Proving Formal Arguments
Work In Progress.
- The language used for his formal argument (symbolically speaking, English, Chinese, and other ambiguous language are to be avoided, but with extensive definitions it's possible to use them)
- The set of valid formulas
- The set of logical axioms
- The set of predicates (functional and relational)
- The structure/interpretation
- The deductions rules
A formula is a concatenation of the symbols of the language. Obviously we do want to throw away and not consider most of the formulas ( e.g AvBvv! Etc.. ). Usually we define a set of recursive rule that define what is a valid formula ( called in the literature well formed formulas). This is a part that actually remove some ambiguity if done properly.
The logical axioms are axioms that are purely syntactic. They work in balance with the deductions rules. You can have no logical axioms and a ton of deduction rules, or no deduction rules and a ton of axioms or a healthy balance of the two.( I tend to work with no axioms and a few deduction rules )
The set of predicates is the relations of the system. They take n-variables and return a truth value.
The structure is the semantic part of the system. As of now everything has no meaning. A structure is an assignment of a domain of discourse ( here it would be the set of all living things, or sentient living beings ... ) and for the predicates the structure assign a certain meaning. So that A(x) represent x is an animal and become true for some of the x in the domain of discourse.
Once you have all of this you have a complete system and start deriving formula and be assured they are valid and with an English interpretation.
I should also mention that premises play the role of non logical axioms. Basically those are axioms that you add on top of the logical ones to study a specific system ( here morality of beings ). You could have set up the axioms of arithmetic and start to study arithmetics, or the axioms of Euclidean geometry and study this field instead.
