Independence-friendly logic

Independence-friendly logic (IF logic; proposed by Jaakko Hintikka and Gabriel Sandu in 1989)^[1] is an extension of classical first-order logic (FOL) by means of slashed quantifiers of the form (\exists v/V) and (\forall v/V), where V is a finite set of variables. The intended reading of (\exists v/V) is "there is a v which is functionally independent from the variables in V". IF logic allows one to express more general patterns of dependence between variables than those which are implicit in first-order logic. This greater level of generality leads to an actual increase in expressive power; the set of IF sentences can characterize the same classes of structures as existential second-order logic (\Sigma^1_1).

For example, it can express branching quantifier sentences, such as the formula \exists c\forall x\exists y\forall z(\exists w/\{x,y\})((x=z \leftrightarrow y=w) \land y \neq c) which expresses infinity in the empty signature; this cannot be done in FOL. Therefore, first-order logic cannot, in general, express this pattern of dependency, in which y depends only on x and c, and w depends only on z and c. IF logic is more general than branching quantifiers, for example in that it can express dependencies that are not transitive, such as in the quantifier prefix \forall x\exists y(\exists z/\{x\}), which expresses that y depends on x, and z depends on y, but z does not depend on x.

The introduction of IF logic was partly motivated by the attempt of extending the game semantics of first-order logic to games of imperfect information. Indeed, a semantics for IF sentences can be given in terms of these kinds of games (or, alternatively, by means of a translation procedure to existential second-order logic). A semantics for open formulas cannot be given in the form of a Tarskian semantics;^[2] an adequate semantics must specify what it means for a formula to be satisfied by a set of assignments of common variable domain (a team) rather than satisfaction by a single assignment. Such a team semantics was developed by Hodges.^[3]

Independence-friendly logic is translation equivalent, at the level of sentences, with a number of other logical systems based on team semantics, such as dependence logic, dependence-friendly logic, exclusion logic and independence logic; with the exception of the latter, IF logic is known to be equiexpressive to these logics also at the level of open formulas. However, IF logic differs from all the above-mentioned systems in that it lacks locality: the meaning of an open formula cannot be described just in terms of the free variables of the formula; it is instead dependent on the context in which the formula occurs.

Independence-friendly logic shares a number of metalogical properties with first-order logic, but there are some differences, including lack of closure under (classical, contradictory) negation and higher complexity for deciding the validity of formulas. Extended IF logic addresses the closure problem, but its game-theoretical semantics is more complicated, and such logic corresponds to a larger fragment of second-order logic, a proper subset of \Delta_2^1.^[4]

Hintikka argued^[5] that IF and extended IF logic should be used as a basis for the foundations of mathematics; this proposal was met in some cases with skepticism.^[6]

Syntax

A number of slightly different presentations of independence-friendly logic have appeared in the literature; here we follow Mann et al (2011).^[7]

Terms and atomic formulas

For a fixed signature σ, terms and atomic formulas are defined exactly as in first-order logic with equality.

IF formulas

Formulas of IF logic are defined as follows:

Any atomic formula \varphi is an IF formula.
If \varphi is an IF formula, then \lnot \varphi is an IF formula.
If \varphi and \psi are IF formulas, then \phi \wedge \psi and \phi \vee \psi are IF formulas.
If \varphi is a formula, v is a variable, and V is a finite set of variables, then (\exists v/V)\varphi and (\forall v/V)\varphi are also IF formulas.

Free variables

The set \mbox{Free}(\varphi) of the free variables of an IF formula \varphi is defined inductively as follows:

If \varphi is an atomic formula, then \mbox{Free}(\varphi) is the set of all variables occurring in it.
\mbox{Free}(\lnot\varphi) = \mbox{Free}(\varphi);
\mbox{Free}(\varphi \vee \psi) = \mbox{Free}(\varphi) \cup \mbox{Free}(\psi);
\mbox{Free}((\exists v/V)\varphi) = \mbox{Free}((\forall v/V)\varphi) = (\mbox{Free}(\varphi) \backslash \{v\})\cup V.

The last clause is the only one that differs from the clauses for first-order logic, the difference being that also the variables in the slash set V are counted as free variables.

IF Sentences

An IF formula \varphi such that \mbox{Free}(\phi) = \emptyset is an IF sentence.

Semantics

Three main approaches have been proposed for the definition of the semantics of IF logic. The first two, based respectively on games of imperfect information and on Skolemization, are mainly used in the definition of IF sentences only. The former generalizes a similar approach, for first-order logic, which was based instead on games of perfect information. The third approach, team semantics, is a compositional semantics in the spirit of Tarskian semantics. However, this semantics does not define what it means for a formula to be satisfied by an assignment (rather, by a set of assignments). The first two approaches were developed in earlier publications on if logic;^[8]^[9] the third one by Hodges in 1997.^[10]^[11]

In this section, we differentiate the three approaches by writing distinct pedices, as in \models_{GTS},\models_{Sk},\models. Since the three approaches are fundamentally equivalent, only the symbol \models will be used in the rest of the article.

Game-Theoretical Semantics

Game-Theoretical Semantics assigns truth values to IF sentences according to the properties of some 2-player games of imperfect information. For ease of presentation, it is convenient to associate games not only to sentences, but also to formulas. More precisely, one defines games G(\varphi, \mathcal M, s) for each triple formed by an IF formula \varphi, a structure \mathcal M, and an assignment s:U\supseteq \mbox{Free}(\varphi)\rightarrow \mathcal M.

Players

The semantic game G(\varphi, \mathcal M, s) has two players, called Eloise (or Verifier) and Abelard (or Falsifier).

Game rules

The allowed moves in the semantic game G(\varphi,\mathcal M,s) are determined by the synctactical structure of the formula under consideration. For simplicity, we first assume that \varphi is in negation normal form, with negations symbols occurring only in front of atomic subformulas.

If \varphi is a literal, the game ends, and, if \varphi is true in \mathcal M (in the first-order sense), then Eloise wins; otherwise, Abelard wins.
If \varphi=\psi_1\land\psi_2, then Abelard chooses one of the subformulas \psi_i, and the corresponding game G(\psi_i,\mathcal M,s) is played.
If \varphi=\psi_1\lor\psi_2, then Eloises chooses one of the subformulas \psi_i, and the corresponding game G(\psi_i,\mathcal M,s) is played.
If \varphi=(\forall v/V)\psi, then Abelard chooses an element a of \mathcal M, and game G(\psi,\mathcal M,s(a/v)) is played.
If \varphi=(\exists v/V)\psi, then Eloise chooses an element a of \mathcal M, and game G(\psi,\mathcal M,s(a/v)) is played.

More generally, if \varphi is not in negation normal form, we can state, as a rule for negation, that, when a game G(\lnot\varphi,\mathcal M,s) is reached, the players begin playing a dual game G^*(\varphi,\mathcal M,s) in which the roles of Verifiers and Falsifier are switched.

Histories

Informally, a sequence of moves in a game G(\varphi,\mathcal M,s) is a history. At the end of each history h, some subgame G(\psi_h,\mathcal M,s_h) is played; we call s_h the assignment associated to h, and \psi_h the subformula occurrence associated to h. The player associated to h is Eloise in case the most external logical operator in \psi_h is \lor or \exists, and Abelard in case it is \land or \forall.

The set h of allowed moves in a history h is \mathcal M if the most external operator of \psi_h is \exists or \forall; it is \{L,R\} (L,R being any two distinct objects, symbolizing 'left' and 'right') in case the most external operator of \psi_h is \lor or \land.

Given two assignments s,t of same domain, and V\subseteq dom(s) we write s\sim_V t if s(w) = t(w) on any variable w\in dom(s)\setminus V.

Imperfect information is introduced in the games by stipulating that certain histories are indistinguishable for the associated player; indistinguishable histories are said to form an 'information set'. Intuitively, if the history h is in the information set I, the player associated to h does not know whether he is in h or in some other history of I. Consider two histories h,h' such that the associated \psi_h,\psi_{h'} are identical subformula occurrences of the form (Qv/V)\chi (Q = \exists or \forall); if furthermore s_h\sim_V s_{h'}, we write h\sim_\exists h' (in case Q = \exists) or h\sim_\forall h' (in case Q = \forall), in order to specify that the two histories are indistinguishable for Eloise, resp. for Abelard. We also stipulate, in general, reflexivity of this relation: if \psi =\chi_1\lor\chi_2, then h\sim_\exists h'; and if \psi =\chi_1\land\chi_2, then h\sim_\forall h'.

Strategies

For a fixed game G(\varphi,\mathcal M,s), write H_\exists for the set of histories to which Eloise is associated, and similarly H_\forall for the set of histories of Abelard.

A strategy for Eloise in the game G(\varphi,\mathcal M,s) is any function that assigns, to any possible history in which it is Eloise's turn to play, a legal move; more precisely, any function \sigma: H_\exists \rightarrow \prod_{h\in H_\exists}A(h) such that \sigma(h)\in A(h) for every history h\in H_\exists. One can define dually the strategies of Abelard.

A strategy for Eloise is uniform if, whenever h\sim_\exists h', \sigma(h) = \sigma(h'); for Abelard, if h\sim_\forall h' implies \sigma(h) = \sigma(h').

A strategy \sigma for Eloise is winning if Eloise wins in each terminal history that can be reached by playing according to \sigma. Similarly for Abelard.

Truth, falsity, indeterminacy

An IF sentence \varphi is true in a structure \mathcal M (\mathcal M\models_{GTS}^+\varphi) if Eloise has a uniform winning strategy in the game G(\varphi, \mathcal M, \emptyset). It is false (\mathcal M\models_{GTS}^-\varphi) if Abelard has a winning strategy. It is undetermined if neither Eloise nor Abelard has a winning strategy.

Conservativity

The semantics of IF logic thus defined is a conservative extension of first-order semantics, in the following sense. If \varphi is an IF sentence with empty slash sets, associate to it the first-order formula \varphi' which is identical to it, except in that each IF quantifier (Qv/\emptyset) is replaced by the corresponding first-order quantifier Qv. Then \mathcal M\models_{GTS}^+\varphi iff \mathcal M\models\varphi' in the Tarskian sense; and \mathcal M\models_{GTS}^-\varphi iff \mathcal M\not\models\varphi' in the Tarskian sense.

Open formulas

More general games can be used to assign a meaning to (possibly open) IF formulas; more exactly, it is possible to define what it means for an IF formula \varphi to be satisfied, on a structure \mathcal M, by a team X (a set of assignments of common variable domain dom(X) and codomain \mathcal M). The associated games G(\varphi,M,X) begin with the random choice of an assignment s\in X; after this initial move, the game G(\varphi,M,s) is played. The existence of a winning strategy for Eloise defines positive satisfaction (M,X\models_{GTS}^+\varphi), and existence of a winning strategy for Abelard defines negative satisfaction (M,X\models_{GTS}^-\varphi). At this level of generality, Game-theoretical Semantics can be replaced by an algebraic approach, team semantics (defined below).

Skolem Semantics

A definition of truth for IF sentences can be given, alternatively, by means of a translation into existential second-order logic. The translation generalizes the Skolemization procedure of first-order logic. Falsity is defined by a dual procedure called Kreiselization.

Skolemization

Given an IF formula \varphi, we first define its skolemization relativized to a finite set U\supseteq \mbox{Free}(\varphi) of variables. For every existential quantifier (\exists v/V) occurring in \varphi, let f_v be a new function symbol (a "Skolem function"). We write Subst(\varphi,v,t) for the formula which is obtained substituting, in \varphi, all free occurrences of the variable v with the term t. The Skolemization of \varphi relative to U, denoted \mbox{Sk}_U(\varphi), is defined by the following inductive clauses:

\operatorname{Sk}_{U}(\varphi )= \varphi if \varphi is a literal.
\operatorname{Sk}_{U}(\psi \lor \chi)=\operatorname{Sk}_{U}(\psi )\lor \operatorname{Sk}_{U}(\chi).
\operatorname{Sk}_{U}(\psi \land \chi)=\operatorname{Sk}_{U}(\psi )\land \operatorname{Sk}_{U}(\chi).
\operatorname{Sk}_{U}((\forall v/V)\psi )=\forall v\operatorname{Sk}_{U\cup \{v\}}(\psi ).
\operatorname{Sk}_{U}((\exists v/V)\psi )=Subst(\operatorname{Sk}_{U\cup \{v\}}(\psi ),v,f_v(y_{1},...,y_{n})), where y_{1},...,y_{n} is a list of the variables in U\setminus V.

If \varphi is an IF sentence, its (unrelativized) Skolemization is defined as \mbox{Sk}(\varphi) = \mbox{Sk}_\varnothing(\varphi).

Kreiselization

Given an IF formula \varphi, associate, to each universal quantifier (\forall v/V) occurring in it, a new function symbol g_v (a "Kreisel function"). Then, the Kreiselization \mbox{Kr}_U(\varphi) of \varphi relative to a finite set of variables U\supseteq\mbox{Free}(\varphi), is defined by the following inductive clauses:

\operatorname{Kr}_{U}(\varphi )= \lnot\varphi if \varphi is a literal.
\operatorname{Kr}_{U}(\psi \land \chi)=\operatorname{Kr}_{U}(\psi )\lor \operatorname{Kr}_{U}(\chi).
\operatorname{Kr}_{U}(\psi \lor \chi)=\operatorname{Kr}_{U}(\psi )\land \operatorname{Kr}_{U}(\chi).
\operatorname{Kr}_{U}((\forall v/V)\psi )=Subst(\operatorname{Kr}_{U\cup \{v\}}(\psi ),v,g_v(y_{1},...,y_{n})), where y_{1},...,y_{n} is a list of the variables in U\setminus V.
\operatorname{Kr}_{U}((\exists v/V)\psi )=\forall v\operatorname{Kr}_{U\cup \{v\}}(\psi )

If \varphi is an IF sentence, its (unrelativized) Kreiselization is defined as \mbox{Kr}(\varphi) = \mbox{Kr}_\varnothing(\varphi).

Truth, falsity, indeterminacy

Given an IF sentence \varphi with n existential quantifiers, a structure \mathcal M, and a list \vec f of n functions of appropriate arities, we denote as (\mathcal M,\vec f) the expansion of \mathcal M which assigns the functions \vec f as interpretations for the Skolem functions of \varphi.

An IF sentence is true on a structure \mathcal M, written \mathcal M\models_{\mbox{Sk}}^+\varphi, if there is a tuple \vec f of functions such that (\mathcal M,\vec f)\models \mbox{Sk}(\varphi). Similarly, \mathcal M\models_{\mbox{Sk}}^-\varphi if there is a tuple \vec f of functions such that (\mathcal M,\vec f)\models \mbox{Kr}(\varphi); and \mathcal M\models_{\mbox{Sk}}^0\varphi iff neither of the previous conditions holds.

For any IF sentence, Skolem Semantics returns the same values as Game-theoretical Semantics.^{[citation needed]}

Team Semantics

By means of team semantics, it is possible to give a compositional account of the semantics of IF logic. Truth and falsity are grounded on the notion of 'satisfiability of a formula by a team'.

Teams

Let \mathcal M be a structure and let V = \{v_1, \ldots, v_n\} be a finite set of variables. Then a team over \mathcal M with domain V is a set of assignments over \mathcal M with domain V, that is, a set of functions s from V to \mathcal M.

Duplicating and supplementing teams

Duplicating and supplementing are two operations on teams which are related to the semantics of universal and existential quantification.

Given a team X over a structure \mathcal M and a variable v, the duplicating team X[\mathcal M/v] is the team \{s(a/v)|s\in X, a\in \mathcal M\}.^[12]
Given a team X over a structure \mathcal M, a function F:X\rightarrow \mathcal M and a variable v, the supplementing team X[F/v] is the team \{s(F(s)/v)|s\in X\}.

It is customary to replace repeated applications of these two operation with more succinct notations, such as X[\mathcal MF/uv] for (X[\mathcal M/u])[F/v].

Uniform functions on teams

As above, given two assignments s,t with same variable domain, we write s\sim_V t if s(w) = t(w) for every variable w\in dom(s)\setminus V.

Given a team X on a structure \mathcal M and a finite set V of variables, we say that a function F:X\rightarrow \mathcal M is V-uniform if F(s)=F(t) whenever s \sim_V t.

Semantic clauses

Team semantics is three-valued, in the sense that a formula may happen to be positively satisfied by a team on a given structure, or negatively satisfied by it, or neither. The semantics clauses for positive and negative satisfaction are defined by simultaneous induction on the synctactical structure of IF formulas.

Positive satisfaction:

\!\mathcal M, X \models^+ R t_1 \ldots t_n if and only if, for every assignment s \in X, \!\mathcal M, s \models R t_1 \ldots t_n in the sense of first-order logic (that is, the tuple \!(s(t_1) \ldots s(t_n)) is in the interpretation R^{\mathcal M} of R).
\!\mathcal M,X \models^+ t_1 = t_2 if and only if, for every assignment s \in X, \!\mathcal M, s \models t_1 = t_2 in the sense of first-order logic (that is, s(t_1) = s(t_2)).
\!\mathcal M,X \models^+ \lnot \phi if and only if \!\mathcal M,X \models^- \phi.
\!\mathcal M,X \models^+ \varphi \wedge \psi if and only if \!\mathcal M,X \models^+ \varphi and \!\mathcal M,X \models^+ \psi.
\!\mathcal M,X \models^+ \varphi \vee \psi if and only if there exist teams \!Y and \!Z such that X = Y \cup Z and \!\mathcal M,Y \models^+ \varphi and \!\mathcal M,Z \models^+ \psi.
\!\mathcal M,X \models^+ (\forall v/V) \varphi if and only if \!\mathcal M,X[M/v] \models^+ \varphi.
\!\mathcal M,X \models^+ (\exists v/V) \varphi if and only if there exists a V-uniform function F: X \rightarrow M such that \!\mathcal M,X[F/v] \models^+ \phi.

Negative satisfaction:

\!\mathcal M, X \models^- R t_1 \ldots t_n if and only if, for every assignment s \in X, the tuple \!(s(t_1) \ldots s(t_n)) is not in the interpretation R^{\mathcal M} of R.
\!\mathcal M,X \models^- t_1 = t_2 if and only if, for every assignment s \in X, s(t_1) \neq s(t_2).
\!\mathcal M,X \models^- \lnot \phi if and only if \!\mathcal M,X \models^+ \phi.
\!\mathcal M,X \models^- \varphi \wedge \psi if and only if there exist teams \!Y and \!Z such that X = Y \cup Z and \!\mathcal M,Y \models^- \varphi and \!\mathcal M,Z \models^- \psi.
\!\mathcal M,X \models^- \varphi \vee \psi if and only if \!\mathcal M,X \models^- \varphi and \!\mathcal M,X \models^- \psi.
\!\mathcal M,X \models^- (\forall v/V) \varphi if and only if there exists a V-uniform function F: X \rightarrow M such that \!\mathcal M,X[F/v] \models^- \phi.
\!\mathcal M,X \models^- (\exists v/V) \varphi if and only if \!\mathcal M,X[M/v] \models^- \varphi.

Truth, falsity, indeterminacy

According to team semantics, an IF sentence \varphi is said to be true (\mathcal M\models^+ \varphi) on a structure \mathcal M if it is satisfied on \mathcal M by the singleton team \{\emptyset\}, in symbols: \mathcal M,\{\emptyset\}\models^+ \varphi. Similarly, \varphi is said to be false (\mathcal M\models^- \varphi) on \mathcal M if \mathcal M,\{\emptyset\}\models^-\varphi; it is said to be undetermined (\mathcal M\models^0 \varphi) if \mathcal M,\{\emptyset\}\not\models^+\varphi and \mathcal M,\{\emptyset\}\not\models^-\varphi.

Relationship with Game-Theoretical Semantics

For any team X on a structure \mathcal M, and any IF formula \varphi, we have: \mathcal M,X \models^+ \varphi iff \mathcal M,X \models_{GTS}^+ \varphi and \mathcal M,X \models^- \varphi iff \mathcal M,X \models_{GTS}^- \varphi.

From this it immediately follows that, for sentences \varphi, \mathcal M\models^+ \varphi \Leftrightarrow \mathcal M\models_{GTS}^+ \varphi, \mathcal M\models^- \varphi \Leftrightarrow \mathcal M\models_{GTS}^- \varphi and \mathcal M\models^0 \varphi \Leftrightarrow \mathcal M\models^0_{GTS} \varphi.

Notions of equivalence

Since IF logic is, in its usual acception, three-valued, multiple notions of formula equivalence are of interest.

Equivalence of formulas

Let \varphi,\psi be two IF formulas.

\varphi\models^+\psi (\varphi truth entails \psi) if \mathcal M,X\models^+\varphi \Rightarrow \mathcal M,X\models^+\psi for any structure \mathcal M and any team X such that dom(X)\supseteq \mbox{Free}(\varphi)\cup \mbox{Free}(\psi).

\varphi \equiv^+\psi (\varphi is truth equivalent to \psi) if \varphi\models^+\psi and \psi\models^+\varphi.

\varphi\models^-\psi (\varphi falsity entails \psi) if \mathcal M,X\models^-\psi \Rightarrow \mathcal M,X\models^-\varphi for any structure \mathcal M and any team X such that dom(X)\supseteq \mbox{Free}(\varphi)\cup \mbox{Free}(\psi).

\varphi \equiv^-\psi (\varphi is falsity equivalent to \psi) if \varphi\models^-\psi and \psi\models^-\varphi.

\varphi\models\psi (\varphi strongly entails to \psi) if \varphi\models^+\psi and \varphi\models^-\psi.

\varphi \equiv\psi (\varphi is strongly equivalent to \psi) if \varphi \equiv^+\psi and \varphi \equiv^-\psi.

Equivalence of sentences

The definitions above specialize for IF sentences as follows. Two IF sentences \varphi,\psi are truth equivalent if they are true in the same structures; they are falsity equivalent if they are false in the same structures; they are strongly equivalent if they are both truth and falsity equivalent.

Intuitively, using strong equivalence amounts to considering IF logic as 3-valued (true/undetermined/false), while truth equivalence treats IF sentences as if they were 2-valued (true/untrue).

Equivalence relative to a context

Many logical rules of IF logic can be adequately expressed only in terms of more restricted notions of equivalence, which take into account the context in which a formula might appear.

For example, if U is a finite set of variables and U\supseteq \mbox{Free}(\varphi)\cup\mbox{Free}(\psi), one can state that \varphi is truth equivalent to \psi relative to U (\varphi \equiv_U\psi) in case \mathcal M,X\models^+\psi \Leftrightarrow \mathcal M,X\models^+\varphi for any structure \mathcal M and any team X of domain U.

Model-theoretic properties

Sentence level

IF sentences can be translated in a truth-preserving fashion into sentences of (functional) existential second-order logic (\Sigma_1^1) by means of the Skolemization procedure (see above). Vice versa, every \Sigma_1^1 can be translated into an IF sentence by means of a variant of the Walkoe-Enderton translation procedure for partially-ordered quantifiers (^[13]^[14]). In other words, IF logic and \Sigma_1^1 are expressively equivalent at the level of sentences. This equivalence can be used to prove many of the properties that follow; they are inherited from \Sigma_1^1 and in many cases similar to properties of FOL.

We denote by T a (possibly infinite) set of IF sentences.

Löwenheim-Skolem property: if T has an infinite model, or arbitrarily large finite models, than it has models of every infinite cardinality.
Existential compactness: if every finite T_0\subseteq T has a model, then also T has a model.
Failure of deductive compactness: there are T,\varphi such that T\models\varphi, but T_0\not\models\varphi for any finite T_0\subset T. This is a difference from FOL.
Separation theorem: if \varphi,\psi are mutually inconsistent IF sentences, then there is a FOL sentence \theta such that \varphi\models^+\theta and \psi\models^+\lnot\theta. This is a consequence of Craig's interpolation theorem for FOL.
Burgess' theorem:^[15] if \varphi,\psi are mutually inconsistent IF sentences, then there is an IF sentence \theta such that \varphi\equiv^+\theta and \psi\equiv^+\lnot\theta (except possibly for one-element structures). In particular, this theorem reveals that the negation of IF logic is not a semantical operation with respect to truth equivalence (truth-equivalent sentences may have non-equivalent negations).
Definability of truth:^[16] there is an IF sentence TRUE(c), in the language of Peano Arithmetic, such that, for any IF sentence \varphi,, \mathbb{N}\models\varphi \Leftrightarrow \mathbb{N}\models TRUE(\ulcorner\varphi\urcorner) (where \ulcorner\urcorner denotes a Gödel numbering). A weaker statement also holds for nonstandard models of Peano Arithmetic (^[17]).

Formula level

The notion of satisfiability by a team has the following properties:

Downward closure: if \mathcal M,X\models^\pm \varphi and Y\subseteq X, then \mathcal M,Y\models^\pm \varphi.
Consistency: \mathcal M,X\models^+ \varphi and \mathcal M,X\models^- \varphi if and only if X=\emptyset.
Non-locality: there are \mathcal M,X,\varphi such that \mathcal M,X\models \varphi\not\Leftrightarrow M,X_{\upharpoonright \mbox{Free}(\varphi)}\models \varphi.

Since IF formulas are satisfied by teams and formulas of classical logics are satisfied by assignments, there is no obvious intertranslation between IF formulas and formulas of some classical logic system. However, there is a translation procedure^[18] of IF formulas into sentences of relational \Sigma_1^1 (actually, one distinct translation \tau_{U,R} for each finite U\supseteq \mbox{Free}(\varphi) and for each choice of a predicate symbol R of arity card(U)). In this kind of translation, an extra n-ary predicate symbol R is used to represent an n-variable team X. This is motivated by the fact that, once an ordering v_1\dots v_n of the variables of dom(X) has been fixed, it is possible to associate a relation Rel_{v_1\dots v_n}(X) = \{(s(v_1),\dots,s(v_n)) | s\in X\} to the team X. With this conventions, an IF formula is related to its translation thus:

\mathcal M,X\models \varphi \Leftrightarrow (\mathcal M,Rel_{v_1\dots v_n}(X))\models \tau_{dom(X),R}(\varphi)

where (M,Rel_{v_1\dots v_n}(X)) is the expansion of \mathcal M that assigns Rel_{v_1\dots v_n}(X) as interpretation for the predicate R.

Through this correlation, it is possible to say that, on a structure \mathcal M, an IF formula \varphi of n free variables defines a family of n-ary relations over \mathcal M (the family of the relations Rel_{v_1\dots v_n}(X) such that \mathcal M,X\models\varphi).

In 2009, Kontinen and Väänänen,^[19] showed, by means of a partial inverse translation procedure, that the families of relations that are definable by IF logic are exactly those that are nonempty, downward closed and definable in relational \Sigma_1^1 with an extra predicate R (or, equivalently, nonempty and definable by a \Sigma_1^1 sentence in which R occurs only negatively).

Extended IF logic

IF logic is not closed under classical negation. The boolean closure of IF logic is known as extended IF logic and it is equivalent to a proper fragment of \Delta_2^1 (Figueira et al. 2011). Hintikka (1996, p. 196) claimed that "virtually all of classical mathematics can in principle be done in extended IF first-order logic".

Properties and critique

A number of properties of IF logic follow from logical equivalence with \Sigma^1_1 and bring it closer to first-order logic including a compactness theorem, a Löwenheim–Skolem theorem, and a Craig interpolation theorem. (Väänänen, 2007, p. 86). However, Väänänen (2001) proved that the set of Gödel numbers of valid sentences of IF logic with at least one binary predicate symbol (set denoted by Val_IF) is recursively isomorphic with the corresponding set of Gödel numbers of valid (full) second-order sentences in a vocabulary that contains one binary predicate symbol (set denoted by Val²). Furthermore, Väänänen showed that Val² is the complete Π₂-definable set of integers, and that it is Val² not in \Sigma^m_n for any finite m and n. Väänänen (2007, pp. 136–139) summarizes the complexity results as follows:

Problem	first-order logic	IF/dependence/ESO logic
Decision	`\Sigma_1^0` (r.e.)	`\Pi_2`
Non-validity	`\Pi_1^0` (co-r.e.)	`\Sigma_2`
Consistency	`\Pi_1^0`	`\Pi_1^0`
Inconsistency	`\Sigma_1^0`	`\Sigma_1^0`

Feferman (2006) cites Väänänen's 2001 result to argue (contra Hintikka) that while satisfiability might be a first-order matter, the question of whether there is a winning strategy for Verifier over all structures in general "lands us squarely in full second order logic" (emphasis Feferman's). Feferman also attacked the claimed usefulness of the extended IF logic, because the sentences in \Pi_1^1 do not admit a game-theoretic interpretation.

Notes

^ Hintikka&Sandu1989
^ Cameron&Hodges 2001
^ Hodges 1997
^ Figueira, Gorin & Grimson 2011
^ e.g. in Hintikka 1996
^ e.g. Feferman2006
^ Mann, Sandu & Sevenster 2011
^ Hintikka&Sandu 1989
^ Sandu 1993
^ Hodges 1997
^ Hodges 1997b
^ The notation s(a/v) is used to denote an assignment that maps v to a, and all other variables to the same element as s does.
^ Walkoe 1970
^ Enderton 1970
^ Burgess 2003
^ Sandu 1998
^ Väänänen 2007
^ Hodges 1997b
^ Kontinen&Väänänen 2009

External links

Tulenheimo, Tero. "Independence friendly logic"
Hodges, Wilfrid. "Logic and Games"
IF logic on Planet Math

Syntax

Terms and atomic formulas

IF formulas

Free variables

IF Sentences

Semantics

Game-Theoretical Semantics

Players

Game rules

Histories

Strategies

Truth, falsity, indeterminacy

Conservativity

Open formulas

Skolem Semantics

Skolemization

Kreiselization

Truth, falsity, indeterminacy

Team Semantics

Teams

Duplicating and supplementing teams

Uniform functions on teams

Semantic clauses

Truth, falsity, indeterminacy

Relationship with Game-Theoretical Semantics

Notions of equivalence

Equivalence of formulas

Equivalence of sentences

Equivalence relative to a context

Model-theoretic properties

Sentence level

Formula level

Extended IF logic

Properties and critique

See also

Notes

External links

Connections

Linked from

Links to (24 total)