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Prior on Aristotle’s Logical Squares

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This paper introduces Prior’s unpublished paper Aristotle on Logical Squares, which is deposited in the Bodleian Library and which discusses Greniewski’s definition of the \(\Box \) operator, which Greniewski introduced in his paper Próba odmłodzeniakwadratu logicznego. It is a unique attempt to formalize the square of opposition. Bendiek’s review, which is an important intermediary between Greniewski’s and Prior’s paper, is also mentioned here. Greniewski’s main motivation was to rejuvenate the traditional square of opposition in order to make a square of opposition more precise. He did not investigate the history of the square of opposition. Prior’s approach differs a great deal from Greniewski’s one because he bases his analyses of Greniewski’s operator on Aristotle’s definitions of squares of oppositions. There are some queries, which arise due to the different aims of Greniewski and Prior and their different approach might have caused Prior’s effort to the approximation of Greniewski’s operator to Aristotle’s definitions of the squares of opposition was unsuccessful.

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  1. 1.

    The use of this symbol might be confusing since the “box” usually symbolizes the necessity. However in this article it will be used as a symbol of the ‘square’, if it is not stated otherwise.

  2. 2.

    Apart from the common symbols as ‘\(\vee \)’ for the inclusive disjunction, ‘\(\wedge \)’ for the conjunction, ‘\(\rightarrow \)’ for the implication and ‘\(\leftrightarrow \)’ for the equivalence, this paper uses the less frequently seen symbols such as ‘\(\oplus \)’ for the exclusive disjunction and ‘\(\vert \)’ for the Sheffer stroke. Further less common symbols will be explained in the parts of this paper where they are dealt with.

  3. 3.

    It did not even appear in Greniewski’s (1955) textbook Elementy logiki formalnej, which was published two years after the publication of Próbaodmłodzeniakwadratu logicznego. Greniewski discusses here the square of opposition but he does not anywhere mention the \(\Box \) operator.

  4. 4.

    This definition differs not only in the replacement of the implications and in the addition of the end of the formula. There is implicitly contained modality within the propositions. It is not expressed in the formalization of these propositions, even though, it can be seen on the operators \(\oplus \)’ and \(\Rightarrow \) which are used in the definition. The formula (p \(\oplus \)’ s) is defined as \(\Box \,(\hbox {p }\oplus \hbox { s})\) and the formula \((\hbox {p }\Rightarrow \hbox { r})\) is defined as \(\Box \,(\hbox {p }\rightarrow \hbox { r})\). The \(\Box \) symbol represents here the ‘box’, the symbol of necessity.

  5. 5.

    In fact there is serious problem in this definition. The formula in Prior’s paper contains a contradiction in the last conjunct. Prior original formula has a form:

    $$\begin{aligned} \Box \left( \begin{array}{ll} c&{}\quad b \\ a&{}\quad d \\ \end{array} \right)= & {} \left[ \left( \left( a\!\leftrightarrow \!b \right) \wedge \left( {a\leftrightarrow {^{\prime }}b} \right) \right) \!\wedge \!\left( \left( {c\!\leftrightarrow \!d} \right) \wedge \left( {c\leftrightarrow {^{\prime }}d} \right) \right) \wedge \left. \left( \left( {c\vee a} \right) \right. \right. \right. \left. \left. \wedge \,\, \lnot \left( {a\vee c} \right) \right) \right] \end{aligned}$$

    However, in the text Prior correctly describe the relationship between c and a as an implication and that it is not vice versa. Hence I decided to include the correct form into the paper.

    This oversight might have been caused by Aristotle’s ambiguous definition on which is this Prior’s attempt based. In addition, it is obvious, that if Prior published this paper, this oversight would be corrected.

    The \(\oplus \) operator is replaced here with the conjunction of two formulas in which appear the \(\leftrightarrow \) operator and the \(\leftrightarrow \)’ operator. The latter means ‘Everything is either an A or a B’. Since Prior uses for it the symbol of equivalence I have followed his example.

  6. 6.

    The same oversight as in the previous formula occurs also here. Prior’s original formula was this:

    $$\begin{aligned} \Box \left( {{\begin{array}{ll} a&{}\quad e \\ i&{}\quad o \\ \end{array} }} \right) = \left[ {\left( {\left( {i\leftrightarrow e} \right) \wedge \left( {i\leftrightarrow {^{\prime }}e} \right) } \right) \wedge \left( {\left( {a\leftrightarrow o} \right) \wedge \left( {a\leftrightarrow {^{\prime }}o} \right) } \right) \wedge \left. {\left( {\left( {a\vee i} \right) \wedge \lnot \left( {i\vee a} \right) } \right) } \right] } \right. \end{aligned}$$


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  2. Aristoteles. (1961). První Anayltiky. Praha: Nakladatelství Československé Akademie věd.

  3. Bendiek, Johannes. (1955). Review: Henryk Greniewski, Versucheiner “Verjungung” des Logischen Quadrates. Journal of Symbolic Logic, 20(1), 83–84.

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  13. Uckelman, S. L. (2012). Arthur Prior and medieval logic. Synthese, 188(3), 349–366.

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I am grateful to my supervisor Jan Štěpán, and I would also like to thank Peter Øhrstrøm who invited me into The Virtual Lab for Prior Studies, Jørgen Albretsen the administrator of The Virtual Lab for Prior Studies, and anonymous referees of an extended abstract and a previous version of this paper for their helpful comments. This work was supported by the student project “A. N. Prior’s modal and temporal logic” No. IGA_FF_2014_037 of the Palacký University Olomouc.

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Correspondence to Zuzana Rybaříková.

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Rybaříková, Z. Prior on Aristotle’s Logical Squares. Synthese 193, 3473–3482 (2016).

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  • A. N. Prior
  • H. Greniewski
  • The square of opposition
  • The \(\Box \) operator
  • Aristotle