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Next: Some Useful Tautologies Up: Brief Logic Previous: Brief Logic


From Wolf:

  1. A proposition is any declarative statement (including mathematical sentences such as equations) that is true or false.
  2. We use the letters tex2html_wrap_inline127 as propositional variables. That is, we let these letters stand for or represent statements.
  3. Five symbols, called connectives, are used to stand for the following words:
    1. tex2html_wrap_inline129 for ``and''. Conjunction.
    2. tex2html_wrap_inline131 for ``or''. Disjunction.
    3. tex2html_wrap_inline133 for ``not''. Negation.
    4. tex2html_wrap_inline135 for ``implies'' or ``if...then''. Conditional or Implication.
    5. tex2html_wrap_inline137 for ``if and only if''. Biconditional or Equivalence.
  4. A statement that is not built up from simpler ones by connectives and/or quantifiers is called atomic or simple. (Quantifiers will be introduced later.) A statement that is built up from simpler ones is called compound.
  5. The truth functions of the connectives are defined as follows:
    1. tex2html_wrap_inline139 is true provided P and Q are both true.
    2. tex2html_wrap_inline145 is true provided at least one of the statements P and Q is true.
    3. tex2html_wrap_inline151 is true provided P is false.
    4. tex2html_wrap_inline155 is true provided P is false, or Q is true (or both).
    5. tex2html_wrap_inline161 is true provided P and Q are both true or both false.
  6. A tautology, or a law of propositional logic, is a statement whose truth function has all T's as outputs.
  7. A contradition is a statement whose truth function has all F's as outputs (in other words, it's a statement whose negation is a tautology).
  8. Two statements tex2html_wrap_inline171 and tex2html_wrap_inline173 are called propositionally equivalent if tex2html_wrap_inline175 is a tautology.

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