I found myself the motivation to translate some statements and either prove them in a specific setting (assumed premises) or at least decide on their provability.
However, I have very little background in formal logic (little bit of propositional and predicate calculus) and find it hard to actually use what I've learnt the way I want. We've only been dealing with exercises that are very artificial in nature and there's no provability involved as we were always given the task to either prove trivial things, or find a counter-example to it.
I am looking at, in particular, a formalisation of the following statement:
The delivery of an argument is independent of its validity, at least in general.
By the '..at least in general., I mean to exclude arguments like:
This statement is delivered by a person named Peter.
Which its validity inevitably rests on the nature of the delivery of the argument.
Here's my naive attempt:
I first want to define a relation that will represent the idea of independence. Let I be a binary relation such that aIb (or I(a,b)) means 'a is independent of b'. I is symmetric. But I am not sure about its reflexive property, and I am thinking that it is non-transitive.
Now if I let a stands for an arbitrary argument, then I want to say: I(delivery of a, validity of a)
Are these functions? I could let v(a) be the function that returns the validity of a. But then the delivery of a wouldn't be a function in the same sense, it is more like a set to me. A set that contains propositions like
- a is delivered by Peter
- a is delivered at noon
- and so on...
I describe this set as D(a) for now.
Then there's still the problem of catching all the Peter exceptions. I first thought of excluding self-referential arguments, but that may be too strong. So I decided to go with:
Let a be an arbitrary argument such that I(a, D(a))
Then we have I(D(a), v(a))
It is a rather trivial thing to say, but I think it is a useful statement to have. But how would I write it in a purely formal way? How would one go about proving it or otherwise? I am not even entirely sure of the keywords that I should be researching into.