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Predicate Logic

From Wolf:

  1. A mathematical variable is a symbol (or combination of symbols like tex2html_wrap_inline281 ) that stands for an unspecified number or other object. The collections of objects from which any particular variable can take its values is called the domain or universe of that variable. Variables with the same domain are said to be of the same sort.
  2. Two symbols, called quantifiers, stand for the following words:
    1. tex2html_wrap_inline283 for ``for all'' or ``for every'' or ``for any''. Universal quantifier.
    2. tex2html_wrap_inline285 for ``there exists'' or ``there is'' or ``for some''. Existential quantifier.
  3. The quantifiers are used in symbolic mathematical language as follows: If P is any statement, and x is any mathematical variable (not necessarily a real number variable), then tex2html_wrap_inline291 and tex2html_wrap_inline293 are also statements. In this case we say that x is quantified.
  4. A mathematical variable occurring in a symbolic statement is called free if it is unquantified and bound if it is quantified. If a statement has no free variables it's called closed. Otherwise it's called a predicate, an open sentence, an open statement, or a propositional function.
  5. A statement of the form tex2html_wrap_inline297 is defined to be true provided P(x) is true for each particular value of x from its domain. Similarly, tex2html_wrap_inline303 is defined to be true provided P(x) is true for at least one value of x from that domain.
  6. For any statement P(x), tex2html_wrap_inline311 is logically equivalent to tex2html_wrap_inline313 . Also, tex2html_wrap_inline315 is logically equivalent to tex2html_wrap_inline317 .
  7. Let P be any statement, x be any mathematical variable, and t any term that denotes a set. (So t could be a single letter standing for a set, or a more complicated expression like tex2html_wrap_inline327 .) Then
    1. tex2html_wrap_inline329 is an abbreviation for tex2html_wrap_inline331 .
    2. tex2html_wrap_inline333 is an abbreviation for tex2html_wrap_inline335 .

Carl Lee
Wed Nov 18 12:16:44 EST 1998