LOGIC: A Request for Clarification of definitions

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Discussion Overview

The discussion centers around the clarification of definitions related to truth and provability within the context of logic. Participants explore the meanings of terms such as "true," "false," "proven," "provable," "unprovable," and "correct," and how these concepts are understood in formal logic and mathematical logic.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Meta-discussion

Main Points Raised

  • One participant questions the nature of truth in logic, asking if something proven is necessarily true and what constitutes provability.
  • Another participant suggests that understanding truth in mathematical logic requires knowledge of language, formulas, structures, and variable assignments.
  • A request for book recommendations on mathematical logic is made, indicating a desire for further reading on the topic.
  • One participant attempts to define the terms, suggesting that "true" can be derivable from axioms or true under all interpretations, while "false" can be derivably false or false under an interpretation.
  • The definitions of "proven" and "provable" are discussed, with one participant suggesting that "proven" refers to statements with a derivation and "provable" refers to statements that can be shown to be provable.
  • Unprovable statements are described as those that can be shown not to be true, while the meaning of "correct" remains uncertain among participants.

Areas of Agreement / Disagreement

Participants express differing views on the definitions of truth and related concepts, with no consensus reached on the meanings or implications of these terms in logic.

Contextual Notes

Some definitions provided are contingent on philosophical interpretations, and there is uncertainty regarding the relationship between truth and the language used to express it. The discussion also touches on the potential need for a formal logical language to articulate these concepts clearly.

Mathbrain
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With the study of logic, lots of words get thrown around that I don't really understand their complete meaning. With a deductive argument the conclusion is true if the premises are true, and an argument is valid if all the inferences (and the conclusion) follow logically from the axioms. These are things taught in any intro to logic class, but the more important question is: "What is truth?" Not just philosophically, but in the realm of logic. If something is proven does that mean it is true? If something is provable, does that mean it is true? Which immediately asks the question, what is provable, and what is proven? This isn't an issue of picking words apart, it's a question of logic. How are these concepts defined in the formal study of logic?

Here's a list of words that I require clarification for, wiki isn't always helpful...
*True
*False
*Proven
*Provable
*Unprovable
*Correct

This is a serious query, I am not interested in getting into an argument on the nature of definitions, please do not consider "What is truth?" to be a profound philosophical question. The issue is what is true from a logical foundation.
 
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To get an understanding of truth as a concept in mathematical logic, you should read about the definitions of language, formula in a language, structure and variable assignment functions. A definition of truth of a formula in a language uses all these terms. I suggest you get yourself a book in mathematical logic.

I suggest this thread to be moved to the math forums logic section...
 
"get yourself a book in mathematical logic"
Any recommendations?

"Mathematical logic" sounds like a better home, but I don't know how to move the thread. Conversely if we leave the thread in logic, is the definition of truth intrinsically linked with the language of the speaker (this case English)? I'm going to assume that you meant a formal language, but I'm not aware of a logical formal language that describes True and False in a logical context. Is it a second-order logic notation?
 
Long time ago, but I'll try and see how far I'll get. I am actually interested in how many flukes I'll make on this one.

*True, derivable either from axioms or true under all interpretations. (Philosophers may differ on the real meaning of truth.)
*False, derivably false or false under an interpretation
*Proven, a statement for which a derivation exists (or all interpretations are proven to be true)
*Provable, as in provable to be true, a statement for which it can be proven that it can be proven
*Unprovable, as in provable not to be true, a statement for which it can be proven not to be true (either since it is false, or it can't be proven true)
*Correct, dunno? As in semantic or syntactic correctness?
 
Doesn't meet criteria for Philosophy or logic, it's not a problem.
 

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