An easier technique that can be used to test argument patterns for validity is the creation of a so-called consistency tree. The concept behind it is that in an invalid argument it will be consistent to have a negated conclusion but in a valid argument it will not be consistent.

What it means to be inconsistent can first be seen by having the strings P and PN. Written one above the other we see a clear contradiction, which is indicated by an x. We say that this branch is closed.
.....P
.....PN
.....x
To work with the tree technique we first represent the truth of any string with a binary signal by breaking it down into two other strings. The key rules for each of our signals are seen below.
PQA..........PQCN...........PQON...........FGDN...........FGE
..P.................P...................PN...............Fa*................Fa*
..Q................QN................QN..............GNa...............Ga
* (using the new name rule)

....PQC........PQO..........PQB..........PQM..........PQAN.........FGD........FGEN
....../\.............../\................/\................/\................../\................/\.............../\
.PN...Q........P..Q..........P..PN.........P..PN.......PN..QN.....FNa..Ga....FNa..GNa
......................................Q..QN.....QN..Q
The idea of branching is that with the signals above, either of two possibilities allows the original string to be true.

We also go ahead and treat expressions ending in V or W, VN or WN in the same way as those ending in D or E by instantiating them, but with W and VN we must observe the new name rule.
FV........FW........FVN..........FWN
Fa.........Fa..........FNa............FNa

Applying consistency trees to examples we have already seen worked through with truth tables, we have the following:

PQARC, RN |- PN
.........PQARC
.............RN
.............P (remembering to negate the original conclusion)
............./\
.....PQAN..R
........./\......x
....PN..QN............. invalid
.....x
Because not all branches are closed, we know that it is consistent to have true premises and a false conclusion. The pattern is invalid.

PQORC, RN |- PN
.........PQORC
.............RN
.............P
.............../ \
.....PQON...R
........PN......x
........QN
.........x..........valid
Since all branches close, we know the pattern is valid. It is inconsistent to have true premises and a false conclusion.

FGD, GHE |- FHE
.............FGD
.............GHE
............FHEN
..............Ga
..............Ha
................/ \
........FNa...Ga
......./ \
...FNa..Hna
..............x......... invalid
Since there is an open branch, again we can say we have an invalid pattern.

FGE, GHD |- FHE
.............FGE
.............GHD
............FHEN
..............Fa
..............Ga
.............../ \
..........GNa...Ha
.............x...../ \
..............FNa...HNa
.................x........x......... valid
Again we have a situation in which all branches close.

Go ahead to see another way in which we could arrange the information in a consistency tree.

Go ahead to the midterm review.

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