1. Answer briefly What are the types of symbols used in symbolic logic?
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Answer:
Varieties of Symbolic Logic
The term ‘symbolic logic’ was introduced by the British logician John Venn (1834–1923), to characterise the kind of logic which gave prominence not only to symbols but also to mathematical theories to which they belonged [Venn, 1881]. He had in mind the two principal manifestations of the time: 1) algebraic representations of the modes in syllogistic logic, which had increased in number from the 1830s by the quantification of the predicate and from the 1860s by some attention to the logic of relations, launched by Augustus De Morgan (1806–1871); and 2) extensions or modifications of the algebra of logic of George Boole (1815–1864) which, as he had shown in his second book The laws of thought (1854), surpassed syllogistic in expressive and deductive power. This whole tradition was deeply influenced by algebras, partly the common tradition in algebraising syllogistic modes, but more closely by newer ones developed in the early 19th century: differential operators, which was a key source for Boole; and functional equations, which bore a strong analogy to the logic of relations, both of which were studied by De Morgan. From the 1870s these two strands were being brought together by the American polymath C.S. Peirce (1839–1914), and from the 1890s they were treated more systematically by a German semi-follower, Ernst Schröder (1841–1902) in his mammoth Vorlesungen über die Algebra der Logik ( 212 volumes, 1890–1905) [Peckhaus, 1997].
Venn's term remains fairly durable, but over the decades its reference has become less clear. For from the late 1870s an alternative kind of symbolic logic was introduced, initially by Gottlob Frege (1848–1925) and about a decade later, and with far more publicity by Giuseppe Peano (1858–1932), who assembled a remarkable cohort of disciples at the University of Turin. The inspiration here came not from algebras but from mathematical analysis, especially the emphasis on rigour and the detailed exhibition of proofs in terms of a developed theory of limits. This approach had been initiated from the 1820s by Augustin-Louis Cauchy (1789–1857), together with the systematic indication of necessary and/or sufficient conditions for the truth of theorems, and the need to formulate definitions carefully and (where appropriate) with generality. However, he never explicitly presented logical principles, either within syllogistic logic or any other tradition of his time.
From the time of Cauchy's death this approach and its attendant practices were being promoted and refined in teaching at the University of Berlin by Karl Weierstrass (1815–1897). Peano's insight was that the refinements to rigour, definition and proof suggested that words were not precise enough, and he brought symbolisation not only to basic notions of the mathematical theories involved (both mathematical analysis and also some of these algebras and geometries, for example) but also of logic itself. There the main notions included logical connectives, which Boole and others had already given symbols, and also the predicate calculus and quantification, which both Frege and Peirce had already pioneered but which Peano popularised in his writings and those of his disciples. From the early 1890s their principal organs were a journal initially called Rivista di matematica, and editions of a compendium of presentations under the title Formulario matematico [Borga, Freguglia and Palladino, 1985].
Peano called this subject ‘mathematical logic’. The name had already been proposed in 1858 by De Morgan, but as a contrast to ‘philosophical logic’, where verbal expression alone was pursued; Venn had proposed ‘symbolic logic’ as a substitute. Peano's sense became widely known, and used by principal followers, especially in Britain by Bertrand Russell (1872–1970) and A.N. Whitehead (1861–1947). However, from 1904 a rather unnecessary alternative name came in, initially as ‘logistique’ in French, to refer both to Peano's and Russell's approaches. But both ‘logistic’ and ‘mathematical logic’ were ambiguous in an important respect, since the Peanists (as they were sometimes known) used their logic to express a mathematical theory in rigorous and axiomatised form; a presentation began with two columns of symbols, one for logical notions and the other for the mathematical ones. By contrast, during 1901 Russell decided that only one column was required: Peano's mathematical logic, together with a logic of relations which algebra had recently and importantly adjoined to it, was sufficient to deliver not only the needed methods of deduction but also the objects of mathematics (Rodriguez-Consuegra 1991). This stance was presented in much (though not complete) detail in their Principia mathematica (1910–1913) [Grattan-Guinness, 2000a, Chs. 6–7]. The position has become known as ‘logicism’; I shall use it for convenience, although it was introduced in the late 1920s by Rudolf Carnap (1891–1970), seemingly to avoid the ambiguity surrounding ‘logistic’.