Section 1.3: Quantifiers, Predicates, and Validity

Abstract:

We now throw variables into the mix, and investigate wffs which describe properties of variables in given domains. We still test their truth values, either for the specific domain in question, or even in all domains (validity).

Predicates and quantifiers

• quantifier: tells how many objects have a certain property.

  Examples:

  • universal quantifier - - ``for all'', ``for every''
  • existential quantifier - - ``there exists'', ``for at least one''

  Have you encountered these quantifiers before?

• predicate: a property of a variable (e.g. x prime)

  We combine the quantifiers and predicates to create expressions such as

  

  which we then must interpret. There is nothing special about the variable x, so this wff is the same as , , etc. We say that x is a dummy variable.

• Truth value may now depend on the Interpretation of an expression:
  • domain of interpretation - non-empty set to which the predicate expression is applied
  • assignment of a property of the objects to each predicate in the expression
  • assignment of particular objects to each constant symbol in the expression

  Example: Ex. 4(b,d)

• scope of a quantifier: the part of an expression to which the quantifier applies; indicated with parentheses or brackets (but these may be neglected if the scope is clear).
• free variable: a variable in a predicate wff outside the scope of a quantifier involving that variable.

Example: Ex. 5(c)

Translation of English statements into predicate wffs

As noted before, this can be a very tricky business, but an important one. Again, the process is not unique: there may be several different ways to do the same job.

Example: Ex. 10/11(a-d)

• Negation of predicate wffs: some cases are standard, e.g.
  • 

    The negation of ``Every x has property A.'' is ``There is an x which doesn't have property A.''

  • 

    The negation of ``There is an x which has property A.'' is ``No x has property A.''

  In general, English makes negation kind of tricky. Watch your step!

Example: Ex. 13/14(c,d)

Validity

The truth value of a predicate wff depends on the interpretation, but there are some for which the wff is true independent of the interpretation. These are called valid predicate wffs.

Whereas we can check the ``validity'' of a propositional wff (just check the truth table to see if it's a tautology), there is no general check for the validity of a predicate wff. In spite of that, there are some valid predicate wffs, as demonstrated in the text:

Example: Ex. 15/18(d,e)