2.2 Used symbols

To express the relation between one action and another, there exist some international icons. The basic operators you must know are $\vee$, $\wedge$, $\neg$, $\Rightarrow$. The others are more complex, but here I put all of them as a reference, to be able to find them if you were searching any of them.

Symbol It's read... Description
$\vee$ or $A\vee B$ is true whenever one of the two, or both, are true.
$\wedge$ and To make $A\wedge B$ true, both $A$ and $B$ have to be true.
$\neg$ not $\neg A$ only is true when $A$ is false.
$\Rightarrow$ implies Shows consequence. The expression $A\Rightarrow B$ says that when $A$ holds, so does $B$. In addition, $A\Rightarrow B$ is considered true except for the case $A$ true and $B$ false. To understand that, think of an $A$ which implies $B$ and ask yourself: is it possible that $A$ is true but not $B$? Anyway, don't worry about that, it's not important right now.
$\Longleftrightarrow$ if and only if $A\Longleftrightarrow B$ is the same as $(A\Rightarrow B)\wedge(B\Rightarrow A)$. It means that from $A$ we can deduce $B$ and viceversa, so they are equivalent.
$\square$ false The empty square represents false (the binary 0). Technically, it represents $\{\}$.
$\blacksquare$ true The filled square represents true (the binary 1). Technically, it represents $\{<>\}$.
$\exists$ exists... $\exists xPx$ can be read there exists an $x$ such that $P$ of $x$. If in our domain, we can find an element (or more) which makes true the property $P$ applied to that element, then the formula is true.
$\forall$ for all... $\forall xPx$ can be read for all $x$, $P$ of $x$. If all elements we are working with make the property $P$ become true, then the formula is true.
$\vdash$ then $\vdash$ is the symbol of the sequent, which is the way of saying ``when all this from the left happens, then it also happens all this from the right''. There exist valid sequents, like $P\wedge Q\vdash P$ or like $P\Rightarrow Q,\ Q\Rightarrow R,\ P\vdash P\wedge R$. But there are also invalid ones, like $P\Rightarrow Q,\ \neg P\vdash\neg Q$. The objective of natural deduction is to prove that a sequent is valid.
$\vDash$ valid $\phi\vDash\varphi$ means that $\varphi$ is logical consequence of $\phi$, but when one writes $A\vDash B$, what we mean is that the sequent $A\vdash B$ is valid, that is, we could somehow prove it, and now is considered true for any interpretation of the predicate symbols.
$\nvDash$ invalid $\phi\nvDash\varphi$ means that $\varphi$ is not logical consequence of $\phi$. If you can find a series of values (model) which make $\phi$ true but $\varphi$ false, then invalidity is proven.
$\Vdash$ satisfiable A set of formulas is satisfiable if there exists a series of values (model) which can make all of them true at the same time.
$\nVdash$ unsatisfiable A set of formulas is unsatisfiable if there isn't any combination of variables (model) which can make all of them become true at the same time.

Daniel Clemente Laboreo 2005-05-17