2.2.2. Argument trees

In the simple form in which we have considered it, the approach to finding proofs described at the end of the last subsection will really work only with conjunction. Among the logical forms we will consider, it is only conjunction whose content is the cumulative content of its components, so in particular, it is the only one whose role as a premise can be simply taken over by its components. However, in the case of other logical forms, we will still have principles like the laws for conjunction as a premise and as a conclusion. The difference will be that the laws will not concern solely premises or conclusions. A law describing the role of a logical form as a premise, for instance, may refer to the role of a component as a conclusion. Proofs involving these other logical forms also force significant modifications to conclusion trees. Such modifications are possible, and we will consider them briefly in later chapters, but the most natural way of implementing the laws for these other logical forms is a different sort of tree which permits consideration of both premises and conclusion at the same time—in short, a tree composed of arguments rather than single sentences.

We will generally find such a tree—an argument tree—growing on its side, with the arguments written vertically.

 

 

 

 

 

 

 

 

A ∧ (B ∧ C)

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(A ∧ B) ∧ C

A
B ∧ C

---

(A ∧ B) ∧ C

A
B
C

---

(A ∧ B) ∧ C

A
B
C

---

A ∧ B

A
B
C

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C

A
B
C

---

A

A
B
C

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B

A
B
C

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C

Fig. 2.2.2-1. An argument tree growing from left to right, using the law for conjunction as premise twice, followed by two uses of the law for conjunction as conclusion, and ending with three arguments whose validity follows from the law for premises.

Moving left to right, we first use the law for conjunction as a premise to analyze premises, replacing them by simpler ones that whose cumulative content is the same. We then analyze the conclusion using the other law. This leads us to replace an argument by two whose conclusions are the conjuncts of the conclusion we analyzed. Finally, at the tips of the branches, we have reached arguments that are valid by the law of premises, and that is indicated by a bullet marking the end of this part of the search for a proof. If every path leading from an argument ends in a bullet, we know that the argument is valid because the paths have lead us to arguments whose validity is tied to its validity by the laws for conjunction as a premise and as a conclusion.

Since an argument tree leads us to see that an argument is valid, it counts as a proof in its own right. But the arguments whose validity is tied by the two laws for conjunction are connected by the use of the patterns of argument Ext and Cnj, so an argument tree points us to a way of using those patterns to construct a conclusion tree. The figure below illustrates this for the argument tree above. The buttons controlling it take the form of a schematic version of that tree with open circles representing arguments for which a proof is not yet provided.

The question marks that appear before the last stage each indicate a conclusion that has not yet been connected with the premises or the conclusions derived from them.

Like conclusion trees, argument trees illustrate important features of deductive reasoning. But, also like conclusion trees, they are cumbersome to write since they take up space and involve rewriting sentences several times. In the next subsection, we will look at a more compact notation for proofs. Of course, any notation for writing out proofs is more than we need to settle questions of entailment involving only conjunction. But the complications introduced by the logical forms we will consider in later chapters make it useful to have some system of notation; and it will help in understanding this notation to introduce it now since the simplicity of conjunction makes it easier to see what is going on. We will write out conclusion trees and argument trees only rarely from this point on, but our more compact notation will have ties to both of them, and it will useful to look at them from time to time since they exhibit quite clearly some features of the compact notation that are disguised by its compactness.