When talking with a mathematics teacher the other day, we discussed these axioms in the context of proving tautologies with semantic tableaux:   $(p\to(q\to p))$ $((p\to(q\to r))\to((p\to q)\to(p\to r)))$ $((\lnot p\to\lnot q)\to(q\to p))$   We wondered if it was possible to show that $\neg(\neg A) \rightarrow A$ from these axioms (along with MP), but it did not seem obvious how to do so and yet this fact is not included as an axiom. Is it provable from the axioms or is this fact simply unprovable in this version of PL?
I'm very new to classical propositional logic, and my lecturer is using a system with the following axioms:  A1. X→(Y→X)  A2. (X→(Y→Z))→((X→Y)→(X→Z))  A3. (¬Y→¬X)→(X→Y)  Use uniform substitution and Modes Ponens (MP) as rule of inference. Deduction Theorem also allowed.  I'd like to deduce the following:  (a) X, Y ⊢ ¬(X→(¬Y)) (conjunction, but expressed in terms of NOT and IMPLIES)  (b) X→((X→Y)→Y)  (c) (X→Y)→(((¬Y)→X)→Y)  (d) (((¬Y)→X)→Y)→(X→Y)  Remark: The last two are to show that the axiom (A3) can be replaced with a different axiom:  B3. (¬Y→¬X)→(((¬Y)→X)→Y).  Solution: Bram28 proves (b) and (c) below. I demonstrate (a) and (d).  (d) is alright; first, I claim that X, ((¬Y→X)→Y) ⊢ Y.  Axiom (A1) gives ⊢X→(¬Y→X) so that ((¬Y→X)→Y)⊢X→Y by Hypothetical Syllogism (HS), which Bram28 shows to be valid.  Now that we have ((¬Y→X)→Y)⊢X→Y just apply the deduction theorem to get what we want: ⊢((¬Y→X)→Y)→(X→Y).  (a) is slightly trickier. Axiom (A1) gives ⊢ Y→(X→Y) so that X, Y ⊢ (X→Y).  Next, by part (b), we have as a theorem that ⊢X→((X→¬Y)→¬Y).  Therefore X, Y ⊢(X→¬Y)→¬Y.  But ⊢(¬¬(X→¬Y))→(X→¬Y) by DN Elim, which Bram28 shows to be valid.  By HS, X, Y ⊢(¬¬(X→¬Y))→¬Y.  By Axiom (A3) and MP, X, Y ⊢(Y→¬(X→¬Y)).  Also, rather trivially X, Y ⊢Y.  Finally by MP we conclude X, Y ⊢¬(X→¬Y) as required.
the_excerpt not displaying Was voldemort's body going to die? Voldemort has gained "immortality" via his horcruxes, but was his body immortal to age? Looking at all the questions about Voldemort and his horcruxes what was his plan to keep his body alive. We know his first Horcux was created at age 16, yet we see him age still so it seems like eventually his body would die of old age and his soul would be left in the half dead state he was in after his spell rebounded off Harry. What was his plan here, why would he not want something such as the elixir of life to keep his vitality?     While the Elixir of Life does indeed extend life, it must be drunk regularly, for all eternity, if the drinker is to maintain his immortality. Therefore, Voldemort would be entirely dependent on the Elixir, and if it ran out, or was contaminated, or if the Stone was stolen, he would die just like any other man. Thereafter, I am convinced, he intended to continue to rely on his Horcruxes: he would need nothing more, if only he could regain a human form. He was already immortal, you see ... or as close to immortal as any man can be.' (Albus Dumbledore)   Horcruxes seem to fail at something important, keeping you alive, they stop you from dying truly yes, but they don't keep you alive. Harry is shown over and over again to be able to beat better wizards because hes has faster reflexes both because of youth and his genetics, but presumably by 50 harry will not be as fast as his 16 year old self even considering their longer life expectancy. Why operate in a body of a 50 year old? I am trying to draw the below given figure. Please help me to plot it in Tikz. I am only able to plot concentric circles. . I would like to have different colors in between different circles. Also, the text should be along the curve. The arrow in between the two concentric circles does not have to overlap with the circle on the right hand side.  I would be grateful if you could help me in plotting this in Tikz. Please let me know if you need more information. Thanks!  \documentclass[conference]{IEEEtran}  \ifCLASSINFOpdf  \else  \fi  \usepackage[usenames,dvipsnames]{color} \usepackage[T1]{fontenc} \usepackage{tikz} \usetikzlibrary{positioning} \usetikzlibrary{arrows} \usetikzlibrary{shapes.multipart} \usepackage[caption=false]{subfig} \usepackage{tabularx,tikz} \usepackage{graphicx} \usepackage{color} \usepackage{pdfpages} \begin{document}  \title{Practice Tex}   \maketitle  \section{Introduction}  \begin{figure} \begin{center} \begin{tikzpicture}     \draw(0,0) circle (0.4cm)   node[draw=none,sloped,above]{cxvzxcv};         \draw(0,0) circle (0.75cm) node[draw=none,sloped,above]{ratio};    \draw(0,0) circle (1.25cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,2.2) {22};    \draw(0,0) circle (2cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,3.8) {33333};    \draw(0,-9) circle (0.4cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-9) {fsgsdgf};      \draw(0,-9) circle (0.75cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-7.8) {fgdsfg};    \draw(0,-9) circle (1.25cm) node[midway,sloped] at (0,-9) {dfsdfad};    \draw(0,-9) circle (2.8cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-6.4) {dfsdfsd};    \draw(0,-9) circle (3.2cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-6) {dfsdfsd};    \draw(0,-9) circle (3.6cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-5.6) {sfdsfd};    \draw(0,-9) circle (4.4cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-4.8) {sdfsf};     \pgfsetlinewidth{3pt}    \draw [-implies][-&gt;] (0,-4) -- (0,-4.6);   \end{tikzpicture}  \end{center} \end{figure} \end{document} Since I upgraded to Windows 10, my system has been consuming RAM excessively    I've been reading a bit and determined it's likely a driver leaking memory. So I got myself the Windows Driver Kit and tracked memory usage with poolmon:    However, I don't really know how to proceed from here. Is the item tagged "smNp" the culprit in this issue? How do I go from there to actually identifying the driver?  I tried some stuff like "C:\Windows\System32\drivers>findstr /s smnp ." but it returned no results. I also took a look at the pooltag.txt file and this is the description I found for it:      So yeah, any help would be appreciated. Thanks in advance. Passing data between two controllers using storyboards Why are my fields initialized to null or to the default value of zero when I've declared and initialized them in my class' constructor? Prove $$\int _0^\infty f^2(x) dx \leq \frac{2}{3}\cdot  \max_{x \in \mathbb R^+} f(x) \cdot  \int _0^\infty f(x) dx$$ for $f(x) \geq 0$ and convex.   I know via Holder's we can get without the constant $\frac{2}{3}$ but that should also be true for $f$ not convex. 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However, on  I found this:      The word "it", however, has an extremely impersonal connotation, even offensive, in common usage and is rarely used in English to refer to an unspecified human being or person of unknown gender. This is because the word "it" connotes that the person being specified is inferior to a person or is an object.   Is to appropriate to refer to a person of unknown sex as it?  Should I rephrase my sentence as follows:     The user defines two variables, x and y. The user then multiplies each variable by a prime number. Get selected element's outer HTML Subject–verb agreement not clear in "Is two bars OK?" Is the following correct?     Can I get some chocolate for myself? Yes, you can but not much.   Is two bars OK?   Is there any rule to explain the sentence above?  The dialogue's taken from Grammar Textbook Round-Up Level #3. Delete all files except specified files/folders using command line? I have a folder containing a lot of folders and different files with the following structures :  NASA │ ├── a │   ├── doc1 │   ├── doc2 │   ├── doc3 │   ├── folder1 │   └── folder2 │ ├── b │   ├── doc1 │   ├── doc2 │   ├── doc3 │   ├── folder1 │   └── folder2 │ ├── c │   ├── doc1 │   ├── doc2 │   ├── doc3 │   ├── folder1 │   └── folder2 │ ├─ x ├─ y └─ z          I want to delete the content of the folder (NASA/) except specified folders and files.   For example I want to keep a folder, b folder and x file.     I tried   :   rm !(a/) -r NASA/   And (as explained in the answer ):  find NASA/ -type f ! -iname "x" -delete   But this is not very straight forward and I have to use a bash script.    Am I missing a more easy way ? How can I do this in a single command? Number of permutations of $n$ elements where no number $i$ is in position $i$ How does $e^{i x}$ produce rotation around the imaginary unit circle? Euler’s formula states that $e^{i  x} = \cos(x) + i \sin(x)$.  I can see from the MacLaurin Expansion that this is indeed true; however, I don’t intuitively understand how raising $e$ to the power of $ix$ produces rotation.  Can anyone give me an intuitive understanding? How to implement Enums in Ruby? Is it a good practice to show a sign-up or registration form as seperate parts rather than in one go? I am designing a registration form for a site with multiple sets of information required from the user. I have categorized the data fields into relevant groups. Now should I present the form sections in one go or follow a wizard-like approach? What things should I keep in mind while deciding on this? I have a simple skull model, to which I applied the subdivision modifier. For the most part everything works as expected, except I keep getting these weird sharp vertices and I can't seem to figure out how to flatten them.    I'm using Blender 2.8. Is this some kind of feature I can disable/enable for particular vertices? 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