Conditional quantifier: Difference between revisions

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In [[logic]], a conditional quantifier is a kind of [[Lindström quantifier]] (or [[generalized quantifier]]) ''Q''<mathsub>Q_A''A''</mathsub> that, relative to a classical model <math>''A</math>'', satisfies some or all of the following conditions ("''X''" and "''Y''" range over arbitrary formulas in one free variable):

{|

*<math>Q_AXX</math> [reflexivity]

|-

*<math>Q_AXY \Rightarrow Q_AX(Y\land X)</math> [right conservativity]

| || || ''Q''_''A'' ''X'' ''X'' || [reflexivity]

*<math>Q_AX(Y\land X) \Rightarrow Q_AXY</math> [left conservativity]

|-

*<math>Q_AXY \Rightarrow Q_AX(Y\lor Z)</math> [positive confirmation]

| align="right" | ''Q''_''A'' ''X'' ''Y'' || ⇒ || ''Q''_''A'' ''X'' (''Y''∧''X'') || [right conservativity]

*<math>Q_AX(Y\land Z)\Rightarrow Q_A(X\land Y)Z</math>

|-

*<math>Q_AXY \Rightarrow Q_A(X\lor Z)(Y\lor Z)</math> [positive and negative confirmation]

| align="right" | ''Q''_''A'' ''X'' (''Y''∧''X'') || ⇒ || ''Q''_''A'' ''X'' ''Y'' || [left conservativity]

*<math>Q_AXY \Rightarrow Q_A\lnot X\lnot Y</math> [contraposition]

|-

*<math>Q_AXY\land Q_AYZ \Rightarrow Q_AXZ</math> [transitivity]

| align="right" | ''Q''_''A'' ''X'' ''Y'' || ⇒ || ''Q''_''A'' ''X'' (''Y''∨''Z'') || [positive confirmation]

*<math>Q_AXY \Rightarrow Q_A(X\land Z)Y</math> [weakening]

|-

*<math>Q_AXY\land Q_AXZ \Rightarrow Q_AX(Y\land Z)</math> [conjunction]

| align="right" | ''Q''_''A'' ''X'' (''Y''∧''Z'') || ⇒ || ''Q''_''A'' (''X''∧''Y'') ''Z''

*<math>Q_AXZ\land Q_AYZ \Rightarrow Q_A(X\lor Y)Z</math> [disjunction]

|-

*<math>Q_AXY \Rightarrow Q_AYX</math> [symmetry].

| align="right" | ''Q''_''A'' ''X'' ''Y'' || ⇒ || ''Q''_''A'' (''X''∨''Z'') (''Y''∨''Z'') || [positive and negative confirmation]

|-

| &nbsp;

|-

| align="right" | ''Q''_''A'' ''X'' ''Y'' || ⇒ || ''Q''_''A'' (¬''X'') (¬''Y'') || [contraposition]

|-

| align="right" | ''Q''_''A'' ''X'' ''Y'' &nbsp; ∧ &nbsp; ''Q''_''A'' ''Y'' ''Z'' || ⇒ || ''Q''_''A'' ''X'' ''Z'' || [transitivity]

|-

| align="right" | ''Q''_''A'' ''X'' ''Y'' || ⇒ || ''Q''_''A'' (''X''∧''Z'') ''Y'' || [weakening]

|-

| align="right" | ''Q''_''A'' ''X'' ''Y'' &nbsp; ∧ &nbsp; ''Q''_''A'' ''X'' ''Z'' || ⇒ || ''Q''_''A'' ''X'' (''Y''∧''Z'') || [conjunction]

|-

| align="right" | ''Q''_''A'' ''X'' ''Z'' &nbsp; ∧ &nbsp; ''Q''_''A'' ''Y'' ''Z'' || ⇒ || ''Q''_''A'' (''X''∨''Y'') ''Z'' || [disjunction]

|-

| align="right" | ''Q''_''A'' ''X'' ''Y'' || ⇒ || ''Q''_''A'' ''Y'' ''X'' || [symmetry].

|}

(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 ∀<mathsub>\forall_A''A''</mathsub>, which can be viewed as set-theoretic inclusion, satisfies all of the above except [symmetry]. Clearly [symmetry] holds for ∃<mathsub>\exists_A''A''</mathsub> while e.g. [contraposition] fails.

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]].