2.2.6. Two perspectives on derivations

The locations of the stage numbers appearing in a derivation reflect the patterns of argument on which the derivation rules are based. The label for a rule always appears to the left of the conclusion of such an argument, and the number of the stage at which the rule was applied appears not only next to the label but also to the right of the premises of the argument. A conclusion tree can be reconstructed from the derivation by beginning with the final conclusion and working backward to the premises from which it was concluded, the premises from which those were concluded, and so on.

If we apply this idea to the example of the last section (which is reproduced below), we get the conclusion tree following it.

The sentence B concluded by Ext at the second stage of the derivation does not appear in the conclusion tree because it is not used as a premise for any later conclusions, something that is marked in the derivation by the fact that it has no stage number to its right.

Looked at in this way, a derivation could be thought of as the result of disassembling a conclusion tree and stacking the pieces up vertically. When reassembling the tree, we paid no attention to the horizontal organization provided by scope lines. The order of the stage numbers played no role either: they could just as well have been arbitrary codes used to mark corresponding parts of the tree so they could be fit together again. Indeed, even the vertical order of the lines of the derivation did not matter. Matching numbers on the left with numbers on the right is all that was necessary to reassemble the tree, and pieces could have been given to us in an unorganized heap. However, all these features of derivations, which are not needed to reconstruct a conclusion tree, do matter for another, and more important, way of looking at derivations, one in which a derivation is associated with a argument tree.

To see this association, first use the stage numbers, scope lines, and the vertical ordering of lines to determine the way the gaps of the derivation develop over time, beginning with the intial gap, eventually dividing, and finally closing. That is shown on the right in the diagram below, where the stages are arrayed left to right and gaps are indicated by circles, with a filled circle used to indicate closure and an empty circle used to indicate a gap that is open. Colors are used to emphasize where and when changes occur.

We used similar notation in Figure 2.2.2-2 to represent the skeleton of an argument tree, and we get an argument tree if we flesh out the skeleton above by including the proximate arguments of the gaps that the empty circles represent, something that is shown below.

 

 

 

 

 

 

 

 

 

 

 

 

(A ∧ B) ∧ C
D

---

C ∧ (A ∧ D)

D
A ∧ B
C

---

C ∧ (A ∧ D)

D
C
A
B

---

C ∧ (A ∧ D)

D
C
A
B

---

C

D
C
A
B

---

A ∧ D

D
C
A
B

---

C

D
C
A
B

---

A

D
C
A
B

---

D

D
C
A
B

---

A

D
C
A
B

---

D

D
C
A
B

---

D

Colored sentences are new additions as the tree grows, and premises are added at the end of the list to match the order of active resources in the derivation above. Comparison with it should give you some sense of the way in the which a derivation amounts to a squashed argument tree: repeated premises and conclusions coincide and new ones are folded in towards the middle.