Difficulty: Intermediate | Prerequisites: Chapter 8 (basic truth tree rules and construction)
This material builds on the basic truth tree rules from Chapter 8. Where Chapter 8 dealt with simple sentences, Chapter 9 applies the same decomposition rules to compound sentences whose parts are themselves compound. If you are not yet comfortable identifying sentence letters, connectives, and building basic trees, revisit Chapter 8 first. This is the bridge between knowing the rules and using them fluently on real logic problems.
When a truth tree line contains a compound sentence whose parts are themselves compound, you apply the decomposition rules to the outermost (main) connective first. The key skill is identifying that main connective by asking: "What was the last step in building this sentence from its parts?" Once you have the main connective, you apply the corresponding rule, and then work inward on the resulting subsentences.
Main connective
The connective that was used last when building a compound sentence up from its parts. It governs the entire sentence and determines which truth tree rule to apply. Think of it as the outermost "glue" holding the whole sentence together.
Antecedent
The left-hand component of a conditional (the part before the horseshoe, ⊃). In simple terms, this is the "if" part of an "if...then" statement.
Consequent
The right-hand component of a conditional (the part after the horseshoe, ⊃). In simple terms, this is the "then" part.
Compound sentence
A sentence built from simpler sentences using one or more logical connectives. Think of it as any sentence that can be broken down further into parts joined by connectives.
Disjunction (v)
A compound sentence using "or" (the wedge, v). Made true by making at least one disjunct true. In a truth tree, a disjunction branches: each disjunct gets its own leg.
Conjunction (&)
A compound sentence using "and" (the ampersand, &). Made true by making both conjuncts true. In a truth tree, a conjunction stacks: both conjuncts are written on the same path, one below the other.
Conditional (⊃)
A compound sentence using the horseshoe (⊃), read as "if...then." Made true either by making the antecedent false or by making the consequent true. In a truth tree, it branches into the negation of the antecedent on one leg and the consequent on the other.
Biconditional (≡)
A compound sentence using the triple bar (≡), read as "if and only if." True when both sides have the same truth value. In a truth tree, it branches into two stacks: one where both components are true, another where both are false.
Negation (~)
The tilde (~) placed before a sentence to reverse its truth value. When negation is the main connective, you look at what it negates and apply the corresponding negated rule (~v, ~&, ~⊃, ~≡, or ~~).
Decomposition rules
The set of truth tree rules that break compound sentences into their components. Each rule corresponds to a main connective (or its negation) and tells you what to write further down the tree to guarantee the sentence's truth.
The main connective is the connective used last when the sentence was assembled from its parts. To find it, ask: "In building this sentence up from smaller parts, what was the last step?"
If the main connective is not a tilde (~), apply the rule for that connective directly.
If the main connective is a tilde (~), look at the connective of the sentence being negated. Then apply the corresponding negated rule (~v, ~&, ~\u2283, ~\u2261, or ~~).
Parentheses are your guide. They tell you which connective governs the whole sentence. The connective that sits outside all parenthetical groupings is the main one.
(A&B)v(~A&C) – The last step was joining A&B and ~A&C with a disjunction (v). Main connective: v. Apply the disjunction rule: branch, with A&B on one leg and ~A&C on the other. Then decompose each conjunction on its own branch.
(A&B)\u2283(CvD) – The last step was forming a conditional from the conjunction A&B and the disjunction CvD. Main connective: \u2283. Apply the conditional rule: branch into ~(A&B) on one leg and CvD on the other.
~[(BvC)\u2283A] – The last step was negating the conditional (BvC)\u2283A. Main connective: ~ (applied to a conditional). Apply the negated conditional rule (~\u2283): stack BvC and ~A on the same path.
(A\u2283B)\u2283[(CvA)\u2283B] – Two horseshoes appear. The parentheses tell you the last step was taking A\u2283B as antecedent and (CvA)\u2283B as consequent to form the outer conditional. Main connective: the second \u2283.
~{[(A\u2261~B)\u2261C]\u2261[C\u2283(~A\u2261B)]} – The last step was applying the outermost negation sign. Main connective: ~. The sentence being negated is a biconditional (\u2261), so apply ~\u2261. That branches into the two components: (A\u2261~B)\u2261C and C\u2283(~A\u2261B), with one side true and the other false on each leg.
Locate the main connective of the sentence.
Determine which rule applies (v, &, \u2283, \u2261, ~v, ~&, ~\u2283, ~\u2261, or ~~).
Ask: "What do I need to write at the bottom of every open path on which this sentence appears to guarantee it is true?"
Write the result (a branch or a stack) at the bottom of every open path containing the sentence.
Check off the sentence. Move to the next unchecked compound sentence.
Continue until every branch is either closed (contains a contradiction) or every compound sentence on it has been checked.
Conjunctions (&) and negated disjunctions (~v) produce stacks (non-branching). Disjunctions (v) and conditionals (\u2283) produce branches. Working the non-branching lines first keeps the tree compact and may close branches earlier, saving work.
If the tree closes (all branches end in a contradiction, marked with \u00d7), the argument is valid.
If any branch remains open, the argument is invalid. Read the counterexample by collecting the truth values of sentence letters and negated sentence letters on that open branch. If a sentence letter does not appear on the branch at all, the counterexample abbreviates multiple assignments where that letter can be either true or false.
Students often apply rules to the wrong connective, picking an inner connective instead of the main one. Always identify the main connective first, using the "last step" question.
Students sometimes try to apply two rules at once, decomposing both the outer and inner connective in a single step. This produces errors. Apply one rule at a time, check it off, then move to the next compound sentence.
When a sentence has a tilde () as its main connective, students sometimes apply the rule for the un-negated connective. A negated conditional (\u2283) is not the same as a conditional (\u2283). The negated version stacks; the un-negated version branches.
Students sometimes quit a tree early when they can see a counterexample forming on one branch, before all compound sentences on that branch have been checked. Always finish checking every compound sentence on every open branch. An early stop can give a false counterexample.
⚠️ Identifying the main connective is the single most important skill for this chapter. Expect exam questions that give you a complex sentence and ask you to name the main connective (Exercise 9-1 style).
⚠️ Validity arguments with complex premises are standard exam fare (Exercise 9-2 style). You will need to construct a full truth tree, identify closed and open paths, and state whether the argument is valid or invalid with counterexamples.
⚠️ When a sentence letter does not appear on an open branch, the counterexample abbreviates two (or more) full counterexamples. The exam may ask you to list them all.
⚠️ Negated compound sentences (~\u2283, ~v, ~&, ~\u2261) are where most students lose marks. Know the difference between, say, a conditional branching and a negated conditional stacking.
True or false: The main connective of (A&B)v(C\u2283D) is the ampersand (&).
Fill in the blank: To find the main connective, ask "In building this sentence up from smaller parts, what was the ______ step?"
True or false: A negated conditional (~\u2283) branches into two legs.
True or false: If an open branch shows ~H and K but does not mention S, the counterexample is ~H&K.
Fill in the blank: You should work ______ (branching / non-branching) lines first to keep the tree compact.
Answers: 1. False (it is v). 2. last. 3. False (it stacks). 4. True, though strictly it abbreviates two counterexamples: ~H&K&S and ~H&K&~S. 5. non-branching.
Q: What is the main connective of A&[Bv(C\u2283D)]?
A: The ampersand (&). The last step in building the sentence was conjoining A with the subsentence Bv(C\u2283D).
Q: What is the main connective of ~[(Hv~K)\u2261F]\u2283(~Rv~F)?
A: The horseshoe (\u2283). The last step was forming a conditional from the negated biconditional on the left and the disjunction on the right.
Q: You encounter ~[(BvC)\u2283A] on a truth tree. What do you write at the bottom of the open path?
A: This is a negated conditional. Stack BvC and ~A on the same path (both must be true for the negated conditional to be true).
Q: You encounter (A&B)\u2283(CvD) on a truth tree. What do you write?
A: This is a conditional. Branch into ~(A&B) on one leg and CvD on the other. Then decompose ~(A&B) and CvD further on their respective branches.
Q: An argument's truth tree has two open branches. One shows ~H and K. The other shows A and ~B and C. Are the arguments valid? What are the counterexamples?
A: The argument is invalid. The first counterexample is H false, K true (with all other letters taking any value). The second is A true, B false, C true (with all other letters taking any value).
This material connects directly to Chapter 8 (basic truth tree rules), which provides the individual decomposition rules you are now applying to nested sentences. It also sets the foundation for Section 9-2, where you will use truth trees for purposes beyond argument validity: testing for contradictions, logical truths, logical equivalence, and consistency.
The skill of identifying the main connective is also central to natural deduction proofs (if your course covers them), because the main connective determines which proof rule to apply.
Truth trees, semantic tableaux, sentence logic, propositional logic, main connective, principal connective, dominant connective, decomposition rules, tree rules, branching rules, stacking rules, complex sentences, compound sentences, nested sentences, validity testing, counterexample, open branch, closed branch, PHIL 101, logic exercises, Chapter 9, truth tree applications