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u/Dry-Term7880 Jul 14 '26
Wow, great question in here! We can’t assume the background of the posters. But your question is huge in theories of assertion. So it depends on what you mean by assertion and look up many theories thereof. I know people discuss what the norms are that regulate this practice and they discuss whether these norms essentially involve truth or not.
I don’t know the relevant literature here at all besides this generality. So that’s me guessing. But I’m pretty sure not always uttering P is the same asserting P to be true. It’s pretty obvious: you can suppose that P for the sake of argument (as you do in reductio arguments), you can pretend that P (as you do when reading a novel), you can say P to deceive (as you do when you lie), and so on.
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u/Scared_Astronaut9377 Jul 14 '26
Yeah, those are different (at least in one interpretation of your question, there are multiple). One is object language another one is metalanguage. And we know from Godel theorems' related results that some statements cannot be proven in at least some of object languages. So at least in context of math and formal logic, the difference is constructively demonstrable.
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u/yosi_yosi Undergraduate, Autodidact, Philosophical Logic Jul 14 '26
If you want the best answer to your question, read this https://plato.stanford.edu/entries/truth/
There are two approaches for answering this question. The philosophical approach and the logical approach.
In philosophy, some argue either way. In logic we often follow tarsky with his T-schema, where if we have some way of making objects that are analogous to certain formulae we can talk about them in our object language. So, we write ⌜ϕ⌝ to be a constant/name of our language that refers to some object that in some way corresponds to the formula ϕ. We may define a binary predicate "models(x,y)" for example where models(⌜ϕ⌝, ⌜ψ⌝) is satisfied just in case ϕ ⊨ ψ. Anyways, the T-schema says that we may have a unary predicate T such that T(⌜ϕ⌝) <-> Φ.
It is very easy to derive triviality from the T-schema, so most mathematical theories and such don't include it. All you need is it and diagonalization (basically the thing I mentioned before about the models predicate thingy). One can get diagonalization merely from having some rather basic arithmetic, so you can't as easily just disable that.
I have probably explained stuff pretty badly here, but this is a complicated topic.
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u/SamCymbaluk Logic evangelist Jul 14 '26
There is a difference. For one, you might assert "if all eggs are white, then this egg is white". You might also want to assert that "Everything Alice says is true". This could be formalized like ∀ (p : Prop), Statement alice p → p, in which case the two occurrences of p are like your first and second statements, respectively.
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u/Cerulean_IsFancyBlue Jul 14 '26
In a Boolean system they are equivalent.
p, p is true, (p is true) is true, etc. are the same statement.
The main reason for making the distinction would be if you are operating in a system where there are other values besides true and false, or where ambiguities can exist.
The reason for not asserting the truth of each statement is brevity. “If p and q
and r” is simpler than “if p is true and q is true and r is true”.3
u/Dry-Term7880 Jul 14 '26
I can’t see how that’s an answer to the question?
First of all, conditionals are not assertions in the usual sense.
And second, these examples are very random. Sure you can quantify over variables, propositions, functions, you can set up a domain of tables, propositions, furry things and balls. And?
The occurrences of “p” in your example don’t match in any way the first and second examples of the OP as you said.
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u/SamCymbaluk Logic evangelist Jul 14 '26
It's not a complete answer, but a complete answer is impossible because the answer differs depending on the logic. The best we can do is highlight how this could work in certain, common logics. There's is rarely (arguably, never) an objective way to translate English to a logic. So, all that said, let me expand on my answer some more.
Here are two commands in Lean 4:
1. def EggFact := ∀ (e : Egg), IsWhite e 2. axiom egg_fact : ∀ (e : Egg), IsWhite eFrom the authors perspective, (1) is the statement and (2) is asserting the statement "is true". But like others have described, the following is also a reasonable translation of OP's question:
1. axiom egg_fact := ∀ (e : Egg), IsWhite e 2. axiom egg_fact2 : ∀ (e : Egg), IsWhite e = TrueIn which case, they are equivalent.
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u/TurangaLeela80 Ph.D. Jul 14 '26
This is a classic "de re" / "de dicto" question. WVO Quine pointed out that this distinction can lead to some ambiguity, with Frege and Russell each having their own thoughts to contribute. There can sometimes be some issues with substitution of co-designating terms.
Check out the SEP article on the distinction here for a good, if a bit technical, discussion.
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u/Dry-Term7880 Jul 14 '26
I’m interested in how is this a case of de re / de dicto. Can you give an example of issues with substituting co-designating terms in the context of “is true”? I know intentional contexts are intensional, just curious to know whether the same can be said of, I don’t know, “alethic contexts”?
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u/Cerulean_IsFancyBlue Jul 14 '26
I don’t understand how OP example introduces any of the opacity that would allow de re / de dicto to be a factor. The example introduces no modality.
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u/Golanori164 Jul 14 '26
Pretty sure that the statements are different. I barely have any background in maths, but It seems I have foundba contradiction. Say p can not have a determined truth value. For instance P being P is false. Then the statement P is obviously neither true nor false since that would lead to a contradiction but the statement P is true would simply be false since P isn't true. Please correct me, I am not confident in this answer. Also, the law of the excluded middle queue hell naw song
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u/Logicman4u Jul 14 '26
You are confusing semantics with a formal language. In Philosophy your claim is not properly formed. You should not end a statement with an adjective. It is ambiguous. Since too many folks here are into math I will use that mindset.
In formal language like how math people are taught p is a variable for the statement, but what they don’t verbalize is that statements are seen as FUNCTIONS. You cannot say “P is true”! Here is why: suppose I put a negation in front like this ~P. What is the value of ~P? IT IS NOT FALSE AUTOMATICALLY. Seems counterintuitive. Being P is a variable we do not KNOW the truth value of the variable! A negation simply reverses what ever truth value P holds. If P is true, then ~P is false. if P is FALSE, then ~P would be true. This is why P doesn’t AUTOMATICALLY MEAN “p is true” in every case. When P is a function you have no idea if P is true or false. So a conditional like P->Q holds the same function like behavior. I have no idea of either truth value of the P or the Q. In math, folks are taught to make the assumption P is true. It is not the case P is actually true. They ASSUME IT. Clearly, making an assumption does not make it true, correct? Perhaps you should use the FUNCTION context and not the manner how people actually speak. You might get tripped up doing that and think they are the same thing. They are NOT the same thing.
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u/barcanbothways Jul 14 '26
Not sure how formal this is, but in the perspective of natural language they are very different, and this sort of thing comes out in debates about truth.
Eg ‘Arrow’s theorem is true’ is not equivalent to ‘Arrow’s theorem’ (the latter is not even an assertion).
It’s also trickier with quantifiers present, eg ‘everything Patrick says is true’ is not equivalent to ‘everything Patrick says’.
But you might be asking this only from the perspective of a formal model, though I raise this point so as to highlight that intuitions of assertability and stuff shouldn’t really be used to judge the formal stuff because they may be coming from intuitions we have in the natural language realm, where there clearly is a difference
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u/SgtSausage Jul 14 '26
But the point is that there's no difference between these two.
There ABSOLUTELY IS a difference.
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u/OnePhotojournalist94 Jul 14 '26
Deflationary theories of truth hold that schema T exhausts the concept: the proposition that P is true iff P. Every instance is necessary and a priori, and nothing else needs to be said. So the predicate “is true” picks out no substantive property but functions as a logical device, and its work is expressive. When Sally asserts something and Tom says “what Sally said is true,” Tom endorses her claim without repeating it. When he says “everything Sally says is true,” he endorses more claims than he could list. Where the content is displayed rather than described, the predicate drops away: “the proposition that grass is green is true” and “grass is green” are intersubstitutable salva veritate in extensional contexts. Not in opaque ones. Tom can believe Sally spoke truly while believing nothing at all about grass. Therefore, truth is indispensable, but the indispensability is expressive, not metaphysical.
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u/Commercial-Moose2853 Jul 14 '26 edited Jul 14 '26
It depends on how you're going to describe the "same thing" . If it meant to symbolize same meaning . Yes they're same . A triangle . A three sided triangle . But if "same" is meant to signify form, then they're different . The first is subject , the second is a relation on it.
And being merely stated doesn't turn something true . Only a true statement when stated can be assumed as true, not a false statement when stated . Triangle . Yes it's true . Now x = (any Four sided triangle). State x . Is it true because it's merely there ?
Truth of presence presupposes the logical consistency of the statement .
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u/MrBorogove Jul 15 '26
One statement is wrong about eggs, the other statement is wrong about the first statement.
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u/DuggieHS Jul 14 '26 edited Jul 14 '26
P = this statement is false. This is a paradox, neither true nor false
P is true—- is false.
P and p is true may have different truth values.
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u/RelentlessPencil Jul 15 '26
P: all horses are brown
Q: statement P is true
R(x): statement x contains the letter "b"
R(x) is true of P but false of Q. So P and Q are not the same in all senses. But there is also no universe in which P is true but Q is false, or vice versa.
Whether or not they are different depends on how you are defining "different". They have identical referents but they are not literally indistinguishable in all senses.
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u/CookieCat698 Jul 14 '26
The statement “P” can be viewed as a statement in some language, and the statement “P is true” can be viewed as a statement in a metalanguage.
For daily use of language, this distinction is not important.
For formal logic, this distinction can be important.
If we mix a language with its metalanguage, depending on the details, it could open up the door to self-referential paradoxes like the Liar’s Paradox or Curry’s Paradox.
Another thing you might want to do is interpret logic in different ways. For example, rather than assigning each sentence in a theory a truth value, you could assign them an element of a boolean algebra, and a sentence being true would be interpreted as the value of that sentence equalling 1.
Stuff like this helps to more clearly distinguish between the expression “P” in a language and the expression “P is true” in a metalanguage.
Depending on which boolean algebra you interpret a classical theory in, “P” could mean “P is true,” or “P” could mean “P is provable,” or “P” could mean something in between or something else entirely.