The universal quantifier
, the conditional, and first-order predicate calculus
By placing an existential quantifier [there exists] before x ("for some x") and an universal quantifier
[for all] before y ("for all y"), we can bind these variables, as may be seen bellow [Bird, 2009]:
The classical Universal Quantifier
is the following way:
The proposed semantics stays close to the standard account: the epsilon-operator substitutes the universal quantifier
present in standard semantics by arbitrarily binding the open world-variable.
Specific topics include evaluating the morphological form of the German universal quantifier
, the corruption of text types as seen in comedy and readers' commentaries in online newspapers, fictional orality as a challenge for the translator, using the Morphilo toolset to deal with the diversity of English historical texts, and communicative space and language use in the age of globalized migration.
Abstraction makes it possible to define the property of being necessarily identical to entity x as bound by the universal quantifier
in K(Q-W) steps (1) and (3).
Use the tactic GENLTAC to simplify sub-goal g1, thereby stripping the outermost universal quantifier
from the conclusion of sub-goal g1 to obtain sub-goal g2.
But if this is the case, 'is universally valid with respect to "..."' is, like the universal quantifier
of first-order predicate logic, an operator that binds a variable to express generality.
Fortunately, for mathematical assertions, the quantification process is syntactically more straightforward and we offer several easy, short and reasonably natural instances as the ones above and also several where mathematical formulas mix symbolic expressions for quantifiers, with natural language quantifiers such as 'for any natural number x, ...' or 'there exists an x such that ...' checking in the process if he is in possession of or has developed a rudimentary but accurate feeling for syntax to concentrate afterwards in semantics, once we have informed him that the traditional symbol for the universal quantifier
"for any" is "[for all]", an inverted letter "A", and for the existential quantifier "there exists" is "[there exist]", a rotated letter "E".
But the other examples allow only for the interpretation in which the existential quantifier takes scope over the universal quantifier
. This contrast indicates that the two scope options in sentences (i) and (ii) do not come result from the raising of the universal quantifier
to adjoin to the matrix clause, because this option is open in all the cases.
Consider for example the interaction between the universal quantifier
and negation in the sentence in (1).
Huang (Asian studies and Chinese and linguistics, Haverford College, Pennsylvania) explores the formal definition of universal quantification, arguing that the formal definition of EVERY, which stands for any distributive universal quantifier
, ought to incorporate a skolem function to capture the paired reading that for every x there is a y, which is present in all universal quantifier