I have an argument with the conclusion: 'The sky is blue and it is not the case that the sky is blue'.  Without knowing the premises of this argument, am I right in saying this argument is invalid since the this conclusion is necessarily false?
I've recently been trying to introduce myself to formal logic, and I've hit a stumbling point:  In the text I'm reading, a valid argument is defined as "an argument in which it is impossible for the premises to be true and at the same time the conclusion false."  The text then later prompts me to declare whether it's possible for an argument to be valid with a contradiction for a conclusion. The answer key then claims that yes, it is in fact possible, but offers no further explanation. I'm also supposed to create an example.  I just can't wrap my head around this one.  Here's the example I've worked up, is this a valid argument?   If grass is green then it is raining.  If water is wet then it is not raining.  Therefore: It is raining and not raining.   My head hurts.
How do I overload the square-bracket operator in C#? DataGridView, for example, lets you do this:  DataGridView dgv = ...; DataGridViewCell cell = dgv[1,5];   but for the life of me I can't find the documentation on the index/square-bracket operator.  What do they call it?  Where is it implemented?  Can it throw?  How can I do the same thing in my own classes?  ETA: Thanks for all the quick answers.  Briefly: the relevant documentation is under the "Item" property; the way to overload is by declaring a property like public object this[int x, int y]{ get{...}; set{...} }; the indexer for DataGridView does not throw, at least according to the documentation.  It doesn't mention what happens if you supply invalid coordinates.  ETA Again: OK, even though the documentation makes no mention of it (naughty Microsoft!), it turns out that the indexer for DataGridView will in fact throw an ArgumentOutOfRangeException if you supply it with invalid coordinates.  Fair warning. A "non-believer" of C# was asking me what the purpose to extension methods was. I explained that you could then add new methods to objects that were already defined, especially when you don't own/control the source to the original object.  He brought up "Why not just add a method to your own class?" We've been going round and round (in a good way). My general response is that it is another tool in the toolbelt, and his response is it is a useless waste of a tool... but I thought I'd get a more "enlightened" answer.  What are some scenarios that you've used extension methods that you couldn't have (or shouldn't have) used a method added on to your own class? How to make a modification take affect without restarting nginx? Apache has a graceful option which can scan for modification in http.conf without restarting Apache. What about nginx? This is an exercise from textbook Analysis I by Amann/Escher.  Let $G$ be a group and $X$ a nonempty set. Then $G$ acts (from the left) on $X$ if there is a function $$G \times X \rightarrow X, \quad(g, x) \mapsto g \cdot x$$ such that: GA1: $e \cdot x=x$ for all $x \in X$. GA2: $g \cdot(h \cdot x)=(g h) \cdot x$ for all $g, h \in G$ and $x \in X$. Let $\mathrm{S}_{m}$ be the set of all permutations of $\{1,2,\cdots,m\}$. Then $(\mathrm{S}_{m},\circ)$ is a group where $\circ$ is function composition operation. Show that $$\mathrm{S}_{m} \times \mathbb{N}^{m} \rightarrow \mathbb{N}^{m}, \quad(\sigma, \alpha) \mapsto \sigma \cdot \alpha := \left(\alpha_{\sigma(1)}, \ldots, \alpha_{\sigma(m)}\right)$$ defines an action of $\mathrm{S}_{m}$ on $\mathbb{N}^{m}$.  I am stuck at proving the GA2, i.e. $\pi \cdot (\sigma \cdot \alpha) = (\pi \circ \sigma) \cdot \alpha$.  My attempt: Clearly, $\operatorname{id}$ is the identity element of $\mathrm{S}_{m}$ and $\sigma \cdot \alpha = \alpha \circ \sigma$. Because $\operatorname{id} \cdot \alpha = \left(\alpha_{\operatorname{id}(1)}, \ldots, \alpha_{\operatorname{id}(m)}\right) = \left(\alpha_1,\cdots,\alpha_m\right) = \alpha$, GA1 follows. For $\pi,\sigma \in \mathrm{S}_{m}$ and $\alpha \in \mathbb{N}^{m}$, we have $$\pi \cdot (\sigma \cdot \alpha) = (\alpha \circ \sigma) \circ \pi = \alpha \circ (\sigma \circ \pi) = (\sigma \circ \pi) \cdot \alpha$$  It is clear that $\sigma \circ \pi = \pi \circ \sigma$ if and only if $(\mathrm{S}_{m},\circ)$ is Abelian, which is not true in general. Please help me figure out where I am wrong! In Star Wars: "The Force Awakens," how could this lightsaber have turned up where it did? In The Force Awakens:      Maz Kanata tells Rey that the lightsaber she just picked up and had visions from belonged to Anakin and Luke and now it's hers.   The lightsaber that Luke got from Obi-Wan Kenobi in A New Hope was lost/destroyed when Luke fought Darth Vader in Star Wars: The Empire Strikes Back.  In Return of the Jedi Darth Vader compliments Luke on building a new lightsaber and tells Luke that his skills are now complete.   How could that possibly be the lightsaber that Anakin used originally, and that was then passed on to Luke, when that lightsaber no longer exists?  Writers notes, director's notes, interviews, cast interviews, etc. are welcome. Thanks. I frequently need to check which packages are installed, and I use the following command:  dpkg -l | grep foo  which gives the following output  Desired=Unknown/Install/Remove/Purge/Hold | Status=Not/Inst/Cfg-files/Unpacked/Failed-cfg/Half-inst/trig-aWait/Trig-pend |/ Err?=(none)/Reinst-required (Status,Err: uppercase=bad) ||/ Name                       Version         Description ii   foo                       &lt;version&gt;         &lt;description&gt;    What does the ii mean?  What other flags are there? How to read the flags? (because the explanation is quite complicated, IMO)   Thanks. Started new job. Rollover previous employer 401k to new 401k, IRA or Roth IRA? I recently started a new job and have a 401k from my previous employer that I don't know what to do with.  Should I roll it into my new employer's 401k, a traditional IRA or a Roth IRA? I'm on IBM AIX 7.1 where cURL and wget are not available. Also, I'm not in a position to install either.  Are there alternatives that would be available that would enable remote API calls using GET/POST? Is password confirmation necessary on a Change Password dialog? On one hand there is no need to ask for password confirmation because it is annoying. If I made a mistake when confirming my password I'll need retype my very complex password again. There is also a button that can show/hide the password so I don't need retype it in order to confirm it.   However, some users don't use the password show/hide button and the 'forgotten password' option will be pain for them.  Twitter removed the password confirmation from registration but left the dual confirmation in place for changing password. Is this right approach? How to cite sources written in a different language than English? Assume one wants to cite a paper written in a different language than the article citing said paper (e.g., ones own article is written in English, but the source is French).   How to properly handle citation in such a case?  Do I cite the original (French) title or a translated title? If I use a translated title, people might have a hard time to find the source; if the original (French) title is cited, people might have a hard time understanding it. Layout selector missing tags of mundanity that are gratuitously applied to posts It's an additional unnecessary burden to the honest stacker during suggested edit duty. A question presents some code then the minor editor adds a tag like 'for-loop' or 'forms'.  These tags are so general that they only have a handful of followers. They serve no useful purpose and are the cause of many too minor edits. Why not permit culling of generic tags? Retrieve globalpicklist/globalvalueset without SOQL Is it possible to retrieve the values of a Global Picklist/GlobalValueSet using Schema class without a SOQL ? Do you have to want to hurt someone for the cruciatus curse to work? Do I have to do something "special" to get the Color -> Color to alpha menu option to work in GIMP? I am trying to convert a background color on a GIF image to transparent by converting the background color to an alpha channel but the Color to Alpha menu option is not selectable.  I've gone through the GIMP documentation and it looks pretty simple :-) I must be missing something rather obvious here! Why is the area under a curve the integral? Getting Started With Java EE I am about to start a web project and have been working almost exclusively with a LAMP stack for the past 5 years.  I would like to build something larger and more scalable and hence have been looking into Java EE.  The division of logic between JSP and EJB seems ideal for what I want to do.  Can anyone recommend resources for getting familiar with Java EE?  And perhaps if anyone knows of technologies that are more scalable than PHP that I should consider as well, I'm all ears.  Thanks a lot! This is an attempt to write a canonical QA for this issue, as per the Meta post:   I expect it to be periodically edited with the goal of becoming a comprehensive information resource.  I have a MacBook Pro that has a key(Shift, Return,   Command, P, etc.) that doesn’t respond when pressed or is stuck in the down position.    It's possible that my favorite beverage was spilled (or some other liquid damage).  It's also possible that I have no idea what happened.  It was working yesterday and suddenly today it's no longer working.  Is there any way to bypass or remap the key?  Can I  have the system ignore it completely?  (This question can also apply to external USB keyboards or the Bluetooth keyboards) Getting "Not found" message when running a 32-bit binary on a 64-bit system Finding a permutation, and number of, from powers of the permutation