Relations and Multiple Quantifiers in Predicate Logic

Thus far we have been using only one-place predicates. These are predicates that only require one name (put differently, these are properties that only require one object) to complete them. For example,

_____ is red = Rx

_____ is heavy = Hx

_____ is a wife = Wx

(∃x)Rx is true if and only if there is at least one thing in the universe that is red. (∃x)Hx is true iff there is at least one thing in the universe that is heavy, and so on. (x)Rx is true iff everything in the universe is red, and so on. 

 But, as we all know, there are predicates that are more than one place. In other words, there are predicates that require more than one name to complete them (put differently, there are properties-- better-called relations--that require more than one object). For example

 ____ is heavier than ____ = Hxy

 ____ is taller than ____ =  Txy

 ____ loves ____ = Lxy

 ____ is between ____, _____ = Bxyz

 (∃x)(∃y)Hxy is true iff there is something in the universe that is heavier than something else in the universe. (∃x)(∃y)Txy is true iff there is something in the universe that is taller than something else in the universe, and so on (in some cases, the 'something else' clause is not needed. For example (∃x)(∃y)Lxy is true iff there is something in the universe that loves something in the universe and that something can be the same thing--if John loves himself then (∃x)(∃y)Lxy is true; it depends on the nature of the predicate). 

 Now things get a bit trickier when we mix quantifiers. In general, the outermost quantifier has a kind of control over the entire statement (i.e. the outermost quantifier has the widest scope, it governs more of the statement). For example,

 (x)(∃y)Lxy 

 This states that for all x, there is a y, such that x loves y. Because the (x) is the outermost quantifier it governs the rest of the statement. So the statement says that for any x we choose there is something that it loves. Put differently, it states that everyone loves someone. 

 (∃y)(x)Lxy

 Note that the only thing we change was the order of the quantifiers. We did not change the order of the variables next to the predicate. So, this statement tells us that there is some y, for all x, such that x loves y. Because the (∃y) is the outermost quantifier it governs the rest of the statement. So the statement says that for some y that we choose everything loves it, loves that individual y. Put differently, it states that someone is loved by everyone. 

 Notice what happens when we switch up the order of the variables. 

 (∃y)(x)Lyx

 This statement is quite different from the one above. It tells us that there is some y, for all x, such that y loves x. That is, there is someone who loves everyone (I wish I could say it was me :-(.




Comments

Post a Comment

Popular posts from this blog

Predicates in search of nouns

According to standard categorical logic, statements of the form All S are P must have some sort of noun-phrase in the predicate.  Thus ‘All dogs are mammals’ is fine, but ‘All dogs are four-legged’ is not.  Why is this a requirement? Standard categorical logic is based on Aristotelian logic and inherits some of Aristotle’s assumptions. So one reason for the noun-requirement is that given Aristotelian metaphysics predicates without nouns are incomplete. To understand why, we first need to keep in mind that  predicates (parts of language) correspond to  properties (parts of the extra-linguistic—that is, non-linguistic—world).  Predicates without nouns are incomplete because, according to Aristotle,  properties  must inhere in objects in order to exist.  There are no free-floating properties—properties without some subject that they attach to.  Thus, statements like ‘All dogs are four-legged’ while grammatic...
Read more

Meta-Logical Themes: Soundness of Propositional Logic

We have been studying propositional logic (PL) for some time now. PL is about whole or complete propositions and their relationships to each other in terms of truth-preservation. That is, PL investigates the relationships between sets of true propositions to other propositions. How can we be sure that if we are given a set of true propositions (called the premises) that we will through a series of steps wind up with another true proposition and not a false one (i.e. how can we be sure that the relationship between the premise set and the conclusion is, in fact, a valid one). Put differently, we want to make sure that the logic we are using will never take us from truths to falsity. We have been using our rules to show that if we start with a set of (assumed to be) true propositions we can derive another proposition that must be true if the premises are. But we have only studied a finite number of arguments. There is an infinite number of arguments. So, how can we be sure that there i...
Read more