Example: There are no round squares.

An “analytic” sentence, such as “Ophthalmologists are doctors,” has historically been characterized as one whose truth depends upon the meanings of its constituent terms (and how they’re combined) alone... The Analytic/Synthetic Distinction

It is obvious that this definition has some limitations. We can know that: "the big bang theory" definitely has a meaning, but this meaning does not make it true. Some things can be known to be true entirely on the basis of their meanings. We can know that "cats are animals" is true based on what those two words mean.

How do we know which ideas are definitely true and which ones are not definitely true?

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    "Incorrect" in what sense? One is free to define words however they wish. Curry is not even defining "true", just "true for {T}". Indeed, many people define "true for me" based on personal judgments and distinguish it from "true" simpliciter, which they expect most others to share. Moreover, "truth is provability" is quite a conventional stance for intuitionists that Curry was sympathetic to. – Conifold May 8 at 18:32
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    @SofieSelnes So as my response to Conifold indicates Incompleteness and Undefinability do not universally apply to mathematics and logic. They only apply to one particular point-of-view of the notion of a formal system. Other notions of formal systems that are equally mathematical and logical can eliminate Incompleteness and Undefinability while maintaining consistency. These systems reject undecidable sentences as simply untrue. – polcott May 8 at 21:06
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    This question is all over the place. Either you base the question on Curry, where it makes no sense to talk about correct or incorrect since the quote does only state one way to frame things and does not even try to state any objective, absolute, or otherwise authoritative definition. It is an argument showing implications of a certain line of thought. Or you once again ask about Truth (capital T) vs. probability, in which case the whole Curry setup is nothing but a fig leaf to poorly conceal another try to pose your theory. Which one is it? – Philip Klöcking May 9 at 20:50
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    Take my points or leave it. If you do not want any help, I cannot help it. I will not discuss anything here. I tried to tell you what makes this question a mess and how you could tidy it up. I did understand the point the first time you implied it. Still, this is an argument you are trying to construct, not a question. That's why I ignored it. If you posed a question proper instead of trying to lecture, there would be interesting ones, including the one I proposed. But please spare us another lecture with seemingly half-informed appeals to literature – Philip Klöcking May 9 at 21:42
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    No, it is your mistake. Tarski does not work in such a context, and his truth is semantic. It is decided by a model, not theory or meta-theory. – Conifold May 13 at 5:36

This is the most basic foundation of analytic truth:
New concepts are created and they are assigned a Name

The axiomatic argument of the Münchhausen trilemma:
states: "accepted precepts are merely asserted rather than defended"

The axiomatic argument, which rests on accepted precepts (i.e. we reach some bedrock assumption or certainty)


We can know with certainty that a gross pay amount of $400 is correct if the number of hours worked was 40 and the $ per hour was 10. The semantic meaning of the product of decimal integer 10 multiplied by decimal integer 40 is defined by its algorithm to be 400.

This example proves that some knowledge is defined by an algorithm that has been assigned to a term. The "product" of two integers is defined by its algorithm.

Analytic propositions are true solely by virtue of their meaning

We can overcome this objection by realizing that when brand new ideas are created or discovered they are assigned to a term. This is the beginning of their foundational basis. The term: "the big bang theory" has been assigned to a connected set of ideas.

At the point in time when the ideas have been assigned to a term they are merely asserted rather than defended. The assignment of these ideas to a term however, is known with logical certainty.

We can also know with complete certainty that the term: "the big bang theory" has been assigned to the category of possibly true rather than definitely true because it is called a theory. So the big bang theory itself cannot be construed as knowledge because knowledge must be definitely true. Analytical knowledge must be definitely true entirely on the basis of its meaning.

The big question is how do we know that an expression of language is definitely true?
We can definitely know that the connected set of ideas the define the concept of arithmetic makes this expression true: "3 > 2".

Isn't that just an example of an expression that is merely asserted rather than defended?
No it is not. The connected set of ideas of the knowledge of arithmetic form a subset of the body of analytical knowledge. It is a set of ideas that fit together to mutually define each other.

None of these ideas are merely asserted they all fit together to specify an algorithm (definite sequence of operations). From this algorithm we can decide whether or not an expression such as "3 > 2" belongs to the set of expressions comprising the knowledge of arithmetic.

When we look at this using mathematical terms we are establishing some expressions of language as axioms of the formal system of that language. For all practical purposes the axioms of a formal system are stipulated relations between the objects of that formal system. These stipulated relations comprise the basic concepts that the formal system operates with. They are the building blocks of the system.

The arithmetic of natural numbers depends upon the concept of an ordered set of numbers. At some point in time someone had to create this concept and assign it to a name. So the way that the axiomatic argument of the Münchhausen trilemma is overcome is that new concepts are created and assigned a name.

| improve this answer | |
  • Comments are not for extended discussion; this conversation has been moved to chat. – Geoffrey Thomas Jun 14 at 18:35
  • @Hypnosifl I updated my post to address your comments. Feel free to make more comments in chat. – polcott Jun 14 at 18:42

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