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.
"The delivery of an argument is independent of its validity, at least in general."
If this comes as intended to mean the same idea as "the delivery of an argument is usually independent of its validity." Then you'll probably want to use a multi-valued or fuzzy predicate logic of some sort to formalize this sentence.