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.

2 thoughts on “Truth Tree Catechism

  1. In Exercise 23, which uses the truth tree catechism above, my student has a question about the closed branch in problem 5. She found two reasons for an inconsistency instead of one. She listed both in her answer, but we saw that you only listed the first one that occurred. Is listing a second contradiction unnecessary even if the inconsistency is there?

  2. Thank you for the good question. If there are two contradictions, you only need to identify one of them. I chose the first, 3×6, but it would be equally correct to identify the second, 4×7, and even acceptable to list both.

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