Conditional Response

Aside from being entertaining, these DIRECTV commercials offer at least two important lessons about logic.

For starters, let’s name the propositions listed in the video:

  • q: your cable is on the fritz
  • r: you get frustrated
  • s: your daughter imitates
  • t: your daughter gets thrown out of school
  • u: your daughter meets undesirables
  • v: your daughter ties the knot with undesirables
  • w:  you get a grandson with a dog collar

So the ad takes us through the following sequence of conditional statements:

\begin{array}{lcl} q & \longrightarrow & r \\ r & \longrightarrow & s \\ s & \longrightarrow & t \\ t & \longrightarrow & u \\ u & \longrightarrow & v \\ v & \longrightarrow & w \end{array}

Let’s be generous and accept that each statement, individually, is true.  Then we’re led sequentially along a nice string of propositions, beginning at q and ending with w.  Actually, there’s one more tacit proposition, p: you have cable.  So the commercial’s (implicit + explicit) logic looks something like this:

p \longrightarrow q \longrightarrow r \longrightarrow s \longrightarrow t \longrightarrow u \longrightarrow v \longrightarrow w

And therein our first logic lesson: conditional statements respect transitivity.  We can follow an unbroken path of propositions all the way from p to w, which means we can replace that whole string of implications with the statement, “If you have cable, then you’ll get a grandson with a dog collar.”  Symbolically:

p \longrightarrow w

We’ve accepted all the statements along the way, so we accept this one as well, which is both funny and logically sound.  DIRECTV has successfully made fun of the cable companies, and we’ve had a chuckle.  And if the commercial were to end there, everything would be hunky dory.  But it doesn’t end there.  It ends on the line, “Don’t have a grandson with a dog collar; get rid of cable…”  Which is to say, “If you don’t have cable, you won’t have a grandson with a dog collar.”  Or…

\neg p \longrightarrow \neg w

But that’s incorrect!  And that’s our second lesson: the technical name for this fallacy is denying the antecedent, or the inverse error.  To give you a more intuitive example, consider the propositions:

  • p: you are a dog
  • q: you are a mammal

p \longrightarrow q: “If you are a dog, then you are a mammal.”  True.

\neg p \longrightarrow \neg q: “If you are not a dog, then you are not a mammal.”  Obviously false.

It might very well be true that having cable leads to a grandson with a dog collar, but that certainly doesn’t mean getting rid of cable is enough to avoid one.

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