Department of Philosophy (PHI)
Argument Structure: Linkage, Convergence and Sequence
Linked arguments
In a purely "linked" argument, each premise requires the help of the other(s) in order to provide any support at all for the conclusion. Consequently, if any premise in such an argument is doubtful, the argument cannot establish its conclusion.
Example. All white horses love apples. Flash is a white horse. So Flash loves apples.
Analysis. The premise about white horses is irrelevant to the conclusion about Flash in the absence of the information that Flash is one of the white horses. Similarly, that Flash is a white horse is irrelevant to the conclusion without the information that things of that kind love apples.
The following diagram for this argument uses numbers to indicate the order of appearance of the statements, a "+" to indicate linkage, and an arrow to indicate that the premise combination 1+2 leads to the conclusion 3:
↓
3
The same diagram would work for any linked argument with two premises in which the conclusion is stated third. If the conclusion were stated first, the correct diagram would depict 2+3 leading to 1. If the conclusion were stated second, the correct diagram would depict 1+3 leading to 2. In a purely linked argument with three premises in which the conclusion is stated third, the correct diagram would show 1+2+4 leading to 3.
The pattern of linkage in the Flash argument is a common one for linked arguments with two premises. A certain verbal unit C that appears in both premises and not in the conclusion (here "white horse(s)") is linked in one premise with verbal unit A (here "Flash") and in the other with verbal unit B (here "love(s) apples"). The conclusion links A and B. Whether the AB link in the conclusion is actually established by the AC and BC links in the premises is an issue that is not resolved by the pattern of units linked. It depends not only on what units are linked in the various statements but on how they are linked. Deductively valid arguments (like the Flash argument), inductively strong arguments (such as the Flash argument with "all" replaced by "nearly all"), and logically worthless arguments can all exhibit the same pattern of linked units. What linkage does indicate is that the premises must work together to provide even the appearance of support for the conclusion, so that the doubtfulness of a single premise is enough to prevent the argument from establishing its conclusion.
A simpler pattern of linkage involves two units A and B that both appear in one premise, while one (say, A) appears in the other premise and the other (B) in the conclusion. An example is "If Al plays then Bob will play, but Al won't play, so Bob won't play" (an invalid but linked argument). More complicated patterns of linkage may involve two or more units that appear in premises but not in the conclusion, as in the form "All Fs are Gs, all Gs are Hs, all Hs are Is, and all Is are Js, so all Fs are Js". Here F is linked indirectly with H via G, and H is linked indirectly with J via I, so F is linked very indirectly with J via G, H and I.
Convergent arguments
In a purely "convergent" argument, each premise supports the conclusion (or appears to) to some extent by itself, independently of the other(s). So even if a premise in such an argument is doubtful, it is possible that the other(s) still establish the conclusion. In a convergent argument that is not purely convergent, not all premises are independent of all others, but the premises form two or more groups each of which is independent of the other groups.
Example: Art embezzled funds from his company. Art also cheated on his 1996 taxes. Therefore Art is dishonest.
The following diagram uses separate arrows to indicate that the two premises independently support the conclusion:
↓ ↓
→ 3 ←
The absence of repeated verbal units (other than "Art", the subject of all three statements) is the only explicit indication of convergence in this argument. But semantic analysis quickly reveals that embezzlement and cheating both involve dishonesty, so that either premise by itself could (if known to be true) establish the conclusion. The connections could be made explicit by adding premises like (4) "all embezzlers are dishonest" and (5) "all cheaters are dishonest". Then we would have a convergent argument with linkage on each "branch" of the reasoning, 1+4 leading along one branch to 3, and 2+5 leading along the other branch to the same conclusion. It would be convergent because it would include two pairs of linked premises, each independent of the other pair, so that the loss of a premise would not doom the argument to failure. But it would not be purely convergent, because the independence of the premises is not universal: for instance, 1 is independent of 2 and 5, but dependent on 4.
One common indication of (impure) convergence is the occurrence of two separate patterns of linkage. The units in the conclusion, say A and B, may be linked via C in one pair of premises, and linked via D in another pair of premises. This is what happens in the expanded "Art" argument above. Premises 1 and 4 link "Art" with "dishonest" via "embezzle", and premises 2 and 5 link "Art" with "dishonest" via "cheat". Doubts about Art's alleged embezzlement and/or about the dishonesty of embezzling would leave the argument from 2+5 to 3 unblemished. To defeat the argument one would have to raise doubts about at least two of the premises, including at least one from each linked pair.
Sequential arguments
In a "sequential" argument, a premise of the main argument is supported by another premise and so plays a double role in the argument as a whole, being a conclusion in one "stage" of the argument and a premise in the next. If the premise supporting this conclusion/premise is doubtful, but the latter is independently known to be true, the main argument may still establish its conclusion.
Example. Cliff doesn’t care whether he lives or dies. Therefore he is willing to take extreme risks. Consequently, he might be willing to accept a dangerous assignment.
Analysis. Because of the two conclusion indicators, 'therefore' and 'consequently', statements 2 and 3 are both conclusions. It appears to me that 3 is based directly on 2 and only indirectly on 1. It is not Cliff's alleged indifference to life per se, but the supposedly resulting willingness to take risks that bears on the conclusion. (After all, if Cliff is indifferent to life itself, why would he not be even more indifferent to employment?) The diagram shows 2 to be both a conclusion and a premise:
↓
2
↓
3
We might doubt the claim (1) that Cliff doesn't care whether he lives or dies, but know on other grounds that (2) he is willing to take extreme risks. Perhaps we have seen him run into a burning building to save a child, although he is normally as cautious as anyone in dangerous situations. As long as (2) is known independently of (1), the argument might still establish (3). Thus sequence makes it possible for an argument whose premises are not all known to be true to succeed in establishing its (main) conclusion.
Hybrid structures
Complex reasoning usually involves two or all three of these structural elements--linkage, convergence and sequence. An elaboration of the Cliff argument might explicitly state that (4) Cliff ran into a burning building to save a child, and perhaps also (5) anyone who runs into a burning building is willing to take risks and (6) most people who are willing to takes risks are people who might be willing to accept a dangerous assignment. Then 4+5 would support 2 independently of 1, and 2 would work together with 6 to support 3:
↓ ↓
→ ↓ ←
2 + 6
↓
3