In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, a conditional quantifier is a kind of
Lindström quantifier (or
generalized quantifier) ''Q''
''A'' that, relative to a classical model ''A'', satisfies some or all of the following conditions ("''X''" and "''Y''" range over arbitrary formulas in one
free variable
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for f ...
):
{,
, -
, , , , , ''Q''
''A'' ''X'' ''X'' , ,
eflexivity, -
, align="right" , ''Q''
''A'' ''X'' ''Y'' , , ⇒ , , ''Q''
''A'' ''X'' (''Y''∧''X'') , ,
ight conservativity, -
, align="right" , ''Q''
''A'' ''X'' (''Y''∧''X'') , , ⇒ , , ''Q''
''A'' ''X'' ''Y'' , ,
eft conservativity, -
, align="right" , ''Q''
''A'' ''X'' ''Y'' , , ⇒ , , ''Q''
''A'' ''X'' (''Y''∨''Z'') , ,
ositive confirmation, -
, align="right" , ''Q''
''A'' ''X'' (''Y''∧''Z'') , , ⇒ , , ''Q''
''A'' (''X''∧''Y'') ''Z''
, -
, align="right" , ''Q''
''A'' ''X'' ''Y'' , , ⇒ , , ''Q''
''A'' (''X''∨''Z'') (''Y''∨''Z'') , ,
ositive and negative confirmation, -
,
, -
, align="right" , ''Q''
''A'' ''X'' ''Y'' , , ⇒ , , ''Q''
''A'' (¬''X'') (¬''Y'') , ,
ontraposition, -
, align="right" , ''Q''
''A'' ''X'' ''Y'' ∧ ''Q''
''A'' ''Y'' ''Z'' , , ⇒ , , ''Q''
''A'' ''X'' ''Z'' , ,
ransitivity, -
, align="right" , ''Q''
''A'' ''X'' ''Y'' , , ⇒ , , ''Q''
''A'' (''X''∧''Z'') ''Y'' , ,
eakening, -
, align="right" , ''Q''
''A'' ''X'' ''Y'' ∧ ''Q''
''A'' ''X'' ''Z'' , , ⇒ , , ''Q''
''A'' ''X'' (''Y''∧''Z'') , ,
onjunction, -
, align="right" , ''Q''
''A'' ''X'' ''Z'' ∧ ''Q''
''A'' ''Y'' ''Z'' , , ⇒ , , ''Q''
''A'' (''X''∨''Y'') ''Z'' , ,
isjunction, -
, align="right" , ''Q''
''A'' ''X'' ''Y'' , , ⇒ , , ''Q''
''A'' ''Y'' ''X'' , ,
ymmetry
(The implication arrow denotes material implication in the metalanguage.) The ''minimal conditional logic'' M is characterized by the first six properties, and stronger conditional logics include some of the other ones. For example, the quantifier ∀
''A'', which can be viewed as set-theoretic inclusion, satisfies all of the above except
ymmetry Clearly
ymmetryholds for ∃
''A'' while e.g.
ontrapositionfails.
A semantic interpretation of conditional quantifiers involves a relation between sets of subsets of a given structure—i.e. a relation between properties defined on the structure. Some of the details can be found in the article
Lindström quantifier.
Conditional quantifiers are meant to capture certain properties concerning conditional reasoning at an abstract level. Generally, it is intended to clarify the role of conditionals in a first-order language as they relate to other
connectives, such as conjunction or disjunction. While they can cover nested conditionals, the greater complexity of the formula, specifically the greater the number of conditional nesting, the less helpful they are as a methodological tool for understanding conditionals, at least in some sense. Compare this methodological strategy for conditionals with that of
first-degree entailment logics.
References
Serge Lapierre. Conditionals and Quantifiers, in ''Quantifiers, Logic, and Language'', Stanford University, pp. 237–253, 1995.
Quantifier (logic)