Questions tagged [philosophy]
Questions involving philosophy of mathematics. Please consider if Philosophy Stack Exchange is a better site to post your question.
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Is there a modal operator, L, that satisfies φ ↔ 𝛙 & ~L𝛙 ⊢ ~Lφ?
I am wondering if there's some modal operator that would satisfy $$φ ↔ 𝛙 \& ~L𝛙 ⊢ ~Lφ.$$ That is:
Given $φ ↔ 𝛙 \& ~L𝛙$
You can get to $~Lφ$
One limitation is that $L$ for sure does not ...
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How should one understand the "universe of sets"?
One way to understand the axioms of $\mathsf{ZFC}$ is to see them as a describing the "universe of sets" $V$, together with the "true membership relation" $\in$. The universe $V$ ...
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Why were evil numbers named "evil"?
According to the entry for Sloane sequence A001969:
Evil numbers: nonnegative integers with an even number of 1's in their binary expansion.
And deeper inside that entry it states that
The terms &...
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Why do the increasing sequences $x_n, y_n$, decrease to $x,y$ to show a bivariate (or univariate) cdf is right continuous?
I have a problem understanding why the concept of right continuity of a cdf has to decrease a sequence $x_n$ or $y_n$ to a limiting value $x$ or $y$ respectively.
I do not understand why in this ...
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Is expression and its result is the same thing?
So, $\frac{1}{2}$ and $0.5$ are just two different ways to address the same object which is rational number $\frac{1}{2}$.
What about more complex expressions? Like $\{a, b\} \cap \{b, c\}$ is just ...
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On series expansions and valid arguments
Hopefully the answer to this question isn't so obvious one way or the other that it ends up just creating more confusion. Briefly, I'm concerned about the potential for subtly illegal moves to creep ...
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Is the deducutive apparatus of a formal system necessarily a set of inference rules?
In the book "Logic" by Paul Tomassi, the author uses the term deductive apparatus to refer to the set of inference rules in propositional logic and first-order logic. The use of this term ...
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Is proof of the law of identity a case of circular reasoning?
I am reading "Logic" by Paul Tomassi. While discussing first-order logic in Chapter $6$ p. $310$, he provides the following justification for the inference rule known as identity ...
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what is the exact meaning of the identity relation? [closed]
I would like some clarification regarding the exact meaning of the identity relation. Specifically, if $a=b$, does this mean
$(1)$ $a$ is the very same object as $b$
or does it leave open the ...
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How do proofs of program termination depend on strength of logical systems?
I'm looking for clarifying insights on the following topic. While there can be no general proof strategy to show that terminating Turing programs do, indeed, terminate, some specific programs can be ...
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On the Consistency of Non-Euclidean Geometry
I've recently found a really old "Philosophy of Math" book in my University library, and in the book it says that the it has been proven that :
if non-euclidean geometry is inconsistent, ...
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Presupposition in logic [closed]
Let S be a presupposition for Q. For example:
S: I used to smoke
Q: I quit smoking
As far as I know, Q has a truth value if and only if S is true. However, my understanding of the concept of ...
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Using set-builder notation in an unusual way for more concise expressions in a definition. Good style?
What do you think about the following usage of the set-builder notation $\{ x \mid P(x) \}$? Let $Y = \{ p_1, \ldots, p_n \}$ be $n$ objects and $A,B$ subsets of $Y$. Define a function from $\{1,\...
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Did Skolem (and others) consider all "legitimate" models to be "actually" countable?
In Thoralf Skolem's Remarks on Axiomatized Set Theory (van Heijenoort translation), Skolem says:
There is no contradiction at all if a set $M$ of the domain $B$ is nondenumerable in the sense of the ...
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Different Proof Systems for the same Logic
Some standard kinds of proof systems include natural deductive systems, sequent systems, axiomatic systems, and so on. There are various approaches to each of these kinds of proof systems. The rules/...
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