Logic symbols

About logic symbols

Logic symbols are the shorthand of formal logic, used in philosophy, mathematics and computer science to write arguments out exactly. A handful of connectives join statements: ∧ for “and”, ∨ for “or”, ¬ for “not”, → for “if … then” and ↔ for “if and only if”. Two quantifiers, ∀ (“for all”) and ∃ (“there exists”), say how many things a statement is about. ⊤ and ⊥ are statements that are always true and always false. The turnstiles ⊢ and ⊨ stand outside the statements altogether: one says a conclusion can be proved from premises, the other that it follows from them by truth alone. Each page here says what one symbol means, how it is read, and which other signs books use for the same thing. The readings are from OpenStax’s Contemporary Mathematics, Oscar Levin’s Discrete Mathematics: An Open Introduction, forall x: Calgary from the University of Calgary, Stanford’s CS103 course and the Stanford Encyclopedia of Philosophy; the dates are from Jeff Miller’s Earliest Uses of Various Mathematical Symbols, published by the University of St Andrews.

One idea, several signs

Logic books don’t share a single notation. forall x: Calgary explains that authors were often forced to use whatever their printers could typeset, and lists the alternatives a reader will meet in other books:

OpenStax’s Contemporary Mathematics writes “not” with the tilde throughout, and the Stanford Encyclopedia of Philosophy writes “and” with &. Most of the basic symbols were introduced between 1880 and 1920 by Ernst Schröder, Giuseppe Peano, Alfred North Whitehead and Bertrand Russell, according to Miller; ∧ and ¬ are later, from Arend Heyting in 1930, and the upside-down A of ∀ from Gerhard Gentzen in 1935.

Logic in circuits and sets

The same ideas have their own drawings in electronics: the AND, OR and NOT gates in Logic gate symbols. In set theory they become operations on sets. OpenStax calls the intersection of two sets a logical AND and their union a logical inclusive OR (Set theory symbols).