Introduction to Conditional Arguments

The first premise of a conditional argument can be expressed in the form "If p, then q," where "p" is the antecedent and "q" is the consequent. The first premise establishes the condition--the relationship between the antecedent and the consequent. Consider the following examples:

  • If Chinua arrives late, he will miss the bus.
  • Chinua will miss the bus if he comes late.
  • Chinua, if he arrives late, will miss the bus.

Notice that the word order can change, but the sentence retain that same meaning, as long as the same phrase is introduced by "if." Logically, all three can be expressed by the claim, "If Chinua arrives late, then he will miss the bus." For economy, we might shorten that to "If arrive late, then miss bus." In this case, p=arrive late, and q=miss bus.

The second premise of a valid conditional argument does one of two things: it affirms the antecedent (p), or denies the consequent (not q). Thus, the two valid second premises for the conditional above are: "he arrived late" (p), and "he did not miss the bus" (not q). And each of these valid conditional arguments has a valid conclusion:

If arrives late (p), then

miss bus (q).

Arrives late (p).


Therefore, miss bus (q).

--or--

If arrives late (p), 

then miss bus (q).

Not miss bus (not q).


Therefore, not arrive late (not p).

In other words, if the original conditional is true, we can draw the following valid conclusions: Chinua arrived late and therefore missed the bus; or Chinua did not miss the bus, and therefore he must not have arrived late. These two valid conditional arguments are expressed by the following paradigms:

(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.

These are the only two valid forms for a conditional argument. The only valid possibilites are a second premise of p, concluding q, and a second premise of not q, concluding not p.


Example 1. "If Chinua arrives late, he will miss the bus. And he does arrives late. Therefore, he misses the bus." This is a valid argument, because it fits one of the two forms for a valid conditional (in this case, modus ponens):

If p (arrives late), then q (misses bus).

P (arrives late)


Therefore, Q (misses bus).


Example 2. "If Chinua arrives late, he will miss the bus. And he does miss the bus. Therefore, he must have arrived late." This is an invalid argument, because it does not fit one of the two valid forms. In a valid conditional, the second premise must be either p or not q. In this case, the second premise ("miss bus") would be q, so no valid conclusion can be drawn. We say this second premise "affirms the consequent," which is invalid. (This may sound like a good argument, but it is easy to see why it is not, because the conditional says nothing about what might happen when Chinua does not miss the bus. Perhaps he arrived on time, or perhaps he got there late and the bus was delayed--we have insufficient information to conclude anything.)


Example 3. "If Chinua arrives late, he will miss the bus. But he does not arrive late. Therefore, he did not miss the bus." This is an invalid argument, because it does not fit one of the two valid forms. In a valid conditional, the second premise must be either p or not q. In this case, the second premise ("not arrive late") would be not p, so no valid conclusion can be drawn. We say this second premise "negates the antecedent," which is invalid. (Again, this may sound like a good argument, but it is easy to see why it is not, because the conditional says nothing about what will happen if Chinua arrives on time. Perhaps he did catch the bus, perhaps he fell asleep and missed it anyway--we have insufficient information to conclude anything.)


Example 4. "If Chinua arrives late, he will miss the bus. But he does not miss the bus. Therefore, he did not arrive late." This is a valid argument, because it fits one of the two valid forms (in this case, modus tollens):

If p (arrives late), then q (misses bus).

Not q (not miss bus).


Therefore, not p (not arrive late).


To continue with this section of Mission: Critical, click on "Exercises," above.