## 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 , , , . 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.

or is true whenever one of the two, or both, are true.
and To make true, both and have to be true.
not only is true when is false.
implies Shows consequence. The expression says that when holds, so does . In addition, is considered true except for the case true and false. To understand that, think of an which implies and ask yourself: is it possible that is true but not ? Anyway, don't worry about that, it's not important right now.
if and only if is the same as . It means that from we can deduce and viceversa, so they are equivalent.
false The empty square represents false (the binary 0). Technically, it represents .
true The filled square represents true (the binary 1). Technically, it represents .
exists... can be read there exists an such that of . If in our domain, we can find an element (or more) which makes true the property applied to that element, then the formula is true.
for all... can be read for all , of . If all elements we are working with make the property become true, then the formula is true.
then 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 or like . But there are also invalid ones, like . The objective of natural deduction is to prove that a sequent is valid.
valid means that is logical consequence of , but when one writes , what we mean is that the sequent is valid, that is, we could somehow prove it, and now is considered true for any interpretation of the predicate symbols.
invalid means that is not logical consequence of . If you can find a series of values (model) which make true but false, then invalidity is proven.
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.
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