Questions tagged [mathematical-logic]
The mathematical-logic tag has no usage guidance.
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Can Internal Set Theory provide a complete system of arithmetic?
Internal Set Theory (IST) is a conservative extension of ZFC that adds three axioms that serve to define a predicate standard such that all numbers are either standard or not. There are finitely many ...
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Mathematics & Logic (Boolean Algebra)
Been toying with the idea of mathematizing logic, essentially finding mathematical operation analogs to logical connectives. My attempt vide infra
For 1 = True = T and 0 = False = F
a) Logical AND (p ∧...
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Does a mathematical object that does not contradict itself have to exist?
I have recently finished the chapter on constructing the real numbers in my Analysis textbook (via Dedekind cuts). At first the natural numbers, then the whole numbers and the rational numbers were ...
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Is there an interesting relationship between formula formation rules and the rules of sequent calculus?
I am happily proceeding to chapter 4 of Ebbinghaus et al.’s Mathematical Logic and able to ask a new range of clarifying questions on first-order logic.
A first-order theory, regardless of its ...
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An Analogy Between The Goal of the Tractatus and Formal Axiomatic Systems
After struggling with a few sections of the Tractatus, as well as the explanations of said sections is Monk's How to Read Wittgenstein and Glock's A Wittgenstein's Dictionary, I've come to a certain ...
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How do skeptics explain axioms not being arbitrary?
I get infinite regress but surely the axioms of ZFC or arithmetic were not so much chosen as discovered and intuited and thought about. They certainly didn't just grab whatever was around them and say ...
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Can two formulas be “valid under the same interpretations”?
Ebbinghaus 2021:
A formula φ is valid (written “⊨ φ”) iff ∅ ⊨ φ.
Thus, a formula is valid if and only if it holds under all interpretations.
Let’s break this down:
First of all, it is stated in the ...
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The validity of the reasoning in Halting Theorem
Here is an example of the Halting Theorem from Wikipedia (Halting Problem)
Christopher Strachey outlined a proof by contradiction that the
halting problem is not solvable.The proof proceeds as
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What difference between the truth of a conditional* and its logical validity?
I am confused . . . Here is a remark on the "classical analysis" of the implication:
On the classical analysis, logical implication is the same, not as the truth of a conditional statement, ...
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Is there a limited number of 'pragmatic' logic rules?
What you have cited is a pragmatic limit, as you have not seen logic
systems with more than 8 or so precepts.
IF there were such a limit to precept quantity,
then YES there would be a limit to the ...
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What is the relationship between the logical operators in a logic, and how can that relationship be generalized?
One common type of logic, first-order logic, is commonly presented as having a certain collection of well-known logical operators, including:
AND
OR
IMPLIES
NOT
etc.
Interestingly, some of the ...
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Extending the use-mention distinction to account for variables and predicates
When we talk about the use-mention distinction, often the following is said:
To use an expression means to refer to its meaning, to mention an
expression means to refer to the expression itself.
I ...
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What would be the algebraic generalization of the concept of “soundness” in mathematical logic?
Is there a corresponding/generalized concept of “soundness” as we abstract logical structures into algebraic ones? In the manner of algebraic logic: en.m.wikipedia.org/wiki/Algebraic_logic
Let us say ...
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Is there a formal logic that distinguishes between a priori and a posteriori truths?
Briefly,
In a previous post, I explored the question of if logical systems have any way of distinguishing between true statements that are obvious or tautological, and thereby not ‘meaningful’, versus ...
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Is it a problem for arithmetic or our representation (or both) that there is incompleteness?
Is this a settled (as much as it can be) philosophical area? I feel like I understand that there will always be incompleteness for a finite set of axioms trying to capture all of arithmetic. But I ...
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