Abstract
In the first half of this paper, we present a fragment of relational syllogisms named RELSYLL consisting of quantified statements with a special set of numerical quantifiers, and introduce a number of concepts that are useful for the later sections, including indirect reduction, quantifier transformations and equivalence of syllogisms. After determining the valid and invalid syllogisms in RELSYLL, we then introduce two Derivation Methods which can be used to derive valid relational syllogisms based on known valid simple syllogisms. We also show that the two Methods are sound and complete for RELSYLL. In the second half of this paper, we discuss ways to extend the Derivation Methods, including the use of more valid syllogisms and the use of existential assumptions. In this way, we are able to derive more relational syllogisms that contain other types of non-classical quantifiers, including “only” and proportional quantifiers. Finally, we state and prove a proposition concerning the relationship between the two Methods.
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Notes
A quantified statement can be seen as a \(0\)-ary predicate.
In this paper, we use the courier font to represent the denotation of a term as well as objects in a model. Hence, \(\texttt {r}\) represents the denotation of \(r\), i.e. a set of ordered pairs, and \(\texttt {Q}\) represents the logical relation denoted by the quantifier \(Q\).
For any binary predicate \(r\), the converse of \(r\) is defined as the binary predicate denoted \(r^{-1}\) such that \((\texttt {x}, \texttt {y}) \in \texttt {r}^{-1}\) iff \((\texttt {y}, \texttt {x}) \in \texttt {r}\). Note that in this paper, we use “iff” to represent “if and only if”.
“At least all but \(n\)” is usually (and more naturally) expressed as “all but at most \(n\)”, whereas “at most all but \(n\)” is usually (and more naturally) expressed as "at least \(n\) ...not".
Note that the non-negative integers \(n\) and \(0\) in “At least all but \(n\) a are b”, “At most \(n\) a are b”, “At least \(0\) a are b” and “At most all but \(0\) a are b”, where \(n\) = the cardinality of a, are considered inappropriate, because these integers make the statements vacuously true.
Formally, \(Xa\) is true iff \(\texttt {x} \in \texttt {a}.\)
In this case, we can in fact make use of the symmetry of \(E\) and the contrapositivity of \(A\) (see Zuber 2007 for the definitions of these two properties) to rewrite the statements first as \(Eb(Xr^{-1})\) and \( A\lnot b(X\lnot r^{-1})\), respectively, and then interchange \(X\) with the type \(\langle 1\rangle \) quantifiers \(Eb\) and \(A\lnot b\), respectively. However, we will not discuss the symmetry / contrapositivity of quantifiers in this paper.
Note that the order in which the premises appear on the left of \(\vdash \) is immaterial.
The reason for the restriction on \(D_4\) and \(D_5\) will be discussed in the next subsection.
The idea of this nomenclature is borrowed from Thom (1977), with substantial modifications.
Remember that the numerical quantifier \(nA\) can be rendered as “all but at most \(n\)” as well as “at least all but \(n\)”.
Recall that \(Ur^{-1}\) is a predicate which means “that which \(r\) \(u\)”.
In what follows, we have reordered the premises to make it easier to recognize the form of the final syllogism obtained. This reordering is not an essential step of the Derivation Methods.
Dekker (2015) used small-case letters to represent quantifiers and represent the syllogism by \(\ddot{a}oo\hbox {-}1\). For consistency with the notation adopted in this paper, we use capital letters and represent this syllogism by \({\ddot{A}}OO\hbox {-}1\).
In what follows, we make use of the outer negation relation between \(E\) and \(I\) to interpret \(\lnot E\) as “at least one”.
Here we assume the interpretation of the proportional quantifiers to be: \(\ge _p\!\!ab\) is true iff \(|\texttt {a} \cap \texttt {b}| / |\texttt {a}| \ge p\) and \(<_p\!\!ab\) is true iff \(|\texttt {a} \cap \texttt {b}| / |\texttt {a}| < p\). If \(\texttt {a} = \emptyset \), the quantified statements \(\ge _p\!\!ab\) and \(<_p\!\!ab\) are meaningless.
Remember that the numerical quantifier \(nO\) can be rendered as “at least \(n\) \(\ldots \) not” as well as “at most all but \(n\)”.
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Appendix: Counterexamples for Invalid Numerical Syllogisms
Appendix: Counterexamples for Invalid Numerical Syllogisms
This Appendix provides a list of counterexamples for invalid simple syllogisms with numerical quantifiers which may be used for constructing counterexamples for a certain type of invalid relational syllogisms in RELSYLL as described in the proof for Proposition 4. Since RELSYLL only includes relational syllogisms that are based on simple syllogisms of Figures 1 to 3, the following list only includes invalid Figure-1 syllogisms for conciseness. Based on the counterexample for an invalid Figure-1 syllogism \(\alpha \) listed below, one can obtain a counterexample for the invalid Figure-2 syllogism that is equivalent to \(\alpha \) by interchanging \(\texttt {b}\) and \(\texttt {c}\), and a counterexample for the invalid Figure-3 syllogism that is equivalent to \(\alpha \) by interchanging \(\texttt {a}\) and \(\texttt {c}\). Also for conciseness, syllogisms that can be invalidated by the same counterexample are placed in the same cell below.
Note that each counterexample listed below is in fact a counterexample for an infinite number of invalid numerical syllogisms, not only because the figures \(n, m, k\) below can be substituted by an infinite number of appropriate non-negative integers, but also because from each invalid numerical syllogism one can deduce more invalid syllogisms with stronger conclusions. For example, the first counterexample listed below is not only a counterexample for \(nAmA(n + m - 1)A\hbox {-}1\), but also one for \(nAmA(n + m - 2)A\hbox {-}1\), because if \((n + m - 1)Aab\) is a false conclusion, then \((n + m - 2)Aab\) must also be a false conclusion.
Type 1: Numerical syllogisms based on valid classical syllogisms but associated with the wrong numerals
Type 2: Numerical syllogisms based on invalid classical syllogisms
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Chow, Kf. Relational Syllogisms with Numerical Quantifiers and Beyond. J of Log Lang and Inf 31, 1–34 (2022). https://doi.org/10.1007/s10849-021-09345-8
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DOI: https://doi.org/10.1007/s10849-021-09345-8