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Navigating Quantified Truth Trees: Strategies and Rules

Learn how to effectively navigate and apply rules in quantified truth trees, understand key strategies to close branches and avoid splitting trees, with practical examples and explanations.

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Navigating Quantified Truth Trees: Strategies and Rules

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  1. Chapter 6Quantified Truth Trees • 6.1 and 6.2: truth tree => NDL Practically useful • 6.3: Models or interpretations => Slate TheoreticallyImportant • 6.4: From truth trees to interpretations Meta-theoretically Interesting

  2. 6.1 Rule for Quantifiers • Church: undecidability, meaning no deterministic mechanics or procedures in checking of validity • Could result in an infinitely long tree • All the rules for sentential truth trees apply.

  3. 6.1.1 Instances • A(c) is an instance of either uA(u) or uA(u) if u is replaced by c uniformly. • uA(u) and uA(u) can be called generalizations of A(c), so to speak. • The quantifier to be eliminated must be the main connective (or negated main connective). • Non-conservative / conservative distinction

  4. Examples • xF(x) Fa • y(Fy→Gb) Fb→Gb (nonconservative) • xyFxy yFay • xy(zFzx↔wFwy) y(zFzc↔wFwy)

  5. Not Instances • xFx Fz Why: z is not a constant but a variable • yFy → Gb Fb → Gb Why: y is not the main connective • xy(Fxy → Fyx) y(Fcy → Fyx) Why: Not a uniform substitution of x by c • x(Fx & yGxy)Fa & yGby Why: a and b are different constants

  6. Rules that Require New Constants:E and ¬U: Checked • Existential (E) ۷uA(u), ۷(check / patch) A(c), must be conservative • Negated Universal (¬U) ۷¬uA(u) ¬A(c), must be conservative

  7. Rules: U and ¬ENonconservative • Universal (U) * uA(u), * dispatch mark A(c), Nonconservative • Negated Existential (¬E) *¬uA(u) ¬A(c), Nonconservative

  8. 6.2 Strategies • Close branches ASAP • Avoid splitting truth trees ALAP • Apply E or ¬U before U or ¬E, i.e., Conservatives before nonconservatives. • Don’t introduce new constants unless you must.

  9. Examples • Go through examples on the blackboard.

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