Order and Ambiguity with Mixed Quantifiers in FOL [or something along those lines…]

by Ken Hemba

[4/22/03]

 I have on several occasions had the impression that Professor Ryckman may have chosen the LPL text as much for its sarcasm and wit as its clarity and completeness.  I had occasion to note this when once, while paging through the index, I came upon the entry “Saturday Night Live” nestled between “satisfiable truth table” and “scope.”  I had to satiate my curiosity, and upon investigation I found that the section in question is one that can often be quite puzzling to introductory Logic students (I will leave it to the reader to investigate the specific reference if interested).

 The issue was with the order of quantifiers in mixed-quantifier sentences, and the (sometimes amusing) ambiguity that results.  Statements like “every man loves a woman” can be interpreted in two ways: first, that every man loves some woman (or other), and second, that there is some (specific) woman who is loved by every man.

 These two interpretations require different formulations in FOL, as they obviously carry quite different implications.  The primary difference between the two is the order in which the existential and universal quantifiers are written.  One may note that the order in which the male and female individuals were mentioned above is similarly reversed.  This was intentional (though not necessary), as these happen to be the orders in which the quantifiers must be written in order to yield each interpretation.

 Here are two tricks that may be helpful in translating such sentences.  First, it is often helpful to ask whether the words “specific” or “or other” could be used in relation to the existentially quantified individual.  If the existential quantifier is referring to some specific individual, it must be written first.  Accordingly, if the existential quantifier is referring to some individual or other, it must be written second.

 As well, it is often helpful to determine who (or what) the sentence is “talking about,” as the related quantifier is the one that must be written first.  In the first interpretation of our sentence above, we are talking about the fixed set of all men, of which each member has the property of loving some woman (or other).  In order to yield this interpretation the (universal) quantifier binding these individuals must come first.  Similarly, in the second interpretation, the subject of the sentence is some (specific) woman who is loved by every man.  In this case the (existential) quantifier binding this individual must be written first.  It is worth noting that these orders remain the same whether the sentence is written with only one quantifier in front (Aristotelian form) or both (prenex form).

 The reader will find these tricks can be helpful both in translating from English to FOL and the reverse. The best way to learn this principle is of course to work it out in specific examples, and the text offers several.  For a more in-depth discussion of this topic, I refer the reader to section 11.5 of LPL.