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
How OEISbot works
Raspberry Pi weather station

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 "regetni" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2019 

Jul 2019

Big Internet Math-Off stickers

Jun 2019

Proving a conjecture

Apr 2019

Harriss and other spirals

Mar 2019

realhats

Jan 2019

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

Tags

propositional calculus dataset pac-man bodmas football palindromes programming folding paper geometry polynomials game of life tennis pythagoras probability mathslogicbot royal baby manchester science festival map projections chebyshev weather station braiding pizza cutting latex oeis speed rugby a gamut of games puzzles go the aperiodical video games european cup reddit people maths world cup trigonometry cross stitch nine men's morris countdown games curvature hats logic reuleaux polygons fractals dragon curves php game show probability draughts data martin gardner dates ternary stickers machine learning accuracy estimation final fantasy christmas captain scarlet gerry anderson menace wool bubble bobble sorting twitter sport light interpolation rhombicuboctahedron statistics books platonic solids javascript misleading statistics raspberry pi frobel inline code radio 4 chess mathsjam matt parker folding tube maps mathsteroids news manchester realhats arithmetic binary national lottery golden spiral noughts and crosses plastic ratio big internet math-off flexagons christmas card hexapawn craft asteroids harriss spiral london underground chalkdust magazine sound golden ratio graph theory python error bars coins triangles approximation london electromagnetic field

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2019