The laws of distribution for syllogisms.

*(English)*Zbl 0993.03006From the introduction: Since at least the seventeenth century, logic textbooks which discuss syllogisms have usually quoted some laws that a valid syllogism must obey. Different authors give different lists, but the following two laws are usually on the menu: (a) The middle term must be distributed in at least one premise. (Some authors add: exactly one premise.) (b) If a term is distributed in the conclusion, then it must be distributed in the premise in which it occurs. (Some authors add: if it is undistributed in the conclusion, then it must be undistributed in the premise in which it occurs.) The laws (a) and (b) are variously known as the laws of distribution or the laws of quantity.

One can justify the laws by checking that they hold for all valid syllogisms – there are only a small finite number to check. But many authors tried to give some general argument which covered all cases. These arguments were always unconvincing; but this was hardly surprising, since the early authors never managed to find suitable definitions of ‘distributed’ and ‘undistributed’. With twentieth century tools there is no problem in writing down sound definitions of these notions – in fact there are several ways of doing it – and then Lyndon’s interpolation theorem [R. C. Lyndon, Pac. J. Math. 9, 129-142 (1959; Zbl 0093.01002)] gives the laws of distribution almost immediately.

This paper is a revised version of the results of a discussion I had with Colwyn Williamson in March 1993. There is nothing original in it, beyond the easy observation that Lyndon’s theorem gives the distribution laws. I wrote it up for publication because I became aware that the facts have not reached print.

One can justify the laws by checking that they hold for all valid syllogisms – there are only a small finite number to check. But many authors tried to give some general argument which covered all cases. These arguments were always unconvincing; but this was hardly surprising, since the early authors never managed to find suitable definitions of ‘distributed’ and ‘undistributed’. With twentieth century tools there is no problem in writing down sound definitions of these notions – in fact there are several ways of doing it – and then Lyndon’s interpolation theorem [R. C. Lyndon, Pac. J. Math. 9, 129-142 (1959; Zbl 0093.01002)] gives the laws of distribution almost immediately.

This paper is a revised version of the results of a discussion I had with Colwyn Williamson in March 1993. There is nothing original in it, beyond the easy observation that Lyndon’s theorem gives the distribution laws. I wrote it up for publication because I became aware that the facts have not reached print.

##### MSC:

03B20 | Subsystems of classical logic (including intuitionistic logic) |

03-03 | History of mathematical logic and foundations |

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\textit{W. Hodges}, Notre Dame J. Formal Logic 39, No. 2, 221--230 (1998; Zbl 0993.03006)

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##### References:

[1] | Arnauld, A., and P. Nicole, La Logique, ou l’Art de Penser , Paris, 1662. |

[2] | Kneale, W., and M. Kneale, The Development of Logic , Clarendon Press, Oxford, 1962. · Zbl 0100.00807 |

[3] | Kretzmann, N., and E. Stump, editors, The Cambridge Translations of Medieval Philosophical Texts , vol. 1, Cambridge University Press, Cambridge, 1988. |

[4] | Lyndon, R., “An interpolation theorem in the predicate calculus,” Pacific Journal of Mathematics , vol. 9 (1959), pp. 129–42. · Zbl 0093.01002 |

[5] | Makinson, D., “Remarks on the concept of distribution in traditional logic,” Noû s, vol. 3 (1969), pp. 103–08. |

[6] | Peirce, C. S., “Qualitative logic,” pp. 323–71 in Writings of Charles S. Peirce , vol. 5, edited by C. J. W. Kloesel, Indiana University Press, Bloomington, 1993 (original 1886). · Zbl 1022.01014 |

[7] | Peter of Spain, Tractatus , called afterwards Summule Logicales , first critical edition from the manuscripts, with an introduction by L. M. de Rijk, Van Gorcum and Co., Assen, 1972. |

[8] | Sánchez Valencia, V., “Monotonicity in medieval logic,” Language and Cognition , vol. 4, Yearbook 1994 of the Research Group for Theoretical and Experimental Linguistics of the University of Groningen, Faculteit der Letteren, Rijksuniversiteit Groningen, pp. 161–74. |

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