£199.99 £139.99 for one year of Unlimited learning. Offer ends on 14 November 2022 at 23:59 (UTC). T&Cs apply

Find out more
Arrow and ‘if … then … ’: an argument for equivalence
Skip main navigation

Arrow and ‘if … then … ’: an argument for equivalence

In this article, we present an argument for the view that arrow and ‘if … then … ’ do in fact have the same truth-table.
Feet in a pair of exactly matching striped socks framing a present in wrapping paper
© University of York

Here is an argument for the view that arrow and ‘if … then … ’ have the same truth-table. This argument uses some of the key resources we’ve developed in this course.

First, note that where ‘If α then β’ is true, that’s enough to guarantee that ‘(α (rightarrow) β)’ is true: we’ve already noted that where ‘If α then β’ is true, it can’t be the case that ‘α’ is true and ‘β’ is false, and that’s the only kind of situation in which‘(α (rightarrow) β)’ can be false. So, ‘If α then β’ entails ‘(α (rightarrow) β)’.

Now, look at this simple argument:

  • Boris did it or Dominic did it
  • Therefore, if it’s not the case that Boris did it, then Dominic did it

Note that this argument is expressed in English and uses ‘If … then … ’. It seems that it is valid. More than that, it seems to be formally valid. Any argument with this shape

  • α or β
  • Therefore, if ~α, then β

will be valid.

But now notice that ‘or’ and vel have the same truth-table. So, we can substitute vel for ‘or’ in the premise, like this:

  • (α (vee) β)
  • Therefore, if ~α, then β

(If you are worried about exclusive-‘or’ here, note that there do seem to be cases of ‘or’ in English which are inclusive, and someone could plausibly say the premise of our example argument intending inclusive-‘or’ and the argument would be valid.)

But now look at the truth-tables for ‘(α (vee) β)’ and ‘(~α (rightarrow) β)’.

Equivalence Table

Figure 1. Truth-table for ‘(α (vee) β)’ and ‘(~α (rightarrow) β)’

What does this show about ‘(α (vee) β)’ and ‘(~α (rightarrow) β)’?

It shows they are logically equivalent. So, as far as valid arguments go, we can swap one for the other and, if our original argument was valid, the argument resulting from the substitution will be valid too.

Making the substitution gives us this:

  • (~α (rightarrow) β)
  • Therefore, if ~α, then β

But now look what we’ve got. We started with an argument in English that we are confident is formally valid. We’ve substituted logically equivalent sentences for the original premise, but the result is that we have an argument (a formally valid argument) from an arrow-sentence to the corresponding ‘If … then … ’-sentence. Putting our two results together, we have:

  • If α then β (models) (α (rightarrow) β)

and

  • (α (rightarrow) β) (models) If α then β

This means ‘If α then β’ and ‘(α (rightarrow) β)’ are logically equivalent. Arrow and ‘If … then … ’ do have the same truth-table.

© University of York
This article is from the free online

Logic: The Language of Truth

Created by
FutureLearn - Learning For Life

Our purpose is to transform access to education.

We offer a diverse selection of courses from leading universities and cultural institutions from around the world. These are delivered one step at a time, and are accessible on mobile, tablet and desktop, so you can fit learning around your life.

We believe learning should be an enjoyable, social experience, so our courses offer the opportunity to discuss what you’re learning with others as you go, helping you make fresh discoveries and form new ideas.
You can unlock new opportunities with unlimited access to hundreds of online short courses for a year by subscribing to our Unlimited package. Build your knowledge with top universities and organisations.

Learn more about how FutureLearn is transforming access to education