Predicate logic

Predicate logic is the logic we are currently using

Logical symbols
Logical symbols vary by author, but usually include the following: Not all of these symbols are required in first-order logic. Either one of the quantifiers along with negation, conjunction (or disjunction), variables, brackets, and equality suffices.
 * Quantifier symbols: $∀$ for universal quantification, and $∃$ for existential quantification
 * Logical connectives: $∧$ for conjunction, $∨$ for disjunction, $→$ for implication, $↔$ for biconditional, $¬$ for negation. Some authors use Cpq instead of $→$ and Epq instead of $↔$, especially in contexts where → is used for other purposes. Moreover, the horseshoe $⊃$ may replace $→$; the triple-bar $≡$ may replace $↔$; a tilde ($~$), Np, or Fp may replace $¬$; a double bar $$\|$$, $$+$$, or Apq may replace $∨$; and an ampersand $&$, Kpq, or the middle dot $⋅$ may replace $∧$, especially if these symbols are not available for technical reasons. (The aforementioned symbols Cpq, Epq, Np, Apq, and Kpq are used in Polish notation.)
 * Parentheses, brackets, and other punctuation symbols. The choice of such symbols varies depending on context.
 * An infinite set of variables, often denoted by lowercase letters at the end of the alphabet x, y, z, ... . Subscripts are often used to distinguish variables: $x_{0}, x_{1}, x_{2}, ....$
 * An equality symbol (sometimes, identity symbol) $=$ (see below).

Other logical symbols include the following:
 * Truth constants: T, V, or $⊤$ for "true" and F, O, or $⊥$ for "false" (V and O are from Polish notation). Without any such logical operators of valence 0, these two constants can only be expressed using quantifiers.
 * Additional logical connectives such as the Sheffer stroke, Dpq (NAND), and exclusive or, Jpq.