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

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 logicEdit

Rules not requiring subproofsEdit

  • 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 subproofEdit

  • Implication by direct proof
 
 
  implies