1 Generally, (a) an instrument or process for carrying out an operation, or (b) a symbol that signals a direction for a particular operation.
2 In formal logic, ‘operator’ is in the broadest sense a collective term for quantifier, logical predicate and logical particle (logical connective); in the narrower sense the collective term (and frequent synonym) for quantifiers: operators are linguistic expressions (or their symbolizations) that serve to specify (=quantify) sets: all, none, every, among others. An operator connects a variable to a complete proposition. One differentiates between the following. (a) The existential operator (also existential quantifier) symbolized by or and read as: ‘there is at least one element x in set S for which it is the case that…’; e.g. Some people are late risers is symbolized by . The existential operator expresses a particular case and is in the truth-functional relation of disjunction (cf. the mnemotechnically motivated symbolization: (small) for disjunction, (big) for existential operator). Through negation the existential operator can be carried over to the universal operator (cf. (b) below): Some people are late risers corresponds to the expression Not all people are late risers (notation:). However, in contrast to the universal operator the existential operator presupposes the existence of the designated objects in the real world (presupposition). (b) The universal operator (or universal quantifier), symbolized by or , and read as: ‘for all elements x in set S it is the case that…’. Everyday language example: All humans are mortal symbolized by where H=humans and M= mortal.
The universal operator expresses a generalization and is in the truth-functional relation of conjunction (cf. the mnemotechnically motivated symbolization: (small) for conjunction, (big) for the universal operator). The proposition All humans are mortal is, for a finite set S, equivalent to an enumeration of all elements, i.e. a and b and c…are mortal. Through this parallel, the distributive reading of the universal operator is confirmed: i.e. ‘for every single element it is the case that’ (in contrast to the collective reading of all). (c) The iota operator symbolized by iota (ι), the ninth letter of the Greek alphabet, or by i, and read as ‘that element x for which it is the case that…’. The iota operator serves to identify a particular entity by means of a characteristic that is true only of this entity (definite description), e.g. to be the composer of ‘The Magic Flute’: ιx(Cx): ‘that element x of the set S that has the characteristic of being the composer of ‘The Magic Flute.’ (d) The lambda operator, symbolized by lambda (λ), the eleventh letter of the Greek alphabet, and read as ‘those xs for which it is the case that…’; e.g. λx(Lx): those people who are late risers. The lambda operator forms class names, i.e. complex one-place predicates, out of propositional functions (=open formulae).
or 
and read as: ‘there is at least one element x in set S for which it is the case that…’; e.g. Some people are late risers is symbolized by 
. The existential operator expresses a particular case and is in the truth-functional relation of 
for disjunction, (big) 
for existential operator). Through negation the existential operator can be carried over to the universal operator (cf. (b) below): Some people are late risers corresponds to the expression Not all people are late risers (notation:
). However, in contrast to the universal operator the existential operator presupposes the existence of the designated objects in the real world (
or 
, and read as: ‘for all elements x in set S it is the case that…’. Everyday language example: All humans are mortal symbolized by 
where H=humans and M= mortal. 
for conjunction, (big) 
for the universal operator). The proposition All humans are mortal is, for a finite set S, equivalent to an enumeration of all elements, i.e. a and b and c…are mortal. Through this parallel, the distributive reading of the universal operator is confirmed: i.e. ‘for every single element it is the case that’ (in contrast to the collective reading of all). (c) The iota operator symbolized by iota (ι), the ninth letter of the Greek alphabet, or by i, and read as ‘that element x for which it is the case that…’. The iota operator serves to identify a particular entity by means of a characteristic that is true only of this entity (
