Skip main navigation

£199.99 £139.99 for one year of Unlimited learning. Offer ends on 28 February 2023 at 23:59 (UTC). T&Cs apply

Find out more

Formal logical necessities (tautologies)

We introduce the idea of a tautology (a claim true in all possible situations) and see how we can identify tautologies using truth-tables.
The word 'tautology' spelt out in letter tiles, split so that the tee of 'tau' is above the tee of 'tology', with two further tees underneath, so there is a column of 4 tees.
© University of York

Suppose we take some basic sentence P (as you’ll see, it won’t matter what specific sentence it is), and put it in a complex sentence, as follows: ‘(P (vee) ~P)’.

Look at the truth-table for this sentence.

P   P (vee) ~ P
T   T T F T
F   F T T F

The sentence is a disjunction: the disjunction of ‘P’ and ‘~P’. Its main connective is the vel. If you look under that vel, you’ll see that the sentence is true no matter what. In all the ways that things could be—whether ‘P’ is true or false— ‘(P (vee) ~P)’ is true. And this is down to its logical form: its structure and the meaning we have stipulated for the connectives vel and tilde.

This makes the sentence a formal logical necessity. A necessary claim is one which is true in all possible situations. Formal logical necessities are claims that are necessary because of their logical form.

Any sentence with the form ‘(α (vee) ~α)’ will be necessary—including those where what appears in place of ‘α’ is a complex sentence. This can be considered a law of logic. This law has a name: the law of the excluded middle.

Formal logical necessities are also called (formal logical) tautologies. (Notice that this sense of ‘tautology’ differs from the usual sense outside of logic, where it relates to redundant multiple expressions of the same idea—as in ‘frozen ice’, and ‘huge big rugby players’. Here it means: has value T in all lines of the truth-table.)

We can also use truth-tables to identify formal logical necessary falsehoods. These are sentences which are false in all circumstances because of their form. Formal logical necessary falsehoods in propositional logic come out false on all rows of their truth-table. (To see an example, construct and complete the truth-table for ‘(P & ~P)’.)

If a claim is true in some possible situations and false in others (and so neither a tautology nor a necessary falsehood), we say it is contingent.

© 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