Like the 3-dimensional mandala, a first order language is built up from its base, which is its set of symbols. Above the base we build the terms, and on the third and most important level we build the formulas.
Are there higher levels? How could we use the language of mandalas to build up deductions?
What about the semantics?
4 responses so far ↓
an125231 // Mar 21st 2013 at 9:41 pm
To build up deductions using the language of mandela, we may take the formula, O is an initial segment of O*O, where O is the constant symbol and is assigned the empty mandala, and deduce that for every free variable x, if x is a mandala, then x is an initial segment of x*x. This is true because the empty mandala is an initial segment of any mandala, so it is an initial segment of the mandala composed of two empty mandalas.
an125231 // Mar 21st 2013 at 11:24 pm
The empty mandala may be substituted by any mandala, making the formula true for any mandala, I think.
an125231 // Mar 25th 2013 at 8:08 pm
The constant symbol in the language of mandala is variable free, so in the formula, O is an initial segment of O*O, O is substitutable by any term, so substituting O with a free variable, x, the formula reads x is an initial segment of x*x, and since x is any free variable, then for all free variable x, x is an initial segment of x*x.
Xin Lin // May 25th 2013 at 12:37 am
Symbols are under terms, terms are under formulas, Formulas are under sentences, (whether the sentences have truth value or not), Sentences are under sets whether the sets are consistent or not, sets are under theorems whether provable or not. And such structure can continue to build on. There might exist a limit when it reaches to only one thing that defines all. and such thing can be represent by the most basic symbols. First order language is a loop.