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Logical Operations

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Logical statements may be joined via operations listed in the following table.

Operation Notation (read as) is true if and only if
Negation $ \lnot A$ (not $ A$) $ A$ is false
Conjunction $ A\land B$ ($ A$ and $ B$) both $ A$ and $ B$ are true
Disjunction $ A\lor B$ ($ A$ or $ B$) $ A$ or $ B$ is true (or both are true)
Antivalence $ A \not\equiv B$ (either $ A$ or $ B$) $ A$ and $ B$ are assigned different truth values
Implication
$ A\Rightarrow B$  
$ B\Leftarrow A$  
($ A$ implies $ B$)  
($ B$ follows from $ A$)  
$ A$ is false or $ B$ is true
Equivalence $ A\Leftrightarrow B$ ($ A$ is equivalent to $ B$) $ A$ and $ B$ are assigned identical truth values

In order to reduce the usage of parentheses in logical formulas, we define that $ \lnot$ is more closely linked to a symbol than $ \land$ and $ \lor$, which in turn are more closely linked than $ \Rightarrow$, $ \Leftrightarrow$ and $ \not\equiv$.

Note that an implication $ A\Rightarrow B$ only requires the truth of $ B$ if $ A$ is true. A false proposition implies anything, hence both true and false implications can be drawn.

Usually, the or-connective is symbolised by a v, derived from the word vel (Latin: or), yet it is also common practice to use the symbol ,,$ +$``; then ,,$ \cdot$``symbolizes the and-connective. Using 0 to refer to the truth value ,,false``while interpreting any other value as ,,true``enables us to determine the truth value of logical formulas via calculation with natural numbers.
Particularly computer-linguists frequently use the English terms NOT (negation), AND (conjunction), OR (disjunction), EXOR or XOR (exclusive or, antivalence) as well as their negations NAND (negated conjunction), NOR (negated disjunction) and NXOR (equivalence).

(Authors: Höllig/Hörner/Kimmerle/Abele)

see also:

[Examples]
  automatically generated 5/25/2009