paulemok wrote: ↑
May 1st, 2018, 3:14 am
Every proposition describes a theory. Every proposition expresses an idea, regardless of whether the idea is true or false. Furthermore, every nonempty set of propositions describes a theory. Every such set expresses a system of ideas that all together form a composite whole, regardless of whether the composite is true or false.
This is where we disagree. A proposition in logic does not express anything, just like a number on its own does not express anything. '2' does not express the idea 'there are two apples on the table'.
'P' does not assert 'Socrates is a man
'. If these terms did stand for ideas, then we would no longer be free to simply assume truth values. Whether P was 'true' or not would be an empirical matter whereas T/F values in logic are purely formal.
Nor could we say what idea a proposition stood for. We express ideas and theories in ordinary language but ordinary language does not work in a logical way. If we try to give an ordinary language equivalent of a proposition we find it never has a simple binary true/false value.
The two given postulates of B are arithmetical propositions about the real number 1.
Then I would ask what you mean by 'the real number 1'
. For example, if you meant 'the way that number is used in maths'
then we would consult a mathematician, or look at books, and discover which one of those two contradictory propositions was correct.
I can write 'Mary is in the park'
and 'Mary is at home'
and there is no contradiction because they are just two sentences. But if they are presented as the answer to the question; 'Where is Mary?
i.e. theories about the location of Mary, then both cannot be true. And because we have an understanding of what we mean when somebody asks 'Where?'
we would know how to find out whether either sentence was true or not.
Me: But if i say a theory is sound then I have to be able to explain why.
I have not said that B is sound. I have said that B exists.
By 'sound' I would understand an argument that is not only logically valid but where the premises are actually true; not just given an assumed formal value 'true' but true in the sense of being a fact about the world.
If 'B' is not said to be 'sound', so not a claim about some fact, yet we also say 'B' does 'exist', then how can I understand what that claim 'exist' might mean? The word 'exist' must have some sense. There must be some reason why we assert 'B' 'exists' as opposed to 'does not exist'. If it is not some sort of factual claim, then what is it?
There are various ways we might say 'B exists'. For example, we might say 'as words on this web page'
. That would be an empirical matter; we could all agree that 'B' existed in that sense, while not agreeing that those sentences made sense, or counted as what is normally understood to be a 'theory'. If instead we claim 'B exists as a theory'
that would be a assertion about language; does 'B' correspond to the way people understand and use the word 'theory'? And so on.
As I write above, this sort of uncertainty about the meaning of 'exist' - a word that seems on first sight to be very simple - applies to all ordinary language. You can only treat logical propositions as simple binary T/F things because they are not in ordinary language. But I think you do not clearly distinguish between the two. You start by describing your sentences as 'propositions' but these become 'postulates' and 'theories' without it being made clear whether all those other words are meant to be synonyms for 'proposition' or claims of something beyond formal logic.