Some explanation of the formatting I'll use to post things here. You may want to use the style I'm using, or parts of it, as a model for how you write up your own homeworks.
I will be composing these web pages in a text markup language that's interpreted by John MacFarlane's excellent program Pandoc. Pandoc's language is based on another, older and more broadly used language, called Markdown. There are only small differences between these: places where MacFarlane cleaned up some problematic parts of Markdown or extended it in some ways that are especially useful for academics. The workflow is that you write your documents in Pandoc's markup language (which I'll also call Pandoc), then use the Pandoc program to read it and output it in a different, usually prettier, format. The reason for doing this is that Pandoc is a very lightweight, easy-for-humans-to-read language. For the most part, it looks like a plaintext email. There are only a few minimal ways in which it dictates that some special formatting should be used. So for instance in Pandoc one writes things like this:
... That's what I *used* to think, but then Peter argued: > Suppose $Fa$ *were* a tautology. > Then $\alpha \horseshoe Fa$ would also be a tautology. I had to admit that Peter was right.and then Pandoc will convert that into HTML (used to generate pretty web pages) or LaTeX (used to generate pretty scientific articles). The end result will look like this:
... That's what I used to think, but then Peter argued:Suppose Fa were a tautology. Then α ⊃ Fa wouldI had to admit that Peter was right.
also be a tautology.
Raw HTML and LaTeX are much harder for humans to write by hand and to read the literal code of. Pandoc lets you get many (not all) of the benefits of those formats without needing to write out all of their complicated details.
One bit of this that I'll call attention to is that Pandoc does let you include some snippets of raw LaTeX code (and also snippets of raw HTML code, though I'll be using this much less). In particular, the bit of the above where I wrote $\alpha \horseshoe Fa$ indicates to Pandoc that what comes between the $s is LaTeX math code.
So what's the upshot of all of this for you?
Well, it's true I would encourage you to also look into using Pandoc for your own writing. I find it very useful and I know a number of other philosophers do too. But that is not the main suggestion I want to make here. I'm only telling you about Pandoc to help you understand the whole documents I'll be posting.
The main suggestion I do want to offer you is that you also adopt the practice of writing up your homeworks in plaintext that includes snippets of LaTeX math code. If you know LaTeX and want to, you can write the whole document up in full LaTeX glory and generate a pretty PDF and send me that. But what you can also do---and what I actually prefer---is that you write up the main document in plaintext, like an email, without any special LaTeX headers or formatting instructions, and just at the point where you want to shift from writing in English prose to including a math formula, you write things like $\alpha \horseshoe Fa$. That's what I'll then read, and understand.
For those of you who already know and are comfortable with LaTeX, this should be very easy. For those of you who don't, this will be a low-impact way to learn some useful parts of LaTeX, without getting in over your head with the whole deal. This skill will be useful to you later in your academic career.
You don't have to do this. Another option is for you to compose your homework in whatever manner you're most comfortable with, and lets you include the special math symbols we'll need for this class, and then generate a pretty PDF output and send me that. Perhaps Word's "Insert Symbol..." dialogue will be enough to get by with. If you do this, though, please don't send me raw Word documents, or RTF documents, or anything else of that sort. Convert it to a PDF and send me that. It's more work for me to display or print any other formats.
I prefer just to get plaintext with LaTeX math code in it, and in the long run I think you'll find it a lot easier to write.
As I said, if you do that, I'll be reading the plaintext with my own eyes, and neither I nor you needs to have any computer program actually process it. So it's okay if you don't get things exactly right, as long as I can understand what you meant. But over time, you should ideally converge towards writing things the same way that I do, which LaTeX (or Pandoc) really would accept and turn into the pretty output you intend.
\emph{This} is in regular text mode, but $\alpha \horseshoe Fa$ is a math formula.or:
\emph{This} is in regular text mode, but \(\alpha \horseshoe Fa\) is a math formula.or:
\emph{This} is in regular text mode, but \math{\alpha \horseshoe Fa} is a math formula.or:
\emph{This} is in regular text mode, but $$\alpha \horseshoe Fa$$ is a math formula.or:
\emph{This} is in regular text mode, but \[\alpha \horseshoe Fa\] is a math formula.or:
\emph{This} is in regular text mode, but \displaymath{\alpha \horseshoe Fa} is a math formula.The first three of these are equivalant, and specify that LaTeX should generate this kind of pretty formatting:
This is in regular text mode, but α ⊃ Fa is a math formula.The last three of these are also equivalent, and specify that LaTeX should generate this kind of pretty formatting:
This is in regular text mode, but α ⊃ Fa is a math formula.In my own writing, I will only use the first (and the last?) of these. So the $...$ is the frame, and the LaTeX math code is what comes inside the frame. If you're not writing full LaTeX documents, you don't have to worry about the formatting code LaTeX uses outside of math mode---like the \emph{This} at the beginning.
Inside of math mode, LaTeX ignores all spaces. So all of these mean the same and will generate the same output:
$\alpha \horseshoe Fa$ $\alpha \horseshoe Fa$ $\alpha\horseshoe Fa$However, you can't write:
$\alpha\horseshoeFa$because that will be interpreted as an \alpha symbol followed by a \horseshoeFa symbol, and the latter is probably not defined. You can write this instead:
$\alpha\horseshoe{}Fa$but it's easiest to just include the space, as in the first style:
$\alpha \horseshoe Fa$
Most of the special symbols and formatting you'll want to use, at least for our purposes, will look like the \alpha in the preceding examples. All of the Greek letters can be generated in that way. \phi gives the small-case φ, and \Phi gives the capital Φ. I won't bother to list all of these.
All of the following characters can be entered directly into math mode: ( ) [ ] - + = ! < > @ ; , . ? /. They aren't interpreted in any special way. The output will contain just that literal character.
: can also be entered directly, but there is also a code for it, \colon.
| can also be entered directly, but there is also a code for it, \vert.
* can also be entered directly, but there is also a code for it, \ast. There is also another code \star which gives a slightly different symbol: ★.
` and " can be entered directly, but you're not likely to be using them in a math formula, and they may come out looking different than you expect. To get a prime after a symbol, just use ' like this: x' gives x′, x'' gives x″.
The following characters are interpreted by LaTeX as part of its special formatting language, and if you want to instead get the literal character, you'll have to specify it with a longer code:
\backslash | gives \ |
\$ | gives $ |
\& | gives & |
\% | gives % |
\# | gives # |
\_ | gives _ |
\textcaret | gives ^ |
\texttilde | gives ~ |
The _ code (without a preceding \) is used to generate subscripts, like this: x_2 gives x_{2}, x_{2} also gives x_{2}, and x_{2y} gives x_{2y}. But x_2y gives x_{2}y.
The ^ code is used to generate superscripts, like this: x^2 gives x^{2}, x^{2} also gives x^{2}, and x^{2y} gives x^{2y}. But x^2y gives x^{2}y.
Notably for our purposes, { and } are also interpreted as part of LaTeX's formatting language, and if you want to instead get the literal characters, you'll have to specify \{ and \} (or \lbrace \rbrace). So to get {1, 2, 3} you'd write \{1,2,3\}. But there's also a different way to do this, you could instead write \set{1,2,3}. To get {x∈Γ | x > 0}, you can write \set{x\in\Gamma \where x > 0}.
To write ordered n-tuples, I recommend using the notation (a,b,c). If for some reason you want to use angle brackets, though, then one way to do it would be < a,b,c >. But that will come out a bit ugly. It looks better if you use \langle a,b,c\rangle, which will look like this: ⟨a, b, c⟩.
The funny ⟦brackets⟧ we use you can get with the code \interp{formula}. A slightly different look is produced by \|formula\| (or \lVert formula\rVert).
Here are the main other symbols we might be using.
\le (or \leq) | gives ≤ |
\sim | gives ∼ |
\lesssim | gives ≲ |
\prec | gives ≺ |
\preceq | gives ⪯ |
\precsim | gives ≾ |
\approx | gives ≈ |
\ll | gives ≪ |
\longrightarrow | gives a long ⟶ |
\longleftrightarrow | gives a long ⟷ |
\Longrightarrow (or \implies) | gives a long ⟹ |
\Longleftarrow (or \impliedby) | gives a long ⟸ |
\Longleftrightarrow (or \iff) | gives a long ⟺ |
\times | gives ⨉ (the Cartesian product symbol) |
\to (or \rightarrow) | gives a short ￫ |
\gets (or \leftarrow) | gives a short ￩ |
\circ | gives ⚪ (the composition symbol) |
\mapsto | gives ↦ |
a \cdot b | gives a⋅b |
a \star b | gives a★b |
\in | gives ∈ |
\emptyset (or \varnothing) | gives ∅ |
\se (or \subseteq) | gives ⊆ |
\subset | gives ⊂ |
\intersect (or \cap) | gives ⋂ |
\union (or \cup) | gives ⋃ |
\Delta \setminus \Gamma | gives Δ ∖ Γ (but I just prefer to write \Delta - \Gamma) |
\pow{\Gamma} | gives 2^{Γ} |
\varpow{\Gamma} | gives ℘(Γ) |
\dots (or \ldots) | gives … |
\neg (or \lnot) | gives ¬ |
\vee (or \lor) | gives ∨ |
\wedge (or \land) | gives ∧ |
\cond (or \horseshoe or \supset) | gives ⊃ |
\bicond | gives ⊂⊃ |
\leftrightarrow | gives ↔ |
\equiv | gives ≡ |
\top | gives ⊤ |
\bot (or \perp) | gives ⊥ |
\exists | gives ∃ |
\forall | gives ∀ |
\Diamond | gives ◇ |
\Box (or \square) | gives □ |
x \Would y | gives x □→ y |
\Circle | gives ◯ |
\Determ | gives △ |
\Indeterm | gives ▽ |
\desc (or \imath or \iota) | gives ⍳ |
\models (or \vDash) | gives ⊨ |
\proves (or \vdash) | gives ⊢ |
\bimodels | gives ⫤⊨ |
\biproves | gives ⊣⊢ |
\forces (or \Vdash) | gives ⊩ |
\N | gives ℕ |
\Z | gives ℤ |
\Q | gives ℚ |
\R | gives ℝ |
\aleph | gives א |
\beth | gives ב |
x \curl y | gives x⌢y |
\corner{\alpha is alert} | gives ⌜α is alert⌝ |
When multiple synonyms are listed, I've listed the choice I prefer first.
Many of the relation symbols can be negated by preceding them with a \not. So for instance, \not\leq will give ≰. For some but not all of these symbols, something like \nleq will also work. But since this doesn't work 100% of the time, I find it easiest to just use the preceding \not.
Some of the LaTeX codes above appear in boldface. What that means is that they aren't official LaTeX instructions, but rather shortcuts that I wrote myself, often just to have more memorable names for a philosopher. If you're just writing plaintext with embedded LaTeX math code for me to read, you can just go ahead and use those same shortcuts. However, if you're actually trying to get a LaTeX program to generate a PDF using these codes, then you'll need to download my shortcuts and include them in your LaTeX document. The shortcuts are available here.
The preceding lists are not at all exhaustive. If you're looking for more symbols, here are some sources you may find useful:
LaTeX is pretty good about figuring out where spaces are needed to make the output look right, but it can't read your mind, and sometimes one needs to tweak it with special instructions to add a bit of extra space (or less frequently, to use less spacing). If you're not really generating pretty PDF outputs, but are just writing raw LaTeX math code for me to read with my human eyes, you don't need to bother with that. I'll just list some of the spacing commands anyway, though, so that if you're ever reading other people's LaTeX code and come across them, you realize what they're doing.
a \, b | adds a little bit of extra space between the a and the b |
a \: b | adds a bit more space |
a \; b | adds a bit more space |
a \ b (that is, a \ character followed by a space) | adds a bit more space |
a \quad b | adds a medium amount of space, three times as much as the preceding |
You may have noticed that lower case letters come out italicized in pretty-formatted LaTeX mode code. LaTeX does this automatically, and it usually looks nice. As I mentioned before, LaTeX also ignores most spacing in math code. It uses its own judgment (or more verbose formatting codes you give it) to decide how things should be spaced. But what this means is that if you suddenly want to lapse into English prose in the middle of a LaTeX math snippet, the results would not come out looking very good. So for example, this LaTeX code: the result was $x^2+1, rather than x^2$ as I expected would come out looking like this:
the result was x^{2}+1, ratherthanx^{2} as I expectedAnd that's probably not what you wanted. To correct this, you could turn off the LaTeX math code, shift into English prose, and then turn on the LaTeX math code again when you're finished, like so: the result was $x^2+1$, rather than $x^2$ as I expected. In this example that is the most natural fix. In some cases, though, it might be easier to use a different technique. Without closing the math code block, you can temporarily "escape" out into regular text mode using the instruction \text{...}. So in our example, we'd instead do: the result was $x^2+1 \text{, rather than } x^2$ as I expected. Either of these would give us the result we were probably after:
the result was x^{2}+1, rather than x^{2} as I expected
Note also that I will be updating the web pages regularly. The bottom of each page will have a timestamp saying when it was last updated. But in general I won't put any notice on the front page saying that something has been updated. If I were to do that, those notices would be displayed almost permanently. So you should get in the habit of checking pages that you've read a couple of days later, to see if there have been any substantial changes. Much of the time, the updates I'll be making will just be formatting fixes and typos. But sometimes they will be meatier.