# Snippet #1

2018-07-14, post № 200

mathematics, #analysis, #formal calculation, #symbolic

# Tau Day MMXVIII

2018-06-28, post № 199

C, programming,

                                     int
O=19,I=19;typedef long double f
;long fac(long n){return n?n--*fac(
n):1;}f pow(f x,int p){return p--?x
*pow(x,p):1;}f sin(f x){f v=0;for
(int           k=~0;++k
<=             O;)k%2&&
(v+=(~-k
%4?-1:1)*
pow(x,k)
/fac(k));
return v
;}main()
{f a=6,b
=7, tau;
for (int
j=~0;++j
<I;)sin(           tau
=a+(b-a)           /2
)>0? (b=          tau
):(a=tau        )  ;
printf (      "%ca"
"u ~ %.*Lf",116,
O,tau);}                     

Try it online!

Happy tau day.

# Truth

2018-06-16, post № 198

mathematics, programming, Python, #logic, #proposition calculus

Proposition calculus deals with statements and the relation between statements, where each of them can only be in one of two states; $\vdash p\lor\lnot p$. Therefore, when working with finitely many connected propositions, one can algorithmically determine all possible truth values of all atomic and thus connected propositions.

Truth is command-line tool which was written to precisely perform those computations; computing a logical expression’s truth value. Download link: truth.py
A list of all supportet operators can be seen by invoking the tool with a --help flag.
This project was inspired by Albert Menne’s Einführung in die Logik [1]; the operator syntax used is similar to his, translated to be 7-bit-ASCII-compatible.

Truth can be used to either verify universally true statements, e. g. tertium non datur and a property of the replication, verum sequitur ex quodlibet.

-(p&-p) <-> 1  ,  1 <- p
1 0010   1  1     1 1  0
1 1001   1  1     1 1  1

Though not only absolute truth, but also complete relational equivalence between two expressions can be shown.

(p->q)|(r>-<s) <-> q|(r|s)&-(r&s)|-p
01 0 1 0 0 0   1  00 000 01 000 110
10 0 0 0 0 0   1  00 000 01 000 001
01 1 1 0 0 0   1  11 000 01 000 110
11 1 1 0 0 0   1  11 000 01 000 101
01 0 1 1 1 0   1  01 110 11 100 110
10 0 1 1 1 0   1  01 110 11 100 101
01 1 1 1 1 0   1  11 110 11 100 110
11 1 1 1 1 0   1  11 110 11 100 101
01 0 1 0 1 1   1  01 011 11 001 110
10 0 1 0 1 1   1  01 011 11 001 101
01 1 1 0 1 1   1  11 011 11 001 110
11 1 1 0 1 1   1  11 011 11 001 101
01 0 1 1 0 1   1  00 111 00 111 110
10 0 0 1 0 1   1  00 111 00 111 001
01 1 1 1 0 1   1  11 111 00 111 110
11 1 1 1 0 1   1  11 111 00 111 101

Complete contravalence can also be shown.

-(p/-p>-<0)|p->q<-r >-< p&-q&r
0 0110 1 0 101 01 0  1  001000
0 1101 1 0 110 01 0  1  111000
0 0110 1 0 101 11 0  1  000100
0 1101 1 0 111 11 0  1  100100
0 0110 1 0 101 01 1  1  001001
0 1101 1 0 010 00 1  1  111011
0 0110 1 0 101 11 1  1  000101
0 1101 1 0 111 11 1  1  100101
Jonathan Frech's blog; built 2021/04/16 20:21:20 CEST