mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Logical contradictions

 2016-10-08 
During my Electromagnetic Field talk this year, I spoke about @mathslogicbot, my Twitter bot that is working its way through the tautologies in propositional calculus. My talk included my conjecture that the number of tautologies of length \(n\) is an increasing sequence (except when \(n=8\)). After my talk, Henry Segerman suggested that I also look at the number of contradictions of length \(n\) to look for insights.
A contradiction is the opposite of a tautology: it is a formula that is False for every assignment of truth values to the variables. For example, here are a few contradictions:
$$\neg(a\leftrightarrow a)$$ $$\neg(a\rightarrow a)$$ $$(\neg a\wedge a)$$ $$(\neg a\leftrightarrow a)$$
The first eleven terms of the sequence whose \(n\)th term is the number of contradictions of length \(n\) are:
$$0, 0, 0, 0, 0, 6, 2, 20, 6, 127, 154$$
This sequence is A277275 on OEIS. A list of contractions can be found here.
For the same reasons as the sequence of tautologies, I would expect this sequence to be increasing. Surprisingly, it is not increasing for small values of \(n\), but I again conjecture that it is increasing after a certain point.

Properties of the sequences

There are some properties of the two sequences that we can show. Let \(a(n)\) be the number of tautolgies of length \(n\) and let \(b(n)\) be the number of contradictions of length \(n\).
First, the number of tautologies and contradictions, \(a(n)+b(n)\), (A277276) is an increasing sequence. This is due to the facts that \(a(n+1)\geq b(n)\) and \(b(n+1)\geq a(n)\), as every tautology of length \(n\) becomes a contraction of length \(n+1\) by appending a \(\neg\) to be start and vice versa.
This implies that for each \(n\), at most one of \(a\) and \(b\) can be decreasing at \(n\), as if both were decreasing, then \(a+b\) would be decreasing. Sadly, this doesn't seem to give us a way to prove the conjectures, but it is a small amount of progress towards them.

Similar posts

Logic bot, pt. 2
Logic bot
Interesting tautologies
How OEISbot works

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 "number" in the box below (case sensitive):

Archive

Show me a random blog post
 2021 

Jan 2021

Christmas (2020) is over
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

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

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2021