lt;br> - $\emptyset$ | # Logic ## Equivalence The below operations are often used interchangeably for a variety of reasons and in different contexts. The guidelines below govern how I would default to reading and using them. These all indicate the *if-and-only-if* operation which is the statement that "either both A and B are true or both are false". | Notation | Reading | Info | | --------------------- | ---------------------- | --------------------- | | $A \leftrightarrow B$ | *material equivalence* | for the current model | | $A \iff B$ | *logical equivalence* | in every model | | $A \equiv B$ | *equivalent* | | ## Partial Equivalence | Notation | Reading | Info | | ----------- | -------------------------- | --------------------------- | | $\leq$ | *less than or equal to* | | | $\leqslant$ | *less than or equal to* | | | $\geq$ | *greater than or equal to* | | | $\geqslant$ | *greater than or equal to* | | | $\lt$ | *less than* | | | $\gt$ | *greater than* | | | $\ll$ | *much less than* | orders of magnitude less | | $\gg$ | *much greater than* | orders of magnitude greater | | $\preceq$ | *at most* | | | $\succeq$ | *at least* | | ## Similarity and Inequality | Notation | Reading | Info | | -------------- | --------------------- | ------- | | $\sim$ | *similar* | | | $\thicksim$ | *similar* | "thick" | | $\nsim$ | *not similar* | | | $\approx$ | *approximately* | | | $\thickapprox$ | *approximately* | "thick" | | $\approxeq$ | *approximately equal* | | | $\cong$ | *congruent* | | | $\ncong$ | *not congruent* | | | $\simeq$ | *similar equal* | | | $\neq$ | *not equal* | | ## And These all indicate the *logical conjunction* operation which is the statement that "both A and B must be true, otherwise false". | Notation | Reading | Info | | ----------- | ------------------------ | -------------------------------------------------------------------------------------------------------------- | | $A \land B$ | *logical and* | To avoid confusion with wedge products (eg of vectors), I **avoid** this notation where possible | | $A \cdot B$ | *logical multiplication* | Makes sense for **numerical** logic where multiplication gives zero (false) or non-zero (true) results | | $A\;\&\;B$ | *and* | This is most similar to the way it is written in English and in most programming languages, so I **prefer** it | ## Or These all indicate the *logical inclusive disjunction* operation which is the statement that "at least one of A or B must be true, otherwise false". | Notation | Reading | Info | | --------------- | ------------------------ | -------------------------------------------------------------------------------------------------------------- | | $A \lor B$ | *logical or* | This pairs with the "$\wedgequot; but is not its opposite, I **avoid** this notation where possible | | $A + B$ | *logical multiplication* | Makes sense for **numerical** logic where addition gives zero (false) or non-zero (true) results | | $A \parallel B$ | *or* | This is most similar to the way it is written in English and in most programming languages, so I **prefer** it | ## Not These all indicate the *negation* operation which is the statement that "if A is true then false, otherwise true". | Notation | Reading | Info | | -------- | --------- | --------------------------------------------------------------------------------------------------- | | $\neg A$ | *negate* | | | $\not A$ | *slashed* | | | $\sim A$ | | This is the same symbol as the one used for similarity, definitely **avoid** this wherever possible | ## Proofs | Notation | Reading | Info | | ---------------- | ------------ | -------------------------------------------------- | | $A \therefore B$ | *therefore* | the state of $B$ is the logical consequence of $A$ | | $A \because B$ | *because* | the state of $A$ is the logical consequence of $B$ | | $A \implies B$ | *implies* | the state of $B$ is implied by $A$ | | $A \impliedby B$ | *implied by* | the state of $A$ is implied by $B$ | ## Modal Logic ### Essential Operators The common $\Box$ and $\lozenge$ operators are used in modal logic to indicate the more essential and the less essential, respectively. These operators are used and reused in nearly every form of modal logic to represent their two basic operations. | Notation | Reading | Info | | ---------- | ------------- | ----------------- | | $\Box$ | *necessarily* | alethic logic | | $\Diamond$ | *possibly* | alethic logic | | $\Box$ | *provable* | provability logic | | $\Diamond$ | *consistent* | provability logic | | $\Box$ | *obligatory* | deontic logic | | $\Diamond$ | *permissible* | deontic logic | | $\Box$ | *always* | temporal logic | | $\Diamond$ | *eventually* | temporal logic | ### Letter Symbols and Additional Operators | Notation | Reading | Info | | ------------------ | --------------- | ----------------------- | | $\boldsymbol O B$ | *obligatory* | deontic logic (abbr) | | $\boldsymbol P E$ | *permissible* | deontic logic (abbr) | | $\boldsymbol I M$ | *impermissible* | deontic logic (abbr) | | $\boldsymbol O M$ | *omissible* | deontic logic (abbr) | | $\boldsymbol O P$ | *optional* | deontic logic (abbr) | | $\boldsymbol N O$ | *non-optional* | deontic logic (abbr) | | $O$ | *obligatory* | deontic logic (letter) | | $P$ | *permissible* | deontic logic (letter) | | $F$ | *forbidden* | deontic logic (letter) | | $G$ | *will always* | temporal logic (letter) | | $F$ | *future* | temporal logic (letter) | | $H$ | *has always* | temporal logic (letter) | | $P$ | *past* | temporal logic (letter) | ### Obscure Operators I haven't found a good reference to these, but there are a handful of examples out there. These are primarily guesswork. | Notation | Reading | Info | | ------------------ | ------------ | ---- | | $\bigcirc$ | *next* | | | $\bigtriangleup$ | *non-con..?* | | | $\bigtriangledown$ | *con..?* | | | $\rhd$ | *future* | | | $\lhd$ | *past* | | ## Others These are the readings used in"boolean" logic. | Notation | Reading | Info | | --------------------- | ------- | ------------------------------------------------------------------------------------------------------------------ | | $A \oplus B$ | *xor* | if either $A$ or $B$ are true but not both | | $A \overline \lor B$ | *nor* | neither $A$ nor $B$ are true | | $A \overline \land B$ | *nand* | both $A$ and $B$ are false | | $A \odot B$ | *xnor* | either both are true or both or false<br>similar to if-and-only-if, but specifically represents the xnor operation | # Numeric Operations | Notation | Reading | Info | | ------------ | ------------------------------------------- | ---------------------------- | | $AB$ | numeric multiplication | Default operation for domain | | $A \cdot B$ | numeric multiplication, vector dot product, | | | $A \wedge B$ | vector wedge product | | | | | | # Attributes | Notation | Reading | Info | | --------------------------------- | ------------------- | ----------------------------------------- | | $\lvert\lvert \vec v\rvert\rvert$ | *magnitude* | distance (length) of a vector from origin | | $\lvert x\rvert$ | *absolute value* | distance of of a scalar from zero | | $f'$ | *derivative* | derivative of a function | | $f''$ | *second derivative* | second derivative of a function | # References ## Formatting ### Helpers - https://wumbo.net - http://detexify.kirelabs.org/classify.html - https://www.atomurl.net/math/ - http://asciimath.org/ ### Documentation - https://help.obsidian.md/Editing+and+formatting/Advanced+formatting+syntax - https://en.wikibooks.org/wiki/LaTeX/Mathematics - https://mirrors.rit.edu/CTAN/info/symbols/comprehensive/symbols-a4.pdf - https://docs.mathjax.org/en/latest/basic/mathjax.html - https://docs.mathjax.org/en/latest/input/tex/extensions/index.html - https://www.tug.org/teTeX/tetex-texmfdist/doc/latex/amsmath/amsldoc.pdf - https://www.ctan.org/tex-archive/info/symbols/comprehensive/?lang=en ### Sets - https://tex.stackexchange.com/questions/253077/how-do-you-create-a-set-in-latex ### Vectors - https://tex.stackexchange.com/questions/396147/row-vector-problem - https://www.physicsread.com/latex-vector-arrow/ ## Notation Using browser inspection tools, Wikipedia can be made to expose the TeX syntax for any of its displayed equations or mathematical characters. - https://en.wikipedia.org/wiki/List_of_mathematical_symbols_by_subject - https://web.archive.org/web/20180619212427/ftp://ftp.ams.org/pub/tex/doc/amsmath/amsldoc.pdf - https://github.com/igorsvara/Latex-Symbols ### Conventions - https://cims.nyu.edu/~tjl8195/survey/results.html ### Coordinates - https://en.wikipedia.org/wiki/Cartesian_coordinate_system?useskin=vector#Notations_and_conventions ### Vectors & Matricies - https://en.wikipedia.org/wiki/Vector_notation - https://en.wikipedia.org/wiki/Row_and_column_vectors - https://math.stackexchange.com/questions/552347/notation-subscript-vs-superscript-for-coordinate-vector-fields - https://en.wikipedia.org/wiki/Matrix_(mathematics) - Labeling rows & columns of matrices and array tables - https://tex.stackexchange.com/questions/30791/array-with-labeling-columns - https://tex.stackexchange.com/questions/223501/matrix-with-rows-and-columns-labeled - https://tex.stackexchange.com/questions/59517/label-rows-of-a-matrix-by-characters - https://math.stackexchange.com/questions/307353/how-do-you-write-represent-the-all-ones-matrix ### Zero Elements - https://en.wikipedia.org/w/index.php?title=Zero_element - https://en.wikipedia.org/wiki/Zero_matrix ### Logic - https://en.wikipedia.org/wiki/List_of_logic_symbols - https://www.actual.world/resources/tex/doc/Modals.pdf - https://bd.openlogicproject.org/ - https://latexref.xyz/Math-symbols.html - https://tex.stackexchange.com/questions/695242/consistent-way-to-typeset-modality-symbols ### Equality and Inference - https://www.geeksforgeeks.org/equality-and-inference-symbols-in-latex/