Chapter Twelve Predicate Logic Truth Trees

1. Introductory Remarks The trees for sentential logic give us decidability—there is a mechanical decision procedure that a machine could follow to determine the validity or invalidity of each argument in sentential logic.

Introductory Remarks, continued The truth trees for predicate logic do not give us decidability as there can be no such decision procedure for predicate logic. This is called Church’s undecidability result.

Introductory Remarks, continued If an argument in predicate logic is valid, a machine will be able to decide it is valid in a finite number of steps. But if an argument is invalid a machine might not be able to show it is invalid in a finite number of steps.

Introductory Remarks, continued Given a tree in predicate logic, three things might occur: 1) All paths will close, so the argument is valid. 2) There will be at least one open path, and no way to apply the tree rules to any line in that path, so the argument is invalid. 3) The tree may seem to grow infinitely, in which case we cannot determine if the argument is invalid.

Introductory Remarks, continued Since we cannot predict if we have an infinitely growing tree we cannot know whether a particular argument that meets condition (3) is invalid or not.

2. General Features of the Method We use a form of indirect proof. We begin by testing an argument by listing its premises and the negation of its conclusion. The only new rules we need are two of the four QN rules and UI and EI. UI and EI will always be applied to constants.

General Features of the Method, continued If we incorporate the identity sign into our symbolism trees become more difficult to construct.

3. Specific Examples of the Method There are four new rules for predicate trees that supplement those for sentential trees: these concern the use of denial, connectives, UI and EI.

Specific Examples of the Method, continued There are two methods for doing predicate trees: A. The adherence to a prescribed order. B. The unrestricted order.

4. Some Advantages of the Trees For longer natural deduction proofs trees will usually involve fewer steps. We can also break down certain sentences more easily than we can in proofs.

5. Example of an Invalid Argument with at Least One Open Path If we apply the tree rules until we can no longer apply them and end with at least one open path we have an invalid argument. We can then read off the truth-values of the atomic sentences and construct a counterexample to the argument.

6. Metatheoretic Results Invalidity in a domain: An argument is invalid in a domain if we can find a counterexample in it. A domain with n members is said to be of cardinality n.

Metatheoretic Results, continued 1. The tree method will mechanically yield a correct decision on every argument on which it yields any decision at all, and it will yield a correct decision on all valid arguments.

Metatheoretic Results, continued 2. We should be able to figure out a method such that with the method of expansion we could know that if we choose a domain of a certain size, a valid argument will show up valid for the domain and hence for all domains of cardinality greater than zero.

Metatheoretic Results, continued 3. If the argument we are testing by trees is invalid, the method may fail, and the method of expression equally fails. Even for some argument we know to be invalid, the truth tree method will not yield a decision.

Metatheoretic Results, continued Points 1-3 are in effect Church’s undecidability results.

7. Strategy and Accounting When predicate truth trees branch, the rules apply serially to each open path.

Key Terms Cardinality of a domain Church’s undecidability result Flowchart for predicate trees Infinitely growing tree Invalid argument Invalidity in a domain