mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Archive

Show me a random blog post
 2019 
 2018 
 2017 
 2016 
 2015 
 2014 
 2013 
 2012 

Tags

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

Archive

Show me a random blog post
▼ show ▼

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 "r" then "a" then "t" then "i" then "o" in the box below (case sensitive):
© Matthew Scroggs 2019