logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

, a conditional quantifier is a kind of

Lindström quantifier In mathematical logic, a Lindström quantifier is a generalized polyadic quantifier. Lindström quantifiers generalize first-order quantifiers, such as the existential quantifier, the universal quantifier, and the counting quantifiers. They were ...

(or

generalized quantifier In formal semantics, a generalized quantifier (GQ) is an expression that denotes a set of sets. This is the standard semantics assigned to quantified noun phrases. For example, the generalized quantifier ''every boy'' denotes the set of sets of ...

) ''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 free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...

): {, , - , , , , , ''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

. 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 In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...

, 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)