Concept
Literal (mathematical logic)
Overview
In mathematical logic, a literal is an atomic formula (also known as an atom or prime formula) or its negation. The definition mostly appears in proof theory (of classical logic), e.g. in conjunctive normal form and the method of resolution.
Literals can be divided into two types:
A positive literal is just an atom (e.g.,
x
{\displaystyle x}
).
A negative literal is the negation of an atom (e.g.,
¬
x
{\displaystyle \lnot x}
).
The polarity of a literal is positive or negative depending on whether it is a positive or negative literal.
In logics with double negation elimination (where
¬
¬
x
≡
x
{\displaystyle \lnot \lnot x\equiv x}
) the complementary literal or complement of a literal
l
{\displaystyle l}
can be defined as the literal corresponding to the negation of
l
{\displaystyle l}
. We can write
l
¯
{\displaystyle {\bar {l}}}
to denote the complementary literal of
l
{\displaystyle l}
. More precisely, if
l
≡
x
{\displaystyle l\equiv x}
then
l
¯
{\displaystyle {\bar {l}}}
is
¬
x
{\displaystyle \lnot x}
and if
l
≡
¬
x
{\displaystyle l\equiv \lnot x}
then
l
¯
{\displaystyle {\bar {l}}}
is
x
{\displaystyle x}
. Double negation elimination occurs in classical logics but not in intuitionistic logic.
In the context of a formula in the conjunctive normal form, a literal is pure if the literal's complement does not appear in the formula.
In Boolean functions, each separate occurrence of a variable, either in inverse or uncomplemented form, is a literal. For example, if
A
{\displaystyle A}
,
B
{\displaystyle B}
and
C
{\displaystyle C}
are variables then the expression
A
¯
B
C
{\displaystyle {\bar {A}}BC}
contains three literals and the expression
A
¯
C
+
B
¯
C
¯
{\displaystyle {\bar {A}}C+{\bar {B}}{\bar {C}}}
contains four literals. However, the expression
A
¯
C
+
B
¯
C
{\displaystyle {\bar {A}}C+{\bar {B}}C}
would also be said to contain four literals, because although two of the literals are identical (
C
{\displaystyle C}
appears twice) these qualify as two separate occurrences.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
In the original texts
Passages from public-domain books where Literal (mathematical logic) is discussed.
Not less true is that which it means spiritually, that in the Soul's liberation from Sin (or in the exodus of the Soul from Sin) it is made holy and free in its powers. But in demonstrating these, the Literal must always go first, as that in whose sense the others are included, and without which it would be impossible and irrational to understand the others. Especially is it impossible in the Allegorical, because…
Dante Alighieri, The Banquet
names the idea; states what it is; opens on it
To present the point more clearly, we cite Paul's distinction of the two covenants. 2 Cor 3, 6. And these two covenants respectively relate to two kinds of people. 4. First, there is the written law commanded of God and composed of written words. It is styled "written" or "literal" because it goes no farther and does not enter the heart, nor are there any resulting works other than hypocritical and extorted ones…
Martin Luther, Epistle Sermons, Vol. 2
a chapter is titled for it; names the idea; opens on it
So, whereas in every other science things are signified by words, this science has the property, that the things signified by the words have themselves also a signification. Therefore that first signification whereby words signify things belongs to the first sense, the historical or literal. That signification whereby things signified by words have themselves also a signification is called the spiritual sense, which…
Aquinas Thomas, Summa Theologica, Part I
names the idea; states what it is
But I conceive it might be at first either a literal Character, and so the whole square Character was composed of so many distinct Letters or Syllables, which composed the Word signified thereby; and so there might be a regular Order of placing these Letters in the Character, that is, that the whole square being divided into so many parts, there was a Rule which was the first, second, third and fourth place: so that…
Royal Society, Miscellanea Curiosa, Vol. 3
names the idea
The same is the case with the figures employed by prophets. Taken literally, such expressions contain wisdom useful for many purposes, among others, for the amelioration of the condition of society; e.g., the Proverbs (of Solomon), and similar sayings in their literal sense. Their hidden meaning, however, is profound wisdom, conducive to the recognition of real truth.
Moses Maimonides, The guide for the perplexed
names the idea
Related ideas
- Propositional variable
- Negation
- Intuitionistic logic
- Propositional logic
- Proof theory
- Function (mathematics)
- Mathematical logic
- Predicate (logic)
- Double negation
- Atomic formula
- Classical logic
- Boolean function
- Recursive definition
Place in the tradition
Part of Outline of logic.
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