Further, to avoid circularity, the notion of truth cannot occur in ϕ.For any sentence p, p is true if and only if ϕ
Putting these together, if we have as one of our sentencesThis sentence is false
then our theory will produce a sentence in the metalanguage that looks likeSnow is white
where s is a sentence in the metalanguage."snow is white " is true iff s
For any sentence p, p is true if and only if ϕ
A name n designates an object o if and only if (( n = "Adam" and o = Adam) or ( n = "Bob" and o = Bob) or( n = "Carol" and o = Carol)...
An object o satisfies a predicate f if and only if ((f="is english" and o is English) or (f="is french" and o is french)
For any sentence p, p is true if and only if ϕ
For any sentence p, p is true if and only if S
"p" is true if and only if p
and we tie meaning down by sticking to one sentence, so that the meaning cannot be ambiguous. We name the sentence on one side, and use it on the other.For any sentence p, p is true if and only if ϕ
"p" is true if and only if p
and points out that if we take a sentence p and produce anther sentence ϕ that satisfies the condition of material adequacy, then ϕ gives the meaning of p.For any sentence p, p is true if and only if ϕ
A name n designates an object o if and only if (( n = "Adam" and o = Adam) or ( n = "Bob" and o = Bob) or( n = "Carol" and o = Carol)...
An object o satisfies a predicate f if and only if ((f="is english" and o is English) or (f="is french" and o is french) — Banno
This sort of construction has an antirealist bent.
So suppose our language were the whole of mathematics, and we adopted a constructivist position, such that a mathematical theorem is true only if there is a proof that it is true. We can adopt the antirealist position that the Goldberg Conjecture, since it is unproven, has the truth value "meh" - is neither truth nor false. — Banno
metalanguage — Banno
Something not quite right there. Did you mean (the Goldbach conjecture is) true XOR false? Any proposition is either true or false (principle of bivalence). — Agent Smith
Suppose we restrict the object language to being about a group of people, Adam, Bob and Carol...
And in the metalanguage we can have a definition of "designates":
A name n designates an object o if and only if (( n = "Adam" and o = Adam) or ( n = "Bob" and o = Bob) or( n = "Carol" and o = Carol)...
Doubtless this looks cumbersome, despite my having skipped several steps, but it gives us
a metalanguage and and object language both talking about the same objects, Adam, Bob and Carol..., and a way to use the same name in both languages. — Banno
I don't think it helps to introduce "meh" as a truth value for undecided arithmetical propositions, because that would distort the existent meaning of arithmetical truth values for both the constructive and classical senses of arithmetic. — sime
Something not quite right there. Did you mean (the Goldbach conjecture is) true XOR false? Any proposition is either true or false (principle of bivalence). — Agent Smith
I'm not sure, but: you mean object language? The interpretation is that fragment of the metalanguage that interprets terms of the object language? — bongo fury
A good read Banno - you simplified it enough that I think I followed along :) — Moliere
if we designate our meta-language to refer to the same objects, does it still, at the same time, function as a meta-language? — Moliere
No and yes. In Kripke's system the truth value of an unproven conjecture would be neither true nor false. Hence, meh. It's a non-classical logic, so the principle of bivalence is dropped. — Banno
Accepting a third truth value basically rejects the principle of bivalence. — Moliere
By logic, do you mean first order logic?Not sure. The logics here are an attempt to make these issues clear. — Banno
We want to add predication. To do this, Tarski developed satisfaction. — Banno
Yep, because the object languagecan talk about Adam and Bob, but can't talk about itself, however the metalanguage can talk about Adam and Bob, and about the sentences of the object language.
So we have Adam, Bob, Carol,...
And in the object language we can write about them: (Adam is English).
And in the metalanguage we can write about them : (Adam is English), and add sentences from the object language: ("Adam is English" is true) — Banno
Quick question. — RussellA
I perceive something in the world that is cold, white and frozen, and I name it "snow".
Therefore, "snow" means something in the world that is cold, white and frozen. — RussellA
By logic, do you mean first order logic? — ssu
Premises might have a few different meaning here. So there are a bunch of rules that set out the game of first order logic - I found this neat summary. There are also axiomatisations, systems in which a specified set of tautologies is assumed. See rules 1 through 8 in this axiomatisation.. This system is both consistent and complete. Only and every true tautology can be deduced from the axioms. The proof of this is called "Gödel's completeness theorem", mostly in order to cause utter confusion.What if our premisses are wrong when we try to make a theory of truth as we try to do it? — ssu
"Snow is white" is true IFF snow is white, and
"Snow is on the ground" is true IFF snow is on the ground, and indeed
"Snow is turquoise with purple polkadots" is true IFF snow is turquoise with purple polkadots
are all true. — Banno
"Snow is turquoise with purple polkadots" is true IFF snow is turquoise with purple polkadots — Banno
But
"Snow is white" is true IFF snow is white, and
"Snow is on the ground" is true IFF snow is on the ground, and indeed
"Snow is turquoise with purple polkadots" is true IFF snow is turquoise with purple polkadots
are all true. — Banno
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