EFFECTIVE COMPLETENESS

An open path of a truth tree is complete when all the statements on the path have been decomposed, i.e., are either ticked compounds or Hterals.

Now we can characterize the truth tree method for determining validity as follows:

A sequent is valid when all the paths of its truth tree are closed.

A sequent is invalid when at least one open path of its truth tree is complete or effectively complete.

Here’s an example of an argument involving effective completeness:

Here we use the Ξ andV rules before applying the V rule, in accordance with the restric­tions on the predicate logic rules. This means that we had to use a different name j in applyingon line 6, since we had already used ³ in applying Ξ on line 5. But this entails that when we come to applyat line 7, we have two choices for the name we use. When we have exhausted both, the path is effectively complete. But it remains open, so the argument is invalid. This is what we should have expected: it is hardly valid to reason from “All photos of ALIENS are FORGERIES” to “If there are any photos of aliens, all photos are forgeries”! (Here I have applied an interpretation to the abstract argument, with a UD of photos, Ax := x is a photo of an alien, Fx := x is a forgery.) The open path shows why: if there’s some example ³ of an alien photo (Ai), then it is a forgery (Fi).

SUMMARY

In predicate logic, our truth tree rules for statement logic need supplementing by decomposition rules for universal and existential quantifications:

EXERCISES 23.1

Prove the validity Ofthefollowing sequents using the truth tree method:

Using the truth tree method, determine whether each of the following abstract arguments is formally valid or invalid:

Using truth trees, determine whether each of arguments 14-18 (= ch. 18, 1-5) is valid. For those that are instances of invalid forms, give a set of possible truth values that makes the form invalid.

14. None of HIS stories are PROBABLE. Improbable stories are not easily BELIEVED. So none of his stories are easily believed. [UD: stories]

15. WARMTH RELIEVES pain. Nothing that does not relieve pain is useful for TOOTH­ACHE. So warmth is useful for toothache.

16. UNIVERSITY students are all EDUCATED. All uneducated people are SHALLOW. Therefore no university students are shallow. [UD: people]

17. No WHEELBARROWS are COMFORTABLE. No uncomfortable vehicles are POPULAR. Therefore no wheelbarrows are popular. [UD: vehicles]

18. Some HEALTHY people are OVERWEIGHT. No unhealthy people are good INSURANCE risks. Therefore some overweight people are not good insurance risks. [UD: people]

19. Using a truth tree, show that the following penevalid argument is valid by supplying the obvious implicit premise that there are Vermont-made beers:

All VERMONT-made beers taste GREAT. All Vermont-made beers are MICRO­BREWED. Therefore at least some micro-brewed beers taste great.

20. Using a truth tree, show that the following penevalid argument is valid by supplying the obvious implicit premise:

No SLOTHS throw CURVE balls. All sloths eat LEAVES. Therefore some crea­tures that eat leaves do not throw curve balls. [UD: creatures]

Prove the invalidity Ofthefollowing sequents using truth trees:

23.2