Independence-friendly logic (IF logic; proposed by Jaakko Hintikka and Gabriel Sandu (philosopher) in 1989) is an extension of classical first-order logic (FOL) by means of slashed quantifiers of the form and, where V is a finite set of variables. The intended reading of 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 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, 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; 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. 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. Hintikka argued 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.

Syntax

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

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:

Free variables

The set of the free variables of an IF formula \varphi is defined inductively as follows: 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 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; the third one by Hodges in 1997. In this section, we differentiate the three approaches by writing distinct pedices, as in. 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 for each triple formed by an IF formula \varphi, a structure \mathcal M, and an assignment.

Players

The semantic game has two players, called Eloise (or Verifier) and Abelard (or Falsifier).

Game rules

The allowed moves in the semantic game 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. More generally, if \varphi is not in negation normal form, we can state, as a rule for negation, that, when a game is reached, the players begin playing a dual game in which the roles of Verifiers and Falsifier are switched.

Histories

Informally, a sequence of moves in a game is a history. At the end of each history h, some subgame 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 we write s\sim_V t if s(w) = t(w) on any variable. 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 are identical subformula occurrences of the form (Qv/V)\chi (Q = \exists or \forall); if furthermore, we write (in case Q = \exists) or (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, then ; and if , then.

Strategies

For a fixed game, 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 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 such that for every history. One can define dually the strategies of Abelard. A strategy for Eloise is uniform if, whenever, ; for Abelard, if implies. 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 if Eloise has a uniform winning strategy in the game. It is false 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 is replaced by the corresponding first-order quantifier Qv. Then iff in the Tarskian sense; and iff 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 begin with the random choice of an assignment s\in X; after this initial move, the game is played. The existence of a winning strategy for Eloise defines positive satisfaction, and existence of a winning strategy for Abelard defines negative satisfaction. 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 of variables. For every existential quantifier occurring in \varphi, let f_v be a new function symbol (a "Skolem function"). We write 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, is defined by the following inductive clauses: If \varphi is an IF sentence, its (unrelativized) Skolemization is defined as.

Kreiselization

Given an IF formula \varphi, associate, to each universal quantifier occurring in it, a new function symbol g_v (a "Kreisel function"). Then, the Kreiselization of \varphi relative to a finite set of variables, is defined by the following inductive clauses: If \varphi is an IF sentence, its (unrelativized) Kreiselization is defined as.

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 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, if there is a tuple \vec f of functions such that. Similarly, if there is a tuple \vec f of functions such that ; and iff neither of the previous conditions holds. For any IF sentence, Skolem Semantics returns the same values as Game-theoretical Semantics.

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 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. It is customary to replace repeated applications of these two operation with more succinct notations, such as for.

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. Given a team X on a structure \mathcal M and a finite set V of variables, we say that a function 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: Negative satisfaction:

Truth, falsity, indeterminacy

According to team semantics, an IF sentence \varphi is said to be true on a structure \mathcal M if it is satisfied on \mathcal M by the singleton team, in symbols:. Similarly, \varphi is said to be false on \mathcal M if ; it is said to be undetermined if and.

Relationship with Game-Theoretical Semantics

For any team X on a structure \mathcal M, and any IF formula \varphi, we have: iff and iff. From this it immediately follows that, for sentences \varphi,, and.

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 be two IF formulas. (\varphi truth entails \psi) if for any structure \mathcal M and any team X such that. (\varphi is truth equivalent to \psi) if and. (\varphi falsity entails \psi) if for any structure \mathcal M and any team X such that. (\varphi is falsity equivalent to \psi) if and. (\varphi strongly entails to \psi) if and. (\varphi is strongly equivalent to \psi) if and.

Equivalence of sentences

The definitions above specialize for IF sentences as follows. Two IF sentences 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, one can state that \varphi is truth equivalent to \psi relative to U in case 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. 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.

Formula level

The notion of satisfiability by a team has the following properties: 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 of IF formulas into sentences of relational \Sigma_1^1 (actually, one distinct translation \tau_{U,R} for each finite 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 of the variables of dom(X) has been fixed, it is possible to associate a relation to the team X. With this conventions, an IF formula is related to its translation thus: where is the expansion of \mathcal M that assigns 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 such that ). In 2009, Kontinen and Väänänen, 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 ValIF) 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: 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.