Where is ‘there is’ in ‘\(\exists\)’? (English) Zbl 1512.03021
Summary: The paper offers a survey of four key moments in which symbolisms for quantification were first introduced: §§11-2 of Frege’s Begriffsschrift (1879); Peirce’s ‘Algebra of Logic’ (1885); Peano’s ‘Studii di Logica matematica’ (1897); and *9 (‘replaced’ by *8 in the second edition) of Whitehead and Russell’s Principia Mathematica (1910). Despite their divergent aims, these authors present substantially equivalent visions of what their differing symbolisms express. In each case, some passage suggests that one (but not the only) way to render one of the symbols into ordinary-language words (German, English and Italian) is to say that there is something or some thing(s) exist(s). Exactly how this comes out varies from language to language, but the point remains the same. As a result, in almost all recent logic manuals that introduce ‘\(\exists\)’, some passage says that the symbol means ‘there is’ or ‘there exists’, and the tradition has grown up of labelling ‘\(\exists\)’ ‘the existential quantifier’. Some considerations are offered for deprecating this reading of ‘\(\exists\)’ and the label adopted for it, and for preferring to read it as ‘for some (at least one)’ and for speaking of the ‘particular quantifier’.
MSC:
| 03A05 | Philosophical and critical aspects of logic and foundations |
| 03-03 | History of mathematical logic and foundations |
| 01A55 | History of mathematics in the 19th century |
| 01A60 | History of mathematics in the 20th century |
References:
| [1] | Brady, G., From Peirce to Skolem: A Neglected Chapter in the History of Logic (2000), Amsterdam: Elsevier, Amsterdam · Zbl 0958.01009 |
| [2] | Carroll, L., Through the Looking Glass (1872), London: Chancellor Press, London |
| [3] | Dummett, M. A. E., Frege: Philosophy of Language (1973), London: Duckworth, London |
| [4] | Frege, G., Begriffsschrift. Einer der arithmetischen nachgebildete Formelsprache des reinen Denkens (1879), Halle: Louis Nebert, Halle |
| [5] | Frege, G., The Foundations of Arithmetic (1884), Austin, Oxford: Basil Blackwell, Austin, Oxford |
| [6] | Frege, G., Conceptual Notation and Related Articles (1972), Bynum, Oxford: Clarendon Press, Bynum, Oxford |
| [7] | Gentzen, G., Untersuchungen über das logisches Schließen, Mathematische Zeitschrift, 39, 176-201 (1935) · Zbl 0010.14501 · doi:10.1007/BF01201353 |
| [8] | Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I, Monatshefte für Mathematik und Physik, 38, 173-198 (1931) · JFM 57.0054.02 · doi:10.1007/BF01700692 |
| [9] | Halbach, V., The Logic Manual (2010), Oxford: Oxford University Press, Oxford · Zbl 1420.03001 |
| [10] | Hamilton, W., Lectures on Logic (1837), Edinburgh: Blackwood, Edinburgh |
| [11] | Johnson, W. E., Logic, (3 Voll.) (1921), Cambridge: Cambridge University Press, Cambridge · JFM 48.0057.09 |
| [12] | Lemmon, E. J., Beginning Logic (1965), London: Nelson, London · Zbl 0158.24406 |
| [13] | Lewis, C. I.; Langford, C. H., Symbolic Logic (1959), New York: Dover, New York · Zbl 0087.00802 |
| [14] | Magnus, P. D. (2005), forallχ:Cambridge, revised by T. Button (2012) and downloadable through his site at people.ds.cam.ac.uk/tectb2/teaching.shtml. |
| [15] | Mitchell, H. O.; Peirce, C. S., On a new algebra of logic, Studies in Logic by Members of the Johns Hopkins University, 72-106 (1883), Boston: Little, Brown, Boston |
| [16] | Peano, G.1897. ‘Studii di logica matematica’, in Atti della Reale Accademia delle Scienze, XXXIII, disp. IIa, 565-584. · JFM 28.0340.01 |
| [17] | Peano, G., Studies in mathematical Logic, Selected Works of Giuseppe Peano, 190-205 (1973), London: Allen and Unwin, London |
| [18] | Peirce, C. S., On the Algebra of logic: A contribution to the Philosophy of Notation, American Journal of Mathematics, 7, 180-202 (1885) · JFM 17.0044.02 · doi:10.2307/2369451 |
| [19] | Peters, S.; Westerståhl, D., Quantifiers in Language and Logic (2006), Oxford: Oxford University Press, Oxford |
| [20] | Prior, A. N., Formal Logic (1955), Oxford: Clarendon Press, Oxford · Zbl 0067.24903 |
| [21] | Prior, A. N., Formal Logic (1962), Oxford: Clarendon Press, Oxford · Zbl 0124.00205 |
| [22] | Prior, A. N.; Hasle, P., Papers on Time and Tense (1968), Oxford: Oxford University Press, Oxford |
| [23] | Quine, W. V. O., Elementary Logic (1941), New York: Harper & Row, New York · Zbl 0027.00501 |
| [24] | Quine, W. V. O., Methods of Logic (1952), London: Routledge and Kegan Paul, London |
| [25] | Russell, B., The Principles of Mathematics (1903), London: G. Allen, London · JFM 34.0062.14 |
| [26] | Russell, B., My Philosophical Development (1959), London: Allen & Unwin, London |
| [27] | Schröder, E., Recension: Frege Begriffsschrift, Zeitschrift für Mathematik und Physik, 25, IV, 81-94 (1880) |
| [28] | Strawson, P. F., Introduction to Logical Theory (1952), London: Methuen, London · Zbl 0049.14602 |
| [29] | Whitehead, A. N.; Russell, B., Principia Mathematica (191013), Cambridge: Cambridge University Press, Cambridge |
| [30] | Whitehead, A. N.; Russell, B., Principia Mathematica (192527), Cambridge: Cambridge University Press, Cambridge |
| [31] | Wittgenstein, L., Tractatus Logico-Philosophicus (1921), London: Routledge and Kegan Paul, London · Zbl 0117.25106 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
