Chapter Six Sentential Logic Truth Trees

1. The Sentential Logic Truth Tree Method People who developed the truth tree method: J. Hintikka— “model sets” E.W. Beth – “semantic tableaux” Richard Jeffrey, in Formal Logic: Its Scope and Limits

The Sentential Logic Truth Tree Method, continued Like truth tables and unlike the method of proofs, the truth tree method provides a mechanical decision procedure for the validity and invalidity of any sentential argument.

The Sentential Logic Truth Tree Method, continued Like proofs and unlike truth tables, the truth tree method is purely syntactical; it does not rely on semantics. Truth trees provide a representation of semantics, a picture of truth conditions.

The Sentential Logic Truth Tree Method, continued The basic principle behind the truth tree method is the reductio proof: Show that the assumption of the negation of the conclusion together with the premises yields a contradiction and so the original conclusion follows validly from the premises.

2. The Truth Tree Rules From truth tables we know that the only lines that we need to look at when testing for the validity of an argument form are those in which the conclusion is false: in those lines, we look to see if all the premises are true. Truth trees give us a new method for doing the same thing: A truth tree pictures truth conditions.

The Truth Tree Rules, continued The tree rule for a logical connective is the picture of the truth table for it. A tree rule for a formula R (p, q) has a branch when there is more than one line of the truth table in which the formula is true. We can always cross of the double negations whenever and wherever they occur.

3. Details of Tree Construction To start a tree, list the premises of the argument we wish to test, and the negation of the conclusion. Break down all the lines that contain connectives according to the rules, until we have listed the truth conditions for all the relevant formulas. The premises and negated conclusion are the trunk of the tree. Each completed branch will picture truth conditions for the wffs in question, and so picture a row in a truth table.

Details of Tree Construction, continued A closed branch will tell us that the conditions that make some of the premises or the negation of the conclusion true make some other of them false. Open branches represent sets of truth conditions that make all the premises and the negation of the conclusion true.

Details of Tree Construction, continued Although logically speaking it makes no difference which statement we start with, it does strategically: We want the smallest tree possible, since that is the least amount of work. So, it is best to save breakdowns that produce branching until the end, hoping that we can cross of the lines before we have to branch.

4. Normal Forms and Trees Every statement form except a contradictory form can be given a logically equivalent expression called its disjunctive normal form (DNF).

Normal Forms and Trees, continued DNFs can be constructed mechanically, and can be used to construct natural deduction proofs mechanically—although this is long and tedious.

DNFs show us that in sentential logic syntax mirrors semantics: the truth trees are just a very efficient form of DNFs. Normal Forms and Trees, continued

5.Constrcting Tree Rules for Any Function Given any truth table you should be able to construct the tree rule for the function that goes on top of the table, even if you do not know what the function is specifically.

Key Terms Closed branch Disjunctive normal form Open branch Truth tree method