Introduction to Conditional Chain Arguments

As we have seen in the section on Conditional Arguments, the two valid forms of a conditonal are:

Modus Ponens:

Affirming the Antecedent

If p, then q.

P.


Therefore, q.

--or--

Modus Tollens:

Negating the Consequent

If p, then q.

Not q.


Therefore, not p.

Such arguments can be developed further by linking them in a chain of conditionals, where the conclusion of each argument is the second premise of the next argument. Consider, for example, the argument:

"If Serge signs the contract, then he will have to fulfill its terms. And if he has to fulfill those terms, then he will have trouble paying his other debts. Serge signed the contract. Therefore, he will have trouble paying his other debts."

This argument can be represented as follows:

If P (signs), then Q (fulfill).

P (signs).


Therefore, Q (fulfill).

And

If Q (fulfill), then R (trouble).

Q (fulfill).


Therefore, R (trouble).

The first conditional affirms the 

antecedent, and is thus valid.




This conditional's antecedent

matches the conclusion

above, and is also valid.

Such chains can go on indefinitely, provided each link makes a valid argument by using the conclusion of the previously link as the antecedent of its conditional premise:

  1. If P then Q.
  2. If Q then R.
  3. If R then S.
  4. If S then T.
  5. If T then U.

And so on. In this chain, when P is affirmed, we can conclude (by modus ponens), that Q, R, S, T, and U are all true; and when U is negated, we can conclude (by modus tollens), not-T, not-S, not-R, not-Q, and not-P. Such a "chain reaction" of conclusions seems quick and easy, but take care to ensure that each of the links is in the proper form before accepting the results.


Chain arguments can be further complicated in two ways: by introducing into the chain "only if" and inverse conditionals. Here is one example of each:

Chain with "Only If"

Conditional


  1. If P then Q.
  2. If Q then R.
  3. R only if S.
  4. If S then T.

Chain with Inverted

Conditional


  1. If P then Q.
  2. If Q then R.
  3. If not S then not R.
  4. If S then T.

In both examples, affirming P affirms Q, R, S, and T, and negating T negates S, R, Q and P. That's because "R only if S" and "If not S then not R" are both equivalent to "If R then S."