stanford Encyclopedia of Philosophy)
Stanford Encyclopedia of Philosophy
# The Emergence of First-Order Logic
First published Sat Nov 17, 2018
For anybody schooled in modern logic, first-order logic can seem an entirely natural object of study, and its discovery inevitable. It is semantically complete; it is adequate to the axiomatization of all ordinary mathematics; and Lindström’s theorem shows that it is the maximal logic satisfying the compactness and Löwenheim-Skolem properties. So it is not surprising that first-order logic has long been regarded as the “right” logic for investigations into the foundations of mathematics. It occupies the central place in modern textbooks of mathematical logic, with other systems relegated to the sidelines. The history, however, is anything but straightforward, and is certainly not a matter of a sudden discovery by a single researcher. The emergence is bound up with technical discoveries, with differing conceptions of what constitutes logic, with different programs of mathematical research, and with philosophical and conceptual reflection. So if first-order logic is “natural”, it is natural only in retro
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