Truth trees

Rationale

Like Truth-tables, truth-trees are a means of graphically representing the logical relationships that may obtain between propositions. Truth-trees and truth-tables complement each other and which method you choose depends on which logical property you are seeking to derive.

Whilst truth-tables have the benefit of being exhaustive - every possible truth assignment is factored into the representation - their complexity grows exponentially with each additional proposition they contain. This can make manually constructing truth tables long-winded and prone to mistakes.

Truth-trees are less onerous but they lack the exhaustive scope of a truth-table. They are more targeted and are best used for demonstrating that something is the case rather than all the possible states that could be the case. For example, a truth tree will tell us that a set S is logically consistent whereas a truth-table will tell us that S is consistent on the following three assignments.

Logical consistency

Recall that a set of propositions is logically or truth-functionally Consistency just if there is at least one assignment of truth conditions which results in all members of the set being true. To identify consistency for a set of three propositions via the truth table approach we would need to construct a truth table with \(2^3\) (8) rows. Assume that this set is consistent on one partial assignment only. This means that 87.5% of our rows are redundant, they are not required to prove the consistency of the set. However we can only know this and we can only be sure of consistency once we have gone through the process of generating an assignment for each row.

Truth trees allow us to reduce the amount of work required and go straight to the assignment that proves consistency, disregarding the rest which are irrelevant.

Truth tree structure and key terms

When using a truth tree to derive logical consistency, the goal is to determine whether there is a truth-value assignment on which all of the sentences of a set are true. If the set is consistent we should be able to derive a partial assignment from the tree that demonstrates consistency.

Each truth tree begins with a series of sentences one on top of the other in a column. We call the sentences that comprise the initial column set members. In constructing the tree, we work downwards from the initial column decomposing set members into their atomic constituents. We a call an atomic sentence that has been decomposed a literal. A literal will either be an atomic sentence or the negation of an atomic sentence. If one of the set members is already a literal, there is no need to decompose it; it can remain as it is.

Once every set member has been decomposed the truth tree is complete. It can then be interpreted in order to derive logical consistency or inconsistency. If the set is consistent, we are able to derive the partial assignment(s) that demonstrate consistency.

The rules for decomposing compound sentences match the truth conditions of the logical connectives. There are rules for every possible connective and the negation of every possible connective however in terms of their tree shape they all correspond to either a conjunction or a disjunction. Disjunctive decomposition results in new branches being formed off the main column (or trunk). Conjunctive decomposition is non-branching which means the decomposed constituents are placed within the trunk of whichever tree or branch they are decomposed within.

As we construct the tree we list each line in the left-hand margin and the decomposition rule in the right-hand margin. When we apply a decomposition rule we must cite the lines to which it applies.

Closed and open branches

Any branch on which an atomic sentence (\(P\)) and the negation of that sentence (\(\sim P\)) both occur is a closed branch. A branch that is not closed is an open branch. No partial assignment is recoverable from a closed branch. An open branch allows truth to ‘flow up’ to the original set members whereas a closed branch blocks this passage.

Completed open branch

A completed open branch occurs when we have an open branch that has been fully decomposed: the branch is open and all molecular sentences have been ticked off such that it contains only literals.

Completed tree

A tree where all its branches are either completed open branches or closed branches.

Closed tree

A tree where all the branches are closed

Open tree

A tree with at least one completed open branch

Deriving consistency

Using the definitions above, we can now define truth-functional consistency and inconsistency in terms of truth trees:

A finite set (\(\Gamma\) ) of sentences is truth-functionally inconsistent if \(\Gamma\) is a closed tree

A finite set (\(\Gamma\) ) of sentences is truth-functionally consistent if \(\Gamma\) is an open tree

Examples

First example

The following is a truth tree for the set \({P \lor Q, \sim P }\):

Interpretation

• We decompose the disjunction at line 1 on line 3. We tick off the compound sentence to indicate that it is now decomposed and no longer under consideration.
• Both P and its negation exist on a single branch (at line 2 and line 3). This makes it a closed branch. We indicate this by the X beneath the branch that is closed, citing the source of the closure by line number.
• The rightward branch is a completed open branch given the decomposition at 3 and the lack of negation of Q. Overall this makes the tree an open tree.

As the set gives us an open tree, it must be truth-functionally consistent. If this is the case we should be able to determine the partial assignment in which each set member is true. Given that Q is not negated the assignment of consistency will contain Q but we have both P and ~P. This means there are two possible assignments where the set is consistent: \(P, Q\) and \(\sim P, Q\). This is confirmed by the truth-table:

P	Q				P	∨	~	P				Q
T	T					T						T     *
T	F					T						F
F	T					T						T     *
F	F					T						F

Any time there is an open tree with a closed branch it will be the case that the negated sentences of the closed branch will appear both as \(S\) and \(\sim S\) in the resultant assignment.

Invoking the truth-table highlights the differences between the two techniques. The values that are derived when we interpret a truth tree are not the truth-functions of the set members but the truth-values for when they are simultaneously true. With truth-tables in contrast, we are deriving the truth functions for every possible truth-value assignment. In other words the values derived from a truth tree correspond to the left hand side of the truth table not the right hand side.

Second example

The following is a truth tree for the set \({A & \sim B, C, \sim A \lor \sim B }\).

Interpretation

• The two molecular set members are decomposed. The disjunction (line 3) results in a branching tree. The conjunction (line 1) results in the continuation of the trunk.
• Both branches are completed making it a completed tree. As each branch is closed this is a closed tree.

As this is a closed tree, the set is not truth-functionally consistent. This is confirmed by the truth table where there is no partial assignment where all set members are true.

A	B	C				A	&	~	B				C				~	A	∨	~	C
T	T	T					F						T						F
T	T	F					F						F						T
T	F	T					T						T						F
T	F	F					T						F						T
F	T	T					F						T						T
F	T	F					F						F						T
F	F	T					F						T						T
F	F	F					F						F						T

Truth tree decomposition rules

So far we have encountered the decomposition rules for conjunction (&D) and disjunction (vD). We will now list all the rules. We will see that for each rule, the decomposition either branches or does not branch which is to say that each rule either has the shape of a conjunction or a disjunction (however the permitted values of the specific disjuncts/conjuncts obviously differ in each case). Moreover there is a parallel rule for the decomposition of the negation of each of the main connectives and these rules rely on logical equivalences

Negated negation decomposition: ~~D

Truth passes only if \(P\) is true

Conjunction decomposition: &D

Truth passes only \(P\) and \(Q\) are both true.

Negated Conjunction decomposition: ~&D

Truth passes if either \(\sim P\) or \(\sim Q\) is true. This rule is a consequence of the equivalence between \(\sim (P & Q)\) and \(\sim P \lor \sim Q\) , the first of DeMorgan’s Laws.

Disjunction decomposition: vD

Truth passes if either \(P\)or \(Q\) are true.

Negated Disjunction decomposition: ~vD

Truth passes if both \(P\) and \(Q\) are false. This rule is a consequence of the equivalence between \(\sim (P \lor Q)\) and \(\sim P & \sim Q\), the second of DeMorgan’s Laws.

Conditional decomposition: ⊃D

Truth passes if either \(\sim P\) or \(Q\) are true. This rule is a consequence of the equivalence between \(P \supset Q\) and \(\sim P \lor Q\) therefore this branch has the shape of a disjunction with \(\sim P\) , \(Q\) as its disjuncts.

Negated Conditional decomposition: ~⊃D

Truth passes if both \(P\) and \(\sim Q\) are true. This is a consequence of the equivalence between \(\sim (P \supset Q)\) and \(P & \sim Q\).

Biconditional decomposition: ≡D

Truth passes if either \(P\) and \(Q\) are true or \(\sim P & \sim Q\) are true. This is an interesting rule because it combines the disjunction and conjunction tree shapes.

Negated biconditional decomposition: ~≡D

Truth passes if either \(P\) and \(\sim Q\) is true or if \(\sim P\) and \(Q\) is true.

Further examples and heuristics for complex truth trees

With truth-trees regardless of which order you decompose the set members, the conclusion should always be the same. This said, there more are more efficient ways than others to construct the trees. You want to find the route that will demonstrate consistency or non-consistency with the shortest amount of work. The following heuristic techniques followed in order, facilitate this:

1. Decompose those molecular sentences the decomposition of which does not produce new branches. In other words that are decompositions of double negations or pure conjunctions.
2. Perform those decompositions that will rapidly generate closed branches.
3. If neither (1) or (2) is applicable, decompose the most complex sentence first.

Here are some examples of these rules applied:

Observe that here we don’t bother to decompose the sentence on line 1. This is because, having decomposed the sentences on lines 2 and 3 we have arrived at a closed tree. It is therefore unnecessary to go any further for if two sentences in the set are inconsistent with each other, adding another sentence is not going to change the overall assignment of inconsistency.

Deriving properties other than logical consistency from truth trees

So far truth trees have been discussed purely in terms of logical consistency however they can be used to derive all the other key truth-functional properties of propositional logic. Given the foundational role of consistency to logic, these properties are expressible in terms of consistency which is what makes them amenable to formulation in terms of truth trees.

Logical falsity

For a given finite set \(\Gamma\), \(\Gamma\) is logically consistent just if all of its members can be true at once. Expressed in terms of truth trees, this is equivalent to an open tree. Contrariwise, \(\Gamma\) is inconsistent if it is not possible for every member of the set to be true at once. This is the same as a tree where all of the branches are closed (i.e. a closed tree).

When we wish to assess logical falsity we are not focused on sets however, we are interested in a property of a sentence. However we can easily construe single sentences as unit sets: sets with a single member. With this in mind and the above accounts of consistency and logical falsity we are equipped to express logical falsity in terms of truth-trees with the following rule:

A sentence \(P\) is logically false if and only if the unit set \({ P }\) has a closed tree

A logically false sentence cannot be true on any assignment. This is the same thing as an inconsistent set. Thus it will be represented in a truth tree as inconsistency which is disclosed via a closed tree.

Logical truth

For a sentence \(P\) to be logically true, there must be no possible assignment in which \(P\) is false. We express this informally by saying it is not possible to consistently deny \(P\). We know that in terms of truth trees an inconsistent set is a closed tree therefore a unit set of \({ P }\) is logically true if \({ \sim P }\) is a closed tree. This is to say: if the negation of \(P\) is inconsistent.

A sentence \(P\) is logically true if and only if the set \({ \sim P }\) has a closed tree

Logical indeterminacy

Indeterminacy follows from the two definitions above; we do not require any additional apparatus. We recall that a sentence \(P\) is logically indeterminate just if it is neither logically true or logically false. Thus the truth tree for an indeterminate sentence is straightforward:

A sentence \(P\) is logically indeterminate if and only if neither the set  \({ P }\) nor the set \({ \sim P }\) has a closed tree

This follows because a closed tree for  \({ P }\) means it is not logically false and an open tree for \({ \sim P }\) means it is not logically true. So if it is neither of these things, \(P\) must be indeterminate.

Logical equivalence

Recall that \(P\) and \(Q\) are logically equivalent just if there is no truth assignment on which one is true and the other is false. We know from the material biconditional shorthand that this state of affairs can be expressed as $P \equiv Q$ and that if this compound sentence is true on every assignment then both simple sentences are equivalent. But ‘true on every assignment’ is another way of saying logically true since there is no possibility of a false assignment. We already know what logical truth looks like as a truth tree: it is a closed tree for the negation of the sentence being tested. Therefore, to test the logical equivalence of two sentences it is necessary to construct a truth tree for the negation of the sentences conjoined by the biconditional (i.e. \(\sim (P \equiv Q)\) )and see if this results in a closed tree. If it does, the two sentences are logically equivalent.

Sentences \(P\) and \(Q\) are truth-functionally equivalent if and only if the set \(\sim (P \equiv Q)\) has a closed tree

Logical entailment and validity

Let’s remind ourselves of the meaning of truth-functional entailment and validity and the relation between the two. \(\Gamma\) \(\vdash\) \(P\) is true if and only if there is no truth-assignment in which every member of \(\Gamma\) is true and \(P\) is false. Entailment is closely related to validity; it is really just a matter of emphasis: we say that \(\Gamma\) are the premises and \(P\) is the conclusion and that this is a valid argument if there is no assignment in which every member of \(\Gamma\) is true and \(P\) is false.

As with the previous properties, to express validity and entailment in terms of truth trees we need to express these concepts in the language of logical consistency. \(\Gamma\) entails \(P\) just if one cannot consistently assert \(\Gamma\) whilst denying \(P\). This is to say that the set \(\Gamma \cup {\sim P}\) is inconsistent. So we just need a closed truth tree for \(\Gamma \cup {\sim P}\) to demonstrate the validity of this set.

A finite set of sentences \(\Gamma\) truth-functionally entails a sentence \(P\) if and only if the set \(\Gamma \cup {\sim P}\) has a closed truth tree.

An argument is truth functionally valid if and only if the set consisting of the premises and the negation of the conclusion has a closed truth tree.