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 "h" then "e" then "x" then "a" then "g" then "o" then "n" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Jul 2020

Happy 3e√3π-43 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

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

Archive

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