THE TRUTH TREE METHOD

In this chapter we examine a different way of analyzing sequents and arguments (and their schemas) using a new format, called a “truth tree.” The diagrams are called trees because they look like a kind of stick-figure tree turned upside down—or, as I prefer to think of them, a tree stump and roots right side up.

The basis of the Truth Tree method for determining formal validity is the same as that of the BTT method: we suppose all the premises are true and the conclusion false. If this supposition pans out, the argument is formally invalid, and as a bonus we find out what set of truth values for the components would make the premises true and conclusion false. If it does not pan out, that is, if every branch of the tree ends in an inconsistency, the argument is formally valid. The difference is that the “forcing” of values that are con­sistent with the initial assignments is explicit, and laid out vertically. Before introducing the method formally, let’s run through a simple example to show the gist of the method. Take the sequent

First, we write out the premises, and beneath these, the negation of the conclusion.

Then we proceed to derive what we can from this, branching every time we have a choice of options:

It’s always best to treat those compound statements first that do not involve any branch­ing possibilities. So first, on line 4 we note that S follows from -∣-∣S by DN. Every time we decompose a compound statement we put a √ to the left of it to indicate that it has been decomposed. Now we have exhausted all the information except that contained in line 2, which entails two possibilities, so the tree (or root) branches. So on line 5 we write down the alternative possibilities, one in each branch, ticking line 2 as we do so. In the left hand branch of the tree we derived -∣R; but this is incompatible with R on line 1. So we have a logical contradiction, denoted by 1, and that branch closes (as indicated by the short line above the 1). The right hand branch does not close, but we have exhausted all the information: there are no compound statements that remain unticked, they have all been decomposed. So S’s being true must be compatible with our starting assumptions. In other words, the premises are consistent with the denial of the conclusion, so the argu­ment is formally INVALID. Moreover, the tree diagram shows us that it is the possibility of S’s being true (as well as R’s being true) that would render the premises true and the conclusion false.

Now let’s see what happens if we investigate a valid sequent by this method. Let’s take (2) A,-∣(A &-∣B) H B

Here A (the first premise) and -∣B (the negation of the conclusion) cannot be decomposed any further.

Let’s look at a couple more examples.

Here lines 4 and 5 represent the decomposition of A & B from line 1. When we decompose -∣B v C from line 2 we get a branching into the two possibilitiesand C on line 6., however, is inconsistent with B above it, so this path through the tree closes.

Then we have to see what to do with line 3. If not both A and C, then by De Morgan’s, it must be not one or not the other, so this gives us another branching intcC on

line 7. ButA is inconsistent with A on line 4, which lies on the path back up the tree to the premises. AndC is inconsistent with C on line 6, which also lies on the same path. So both these remaining paths close. Again, we conclude that the premises are incompat­ible with the negation of the conclusion, so that the sequent is VALID.

Now let’s look at an argument form:

Here lines 3 and 4 exploit another of De Morgan’s Laws, that not either p or q is equivalent to neitherp nor q, i.e., notp (line 3) and not q (line 4).

To summarize the truth tree method of determining the validity of a sequent:

• Write each premise on a separate numbered line; then write the negation of the conclusion on the last Hne of the “stump,” and underline (draw in the “ground”).

• Now on succeeding lines, conclude what follows from each of the statements of the trunk and their consequences in terms of the truth or falsity of the component statements. Tick each compound statement when it has been decomposed.

• Once a tree has branched, if you apply a decomposition rule to a compound state­ment above the branching, you must do the decomposition in all the different branches on the same horizontal line.

• Statements that cannot be further decomposed are called literals; they will be either component statements or negations of component statements.

• A path through the tree is the collection of all the statements from the bottom of a branch up to the top of the trunk (first premise).

• Inspect each path of the tree as you proceed. If on the same path there occurs both a statement and the negation of that statement, then the path closes, as indicated by a short line at the bottom of the path with aunder it. Any path that does not close is said to be open.

• Proceed in this way until all the statements on any remaining open paths are either ticked compounds or literals. Any such remaining open path is then said to be a complete open path.

• When all the paths of a tree are either closed or complete open paths, the tree is said to be complete.

• If all the paths of a complete tree are closed, then the sequent is valid.

• If there are any complete open paths in a tree, then the sequent is invalid. The combinations of literals that make the premises true and the conclusion false can be read off each complete open path of the tree: all positive component statements on the path have the value T, and all negations on the path yield the value F for the statements they negate.

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