2 posts tagged “tautology”
Yes, I did ask for it and did give them a drawing of the tautology, but isn't it just great, anyway? Truth has never been so yummy!
The tautology reads as follows: If yellow implies green and green implies orange, then yellow implies orange. (colours refer to the colours of the Gummibärchen)
Okay, so this one is crocheted, but a quilt in this style would be nice, too, and "theorem quilting" has a nice ring.
When I learned about Peirce's existentail graphs, almost the first thing that came to my mind was that is was easy to replace letters by colours and make it something crocheted. This first sample is Leibniz' Theorem, i.e. if yellow implies red and blue implies green, then yellow and blue imply green and red. Which, of course, is a tautology.
Peirce had lines (called cuts) drawn around negated things. Two lines are non-negated, i.e. positive again. So, symbols surrounded by one cut are in a negative context, those surrounded by two cuts (or none) are in a positive context. Though Peirce wanted to use it for reasoning, actually he gave a sound and complete propositional calculus. (There is a beta version with quantification and a gamma version which incorperates higher order logic.) Writing down a symbol means to assert it. Writing down to symbols asserts both; i.e. asserting the conjunction of the two symbols. Since the deMorgan laws are valid in this calculus, you can write every formula just with conjunction and negation.
Since I did not want to crochet lines, I changed the negative context into black. Which, incidentally, makes the thing easier to read - you don't have to count lines to determine the context.
It would be nice to find a way to crochet proofs, too. Maybe a 24-step-proof makes a nice afghan...
When I learned about Peirce's existentail graphs, almost the first thing that came to my mind was that is was easy to replace letters by colours and make it something crocheted. This first sample is Leibniz' Theorem, i.e. if yellow implies red and blue implies green, then yellow and blue imply green and red. Which, of course, is a tautology.
Peirce had lines (called cuts) drawn around negated things. Two lines are non-negated, i.e. positive again. So, symbols surrounded by one cut are in a negative context, those surrounded by two cuts (or none) are in a positive context. Though Peirce wanted to use it for reasoning, actually he gave a sound and complete propositional calculus. (There is a beta version with quantification and a gamma version which incorperates higher order logic.) Writing down a symbol means to assert it. Writing down to symbols asserts both; i.e. asserting the conjunction of the two symbols. Since the deMorgan laws are valid in this calculus, you can write every formula just with conjunction and negation.
Since I did not want to crochet lines, I changed the negative context into black. Which, incidentally, makes the thing easier to read - you don't have to count lines to determine the context.
It would be nice to find a way to crochet proofs, too. Maybe a 24-step-proof makes a nice afghan...
Some less rambling information on Peirce's graphs can be found here:
