Proving Formal Arguments
For a great, accessible introduction to formal logic, including how to prove that valid formal arguments are indeed valid and construct counterexamples that demonstrate the invalidity of invalid arguments, see Paul Teller's A Modern Formal Logic Primer, available free online here[1].
Work In Progress.
This is a short introduction on how to build and prove formal arguments :
Motivation
There are specific concepts to define in order to have certain guarantees. A simple example will be that you can define a system of formal logic on your own, but then it's up to you to demonstrate that your system is sound (if T ⊢ ϕ then T ⊨ ϕ) and adequate (if T ⊨ ϕ then T ⊢ ϕ). By using one of the system already worked on decades ago we have those guarantees.(in brief, everything you can derive is True, and everything that is true has a derivation)
Setting up the stage
- 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.
Wiki usage
In this wiki we use the preferred system of natural deduction. We use no axioms and a set of deduction rules. We do include the law of excluded middle unless otherwise specified ( non intuitionistic logic). Other type system are possible ( Hilbert type system , intuitionistic logic, kleen systems etc ...) but here we made this conscious choice because it mimic how the layman try to reason. For a full list of deduction rules see the wikipedia entry : Natural deduction
Note on symbols
T ⊢ ϕ : ϕ is provable/derivable in the set of premises T T ⊨ ϕ : ϕ follow semantically from the set of premises T
