I am confused on the usage of the word "interpretation" and/or "model" when it comes to propositional logic versus first-order logic because there are so many conflicting / unclear notions that I would like to clear up.  My current understanding is as follows and I would like any corrections / elaborations on where I am confused:  I am using the definition of a "model" being an interpretation that satisfies a formula / set of formulas. In the case of a theory we can treat its axioms as a set of formulas, and so a "model of a theory" meaning any interpretation that satisfies the set of axioms of that theory.   In propositional logic an "interpretation" is an arbitrary assignment of true/false values to all the atomic propositions in the alpha set. For example $p_0 = T, p_1 = F, p_2 = T, p_3 = F, ...$ and so on. This tells us which row of any given truth table we should look at when evaluating the "truth value" of any fixed proposition.   But then in first-order logic, it seems like an "interpretation" is no longer some specific assignment of values to non-logical terms, but rather entire number systems like "the natural numbers," which would also be a "model" of, for instance, peano arithmetic, which satisfies its axioms.  Why is this? Why wouldn't we say "boolean variables" model propositional logic then? Why wouldn't we say some specific assignment of values satisfy first-order logic / PA / etc?   Why is interpretation seemingly being used differently in both cases? If an interpretation is what we'd call the specific T/F assignments in propositional logic, then what do we call the choice of a boolean system in the first place?  And more of a side question, but then what of propositional logic systems like natural deduction which have no axioms at all? What models "satisfy" it if there's no set of axioms to represent the theory?
I am trying to understand model vs. interpretation of sentences vs. theories, with respect to a propositional calculus.  Is a model of a wff just a sentence that is true under some interpretation?   Or a set of wffs that are all true under some interpretation?   Or when an interpretation satisfies a wff (or set of wffs)?   Does it make sense to say that an interpretation is a model if all valid propositions are true? Or an interpretation being a model iff all theorems from the theory are still valid under the interpretation?
Intuition of pdf of a continuous random variable What is the intuition behind the probability density function of a continuous random variable? Integrating it within two points provides the probability that is associated between two points, but if you plug a single value into $f(X=x)$, it outputs a value. What is the intuitive understanding of this value for a given x?   For example, in the above image, the value of the red curve at x=0 is 0.4. How can 0.4 be interpreted? I see in my company's IT installations that there are numerous disks with one partition occupying all of the space on disk. There is typically something like a /dev/sdb with a partition /dev/sdb1, and the size of /dev/sdb1 is the size of the whole block device. The partition is then mounted with the required file-system format, mountpoint etc.  I do not understand why we would use a partition rather than the whole disk for such (1-partition) configurations. Is there any reason why a partition would be used in this case rather than a whole disk ? Are there any best practices that recommend this approach ?  For example, we could create a Filesystem on a whole raw block device and mount it without needing any partition:  mkfs.ext4 -E stride=16,stripe-width=64 /dev/xvde mount /dev/xvde /mnt/abc   That creates a mountpoint without a specified partition. As a verification:  # sfdisk -l /dev/xvde  Disk /dev/xvde: 6527 cylinders, 255 heads, 63 sectors/track  # fdisk -l /dev/xvde  Disk /dev/xvde: 53.7 GB, 53687091200 bytes, 104857600 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes   For something with a partition, we get the list of partitions under it:  # fdisk -l /dev/sdb  Disk /dev/sdb: 2000.4 GB, 2000398934016 bytes, 3907029168 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk label type: dos Disk identifier: 0x00082e8c     Device Boot      Start         End      Blocks   Id  System /dev/sdb1            2048  3907026943  1953512448   83  Linux   My fundamental question is this: Is there some advantage in creating a partition and then creating a filesystem on that partition, as compared to creating a filesystem on the whole disk and then mounting the whole disk ? I am running Ubuntu 14.04 on think-pad T450.  When using an external monitor with higher resolution than the in-built laptop screen, the text of dialog boxes are getting messed up.  Looks like I need to replace the Graphic card driver. Anyone found a solution to this?  Updated to add requested information:  lspci -nnk | grep '\[03' -A2   gives the following:  00:02.0 VGA compatible controller [0300]: Intel Corporation Broadwell-U Integrated Graphics [8086:1616] (rev 09)     Subsystem: Lenovo Device [17aa:5034] Kernel driver in use: i915   Below is a screenshot, see how the letters are messed up on the screen. Don't remember the exact start to the story, but it went something like "walking along a path" or something, and when he hears what sounds like a shotgun blast behind him, he then feels something strike him in the butt. He passes out, and when he awakens, he discovers that the "shotgun blast" was actually a miniature spaceship that exploded and shot miniature aliens into his body. The aliens prove to be very beneficial because they clean up all the toxins and other poisons in his body. They can be transmitted through kissing or sexual activity.  Any help with the name of the story would be greatly appreciated.  Dan MongoDB could not connect local server I had mongdb installed and evrything was working fine. I wanted to delete the mongodb lock. By mistake the directory /var/lib/mongodb/ got deleted.  So, I uninstalled mongo db and installed again. But now i always keep getting this error:  Error: couldn't connect to server 127.0.0.1:27017 at src/mongo/shell/mongo.js:145 exception: connect fail   The error remains same even if I try to login from terminal or rockmongo or my application. I am using ubuntu12.04 and have latest version of mongo. Consider a block cipher with a key of size 128 bits but a small block size, say 32 bits.  Is this kind of block cipher secure?  I would like to know to what extent I can use a small block cipher, like a 32 bit block cipher. What are the constraints so that the use is acceptable in a real life application.  For instance, if we use a one-time key, that is, to encrypt only one message. Besides, the message could be small.  Suppose that when we want so send one file, we perform a key agreement so that we use the key for one message. This scenario is not efficient, but the CPA find-then-guess game is not adequate in this case. Related:   I know I must describe my past and current research in my personal/research statement or statement of purpose. But do I use technical terms?  I was given the advice that since the people reading my statement are mathematicians (I am applying to US for applied mathematics), they should know what I am talking about.  Although I am applying to applied mathematics, it would also be nice to hear from other fields (biology, computer science, engineering etc) about this. "In" and "on": How can I decide which one to use for vehicles? Examples:   In a car, van, etc. On a bus, boat, motorcycle, etc.   How can one decide which preposition to use? Is memorization the only way or is there a better way?  Note: People generally explain this by either distinguishing between open and closed vehicles or between large and small vehicles. However, the examples I've given defeat both explanations. In the following code I loop through a map and test if an element needs to be erased.  Is it safe to erase the element and keep iterating or do I need to collect the keys in another container and do a second loop to call the erase()?  map&lt;string, SerialdMsg::SerialFunction_t&gt;::iterator pm_it; for (pm_it = port_map.begin(); pm_it != port_map.end(); pm_it++) {     if (pm_it-&gt;second == delete_this_id) {         port_map.erase(pm_it-&gt;first);     } }   UPDATE: Of course, I then  which I didn't think would be related but answers my question. Intuition/Real-life Examples: Pairwise Independence vs (Mutual) Independence Would someone please advance/discuss some real-life situations falsities $1, 2$? I'd like to intuit why these are false. As a neophyte, since I still need to compute the probabilities for the examples in the two answers to apprehend them, I haven't naturalised these concepts yet. Thus, I beg leave to ask about other real-life examples which require less or no computations.  I tried and would appreciate less numerical examples than  and , and  Examples 1.22 and 1.23 on P39 of Introduction to Pr by Bertsekas.  $1.$ Pairwise Independence $\require{cancel} \cancel{\implies}$ (Mutual) Independence.      $2.$ Pairwise Independence $\require{cancel} \cancel{\Longleftarrow}$ (Mutual) Independence.         P38 defines (Mutual) Independence :    For all $S \subseteq \{1, ..., n - 1, n\} $ and events $A_i$, $Pr(\cap_{i \in S} A_i) = \Pi_{i \in S} A_i.$ Setting CSS pseudo-class rules from JavaScript I'm looking for a way to change the CSS rules for pseudo-class selectors (such as :link, :hover, etc.) from JavaScript.  So an analogue of the CSS code: a:hover { color: red } in JS.  I couldn't find the answer anywhere else; if anyone knows that this is something browsers do not support, that would be a helpful result as well. Defining scalable "white curly brackets" {| and |} (⦃ and ⦄) I would like to construct what Unicode calls "white curly brackets" that are in addition scalable. They look like ⦃ (≈{|) and ⦄ (≈|}) and have Unicode codepoints U+2983 and U+2984.  unicode-math has them as \lBrace and \rBrace, but (1) I don't presently use XeTeX or LuaTeX and (2) their design (shown in  on pages 3-4) is not to my liking: the brace part is too fat for my taste, and I would like the vertical bar to not leave any protrusions. I am trying to have them visually match the parentheses \llparenthesis, \rrparenthesis, \llbracket, and \rrbracket from stmaryrd.  Right now I am using a makeshift definition, which is sufficient for now, but the symbols don't scale (in general or with \left/\right). Here is sample code:  \documentclass{article} \usepackage{amsmath} \usepackage[only,llbracket,rrbracket,llparenthesis,rrparenthesis]{stmaryrd} % for comparison with 4 similar parenthesis symbols \usepackage{accsupp} % for ensuring the right Unicode codepoint upon pasting  \newcommand*{\llbrace}{%   \BeginAccSupp{method=hex,unicode,ActualText=2983}%     \textnormal{\usefont{OMS}{lmr}{m}{n}\char102}%     \mathchoice{\mkern-4.05mu}{\mkern-4.05mu}{\mkern-4.3mu}{\mkern-4.8mu}%     \textnormal{\usefont{OMS}{lmr}{m}{n}\char106}%   \EndAccSupp{}% } \newcommand*{\rrbrace}{%   \BeginAccSupp{method=hex,unicode,ActualText=2984}%     \textnormal{\usefont{OMS}{lmr}{m}{n}\char106}%     \mathchoice{\mkern-4.05mu}{\mkern-4.05mu}{\mkern-4.3mu}{\mkern-4.8mu}%     \textnormal{\usefont{OMS}{lmr}{m}{n}\char103}%   \EndAccSupp{}% }   \begin{document}  \noindent ok: \(\displaystyle\llbrace abc \rrbrace\) \(\textstyle\llbrace abc \rrbrace\) \(\scriptstyle\llbrace abc \rrbrace\) \(\scriptscriptstyle\llbrace abc \rrbrace\)  \noindent ok: \(\displaystyle \llbrace \frac{a}{b} \rrbrace\) \(\textstyle\llbrace \frac{a}{b} \rrbrace\) \(\scriptstyle\llbrace \frac{a}{b} \rrbrace\) \(\scriptscriptstyle\llbrace \frac{a}{b} \rrbrace\)  \noindent incorrectly scaled (\verb|\scriptsize|): {\scriptsize\(\llbrace abc \rrbrace\)} (large magnification required)  \noindent incorrectly scaled (\verb|\tiny|): {\tiny\(\llbrace abc \rrbrace\)}  \noindent \verb|\left| and \verb|\right| not possible before \verb|\llbrace| and \verb|\rrbrace|: \\ \indent \(\llbrace \left\llbracket\left\{\left[(a) \cdot bc^{2^3}\right]^4\right\}^5\right\rrbracket \rrbrace\) % Note: This is also not possible with \llparenthesis and \rrparenthesis from stmaryrd.  \noindent for comparison: \(\llparenthesis x \rrparenthesis, \llbracket x \rrbracket, \llbrace x \rrbrace; (x), [x], \{x\}\)  \end{document}   Lines 1 and 2 look just right with the present setup. Lines 3 and 4 demonstrate that my definitions don't scale. Lines 5a/b gives an example of where I would like to be able to use \left and \right. Line 6 juxtaposes different types of parentheses for visual comparison.  Is there an easy way to scale the glyphs I have constructed myself? Or is designing new glyphs the only way? While the vertical part of my glyphs doesn't leave any protrusions, the "brace part being too fat" part of my problem isn't being addressed by my code. With this in mind: I recognize that it might not be that easy to make the brace part of the glyph thinner (though if you can, more power to you), but I think that any solution that addresses the scaling problem for any manually constructed glyph cluster will be a valuable contribution to the community.    For a slightly different setup (original example code) with  \RequirePackage{fix-cm} \documentclass[12pt]{memoir} \usepackage{fixltx2e}[2006/09/13] \usepackage{amsmath} \usepackage{txfonts} \usepackage[only,llbracket,rrbracket,llparenthesis,rrparenthesis]{stmaryrd} \usepackage{accsupp}   in the preamble, one needs \mkern-5.7mu in the fourth argument to \mathchoice for "correct" scaling. To see what goes wrong in that case, it's best to include a line such as discontinuous small symbols in a footnote\footnote{\(\scriptscriptstyle\llbrace \frac{a}{b} \rrbrace\)}. This might help a potential answerer test things out. Windows 7 Audio Too Loud In Windows 7 (64-bit, Ultimate) the default audio output is extremely high. This has been an issue for a while but I was able to get around it as my previous headphones took more power to drive so it actually worked in my benefit.  However, now I have a set of really sensitive headphones and they're being over-powered. So much so that if I put it down at around 5% of total output it's bearable.  Is there a way I can sort of limit the system's volume so that it's incapable of going extremely loud and then have the rest of it just behave as normal?  Things I've tried:   Uninstalling and re-installing the audio device and drivers. Limiting each application individually. Searching for third-party software (so far nothing looks promising).   The first has no effect. The second works find but I have to do it for every application individually (which can be a pain) and many of them reset their volume every time they're launched, which makes it almost useless. Some even reset the volume while I'm playing a game!  So does anyone have any suggestions on how to make my volume reasonable?  The device is, apparently, a Cirrus Logic CS4206A (AB 24). The computer is an 2010 iMac. I try to import .obj files from this project:  - the files contain Polygon Mesh Data.  When I import the obj file into blender, the outliner shows a new mesh added, but nothing is visible in the 3D viewport:      Any help will be appreciated. Convert from MySQL datetime to another format with PHP What are our policies regarding self-harm and suicide? What should I do as a Moderator when I encounter situations regarding self-harm and suicide? C# Textreader and TextWriter are the another way to read and write file respectively...      I added a text box and button I called tbUpdate and btUpdate respectively...      In the right hand column, we’ll use three TextBlocks; the first and third as fixed labels (with the values Likeability: and % respectively)...      WorkerSupportsProgress and WorkerSupportsCancellation which hold true/false values that let the backgroundworker report progress or cancel an async progress respectively...   While learning many the many different aspects of development, I have not come across a book or series where the word respectively is not used in explaining an example. Honestly I guess it's implied that we know what it means. To me I always kind of skim over it because I can usually figure out what they're saying through the surrounding context. Can anyone explain this? If Durmstrang and Beauxbatons were hidden from common knowledge, how did people from other schools go to them for the Triwizard tournament? So in GoF, they state that Beauxbatons and Durmstrang are hidden from view and don't have their locations public.     “Durmstrang’s another wizarding school?” said Harry      “Yes,” said Hermione sniffily, “and it’s got a horrible reputation.      According to An Appraisal of Magical Education in Europe, it puts a   lot of emphasis on the Dark Arts.”      “I think I’ve heard of it,” said Ron vaguely. “Where is it? What   country?”      “Well, nobody knows, do they?” said Hermione, raising her   eyebrows.      “Er — why not?” said Harry.      “There’s traditionally been a lot of rivalry between all the magic   schools. Durmstrang and Beauxbatons like to conceal their whereabouts   so nobody can steal their secrets,” said Hermione matter-offactly.   So how do students get to the other school for the Triwizard tournament? I have this MySQL query:  SELECT DAYOFYEAR(`date`)  AS d, COUNT(*)  FROM  `orders`  WHERE  `hasPaid` &gt; 0 GROUP  BY d ORDER  BY d   Which returns something like this:  d  | COUNT(*) | 20 |  5       | 21 |  7       | 22 | 12       | 23 |  4       |   What I'd really like is another column on the end to show the running total:  d  | COUNT(*) | ??? | 20 |  5       |   5 | 21 |  7       |  12 | 22 | 12       |  24 | 23 |  4       |  28 |   Is this possible? Prevent back button after logout