Daily Archives: February 21, 2019

Truth Tree Catechism

Q: What is a truth tree?
A: A truth tree is a diagram that shows a set of compound propositions decomposed into literals following standard decomposition rules.

Q: What is a literal?
A: A simple proposition symbolized as a constant or variable, or the negation of the same.

Q: What does it mean to decompose a compound proposition?
A: It means to show the components that must be true for the decomposed proposition to be true. A fully decomposed proposition is broken down into literals.

Q: Why do some compound propositions branch when decomposed?
The branching shows that there is more than one way for the proposition to be true.

Q: What does consistency mean?
A: Consistent propositions can all be true at the same time.

Q: How does the truth tree show consistency?
A: If the propositions in the set are fully decomposed into literals on at least one branch without contradiction, the propositions are consistent.

Q: What does it mean to recover the truth values?
A: It means to show the truth values of the component propositions that make every proposition in the given set true.

Q: What does SM mean?
A: It stands for Set Member; a label for a proposition in the given set.

Q: What is the meaning of the number and the symbols at the end of a row?
A: It is the justification for the decomposition, showing the number of the compound proposition that is decomposed, and the abbreviation of the rule used to decompose it.

Q: What is the meaning of the Ο at the bottom of a truth tree branch?
A: It designates an open branch, meaning that there are no contradictions on that branch.

Q: What is the meaning of the numbers separated by an Χ at the bottom of a branch?
A: The X designates a closed branch; the numbers are the line numbers of the propositions that contradict on that branch.

Q: What is the benefit of using truth trees?
A: Truth trees do the same things as truth tables — showing consistency, equivalence, validity, etc. — but in a visual way. They are a tool used in higher-level logic.