Skip main navigation

Using arrow in looking at arguments involving ‘if … then … ’

In this article, we look at how we can use arrow in looking at arguments involving ‘if … then … ’, if we are careful.
A sign with an arrow pointing right, marked with an ‘i' for ‘information’ and a question-mark
© University of York

There is a lot more that could be said about whether arrow and ‘If … then … ’ have the same truth-table. But we don’t have space to take that further here. What we can note is that arrow can be a useful tool to use in looking at arguments involving ‘If … then … ’, if we are careful.

Where an ‘If … then … ’-claim is true, the corresponding arrow-claim will have to be true. (It seems right to say that where ‘If P, then Q’ is true, it can’t be the case that ‘P’ is true and ‘Q’ is false. And that rules out the one kind of situation in which ‘(P (rightarrow) Q)’ is false). This means that if we’re looking at an argument with (positive, i.e. not negated) ‘If … then … ’ premises and the version with arrow-sentences substituted tests as valid, then the original argument will be valid.

Remember what we said earlier about the parallels between key inferences involving arrow and inferences involving ‘If … then … ’. The form ‘(P (rightarrow) Q), P; therefore, Q’ is valid. And so is the form ‘If P then Q, P; therefore Q’. The form ‘(P (rightarrow) Q), ~Q; therefore, ~P’ is valid. And so is ‘If P then Q, It’s not the case that Q; therefore It’s not the case that P’. So we don’t need to worry about using arrow to investigate an argument just because we see structures like these involving ‘If … then … ’.

We do have to be more wary if the premises of an argument expressed in natural language include negated conditionals (e.g. ‘It is not the case that if P then Q’). Here’s the reason. The sentence ‘~(P (rightarrow) Q)’ is only true where ‘P’ is true and ‘Q’ is false, but it seems like we think that some sentences of the form ‘It is not the case that if P then Q’ can be true without it having to be that ‘P’ is true and ‘Q’ is false. This means we should hold back from evaluating arguments with this kind of sentence as a premise.

© 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