Mathematical Proof and the Principles of Mathematics/Logic/Rules of inference summary

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This is a list of the rules of inference given in previous sections. In the notation being used, a solid horizontal bar means that the statement below is a valid deduction from the statement(s) above. A vertical bar with a horizontal bar connected to it means that whatever is to the right of the vertical bar is a subproof, and whatever is above the horizontal bar are assumption(s) and whetever is below the horizontal bar is what has been derived. The names given are just placeholders and no guarantees are made that they are standard in any way.

Propositional logic[edit]

Rules not requiring subproofs[edit]

  • Iteration
  • Use of contradiction
  • Disjunction by first case
or
  • Disjunction by second case
or
  • First use of conjunction
and
  • Second use of conjunction
and
  • Implication from the conclusion
implies
  • Implication from false assumption
not
implies
  • Double negation
not not
  • Equivalence to implication
iff
implies
  • Equivalence to converse
iff
implies
  • Conjunction by components
and
  • Use of disjunction, first alternative false
or
not
  • Use of disjunction, second alternative false
or
not
  • Use of implication, from premise (modus ponens)
implies
  • Use of implication, from false conclusion (modus tollens)
implies
not
not


Rules requiring one subproof[edit]

  • Implication by direct proof
implies