mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Interesting tautologies

 2020-05-03 
This is a post I wrote for The Aperiodical's Big Lock-Down Math-Off. You can vote for (or against) me here until 9am on Tuesday...
A few years ago, I made @mathslogicbot, a Twitter bot that tweets logical tautologies.
The statements that @mathslogicbot tweets are made up of variables (a to z) that can be either true or false, and the logical symbols \(\lnot\) (not), \(\land\) (and), \(\lor\) (or), \(\rightarrow\) (implies), and \(\leftrightarrow\) (if and only if), as well as brackets. A tautology is a statement that is always true, whatever values are assigned to the variables involved.
To get an idea of how to interpret @mathslogicbot's statements, let's have a look at a few tautologies:
\(( a \rightarrow a )\). This says "a implies a", or in other words "if a is true, then a is true". Hopefully everyone agrees that this is an always-true statement.
\(( a \lor \lnot a )\). This says "a or not a": either a is true, or a is not true
\((a\leftrightarrow a)\). This says "a if and only if a".
\(\lnot ( a \land \lnot a )\). This says "not (a and not a)": a and not a cannot both be true.
\(( \lnot a \lor \lnot \lnot a )\). I'll leave you to think about what this one means.
(Of course, not all statements are tautologies. The statement \((b\land a)\), for example, is not a tautology as is can be true or false depending on the values of \(a\) and \(b\).)
While looking through @mathslogicbot's tweets, I noticed that a few of them are interesting, but most are downright rubbish. This got me thinking: could I get rid of the bad tautologies like these, and make a list of just the "interesting" tautologies. To do this, we first need to think of different ways tautologies can be bad.
Looking at tautologies the @mathslogicbot has tweeted, I decided to exclude:
After removing tautologies like these, some of my favourite tautologies are:
You can find a list of the first 500 "interesting" tautologues here. Let me know on Twitter which is your favourite. Or let me know which ones you think are rubbish, and we can further refine the list...

Similar posts

Logical contradictions
Logic bot, pt. 2
Logic bot
A surprising fact about quadrilaterals

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "sexa" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Jul 2020

Happy 3√3+e-√2-5 Approximation Day!

May 2020

A surprising fact about quadrilaterals
Interesting tautologies

Mar 2020

Log-scaled axes

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

the aperiodical captain scarlet matt parker braiding manchester people maths golden ratio manchester science festival royal baby sobolev spaces mathslogicbot palindromes nine men's morris noughts and crosses rhombicuboctahedron sorting trigonometry ternary plastic ratio reddit fractals london estimation logic computational complexity speed triangles final fantasy binary game of life radio 4 graphs graph theory european cup christmas card signorini conditions inverse matrices matrix of cofactors books sport propositional calculus gaussian elimination london underground puzzles programming tennis stickers geogebra folding paper reuleaux polygons dragon curves chess geometry frobel matrix of minors cambridge gerry anderson wool big internet math-off probability wave scattering video games raspberry pi boundary element methods asteroids rugby accuracy determinants pythagoras dates pac-man matrix multiplication folding tube maps talking maths in public finite element method arithmetic simultaneous equations sound pizza cutting squares pi approximation day statistics python weak imposition harriss spiral curvature martin gardner dataset chalkdust magazine realhats data football misleading statistics matrices inline code platonic solids go error bars world cup pi hexapawn chebyshev ucl weather station map projections light data visualisation preconditioning cross stitch php national lottery coins a gamut of games twitter mathsjam christmas oeis advent calendar approximation craft electromagnetic field phd game show probability mathsteroids menace machine learning polynomials bubble bobble tmip bempp hannah fry exponential growth convergence interpolation flexagons numerical analysis hats royal institution bodmas golden spiral javascript countdown latex draughts quadrilaterals logs games news

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020