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

Archive

Show me a random blog post
 2020 

Jul 2020

Happy τ+e-6 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

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

Archive

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