Chapter Six

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Chapter Six

• Sentential Logic Truth Trees

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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).

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Normal Forms and Trees, continued

• DNFs can be constructed mechanically, and can be
  used to construct natural deduction proofs
  mechanicallyalthough this is long and tedious.

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Normal Forms and Trees, continued

• DNFs show us that in sentential logic syntax
  mirrors semantics the truth trees are just a
  very efficient form of DNFs.

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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.

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Key Terms

• Closed branch
• Disjunctive normal form
• Open branch
• Truth tree method