Derivations in Predicate Logic
Peter Suber, Philosophy Department, Earlham College
  1. All the derivation rules we've learned so far apply in predicate logic: the 9 rules of inference, the 10 rules of replacement, plus conditional and indirect proof. So keep practicing them; you still need them. (For help, see my earlier hand-out of tips.)

  2. Of the four new rules, UI, EI, UG, and EG, two are easy (UI and EG), and two are difficult and become familiar only with practice (EI and UG). (For help, see my earlier hand-out on these four rules.)

  3. Remember that UI, EI, UG, and EG apply only to entire lines of proof, not to components of larger compounds.

  4. The QN rules apply either to whole lines or to components within larger compounds.

  5. The general strategy for predicate logic derivations is to work through these three phases: (1) instantiate the premises, (2) work with what you have then, using the original 19 rules plus CP and IP, and (3) then generalize as needed to put the right quantifiers on the conclusion.
  6. One reason why it's usually necessary to instantiate your premises is that the original 9 rules of inference usually do not apply to quantified statements. They only apply to simple statements and truth-functional compounds.
  7. The original 10 rules of replacement apply to components within larger compounds, even components of predicate logic expressions.
  8. Keep some clear statement of UI, EI, UG, and EG at hand when doing derivation exercises, at least at first. (Use the inside back cover of Copi's book, or my hand-out, or your own notes.) Whenever you need to drop a quantifier, look up the relevant instantiation rule to see whether you can comply with all restrictions. Whenever you need to add a quantifier, look up the relevant generalization rule to see whether you can comply with all restrictions. It can help to write down the inference you want to draw (adding or dropping a quantifier) and then look up the relevant rule. After a handful of practice derivations for each rule, you should be able to apply them without the reference sheet.

  9. It is usually helpful to instantiate all quantified expressions to the same variable or constant. Then the instantiated statements are more likely to "bind" with one another under the inference rules.
  10. Instantiate existential quantifiers before universal quantifiers.
  11. If you have two or more existential quantifiers, realize that you cannot instantiate them to the same variable. Before you instantiate them to different variables, see whether you can do the proof without instantiating one or more of them at all.

  12. Don't instantiate negated quantifiers. Copi forgot to tell you that this is illegal.
  13. The restriction on UG inside the scope of an assumption should be understood precisely or you will fail to take some permissible inferences.
  14. When generalizing, the new quantifier goes at the far left of the expression, putting the entirety of the original expression within its scope. (This follows from rule 3 above.)
  15. A special case of the previous rule is that, when generalizing a negated statement, the negation sign stays to the right of quantifier. When you're finished, the negation sign should be inside scope of the quantifier, not vice versa.
  16. We now have three kinds of assumption, and they all discharge differently. Each can nest inside any combination of the others. So keep track of which is which, and discharge each one properly.
  17. In working with polyadic predicates, generally avoid generalizing different variables to the same variable in the same expression.

This file is an electronic hand-out for the course, Symbolic Logic.

Most of the logic symbols in this file are GIFs. See my Notes on Logic Notation on the Web.

Ribbon] Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U.S.A. Copyright © 1997, Peter Suber.