Independence-friendly logic


Independence-friendly logic is an extension of classical first-order logic by means of slashed quantifiers of the form and . The intended reading of is "there is a which is functionally independent from the variables in ". 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. 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 depends only on and, and depends only on and. 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 .
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. 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 rather than satisfaction by a single assignment. Such a team semantics was developed by Hodges.
IF 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.
IF logic shares a number of metalogical properties with first-order logic, but there are some differences, including lack of closure under 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 .
Hintikka has argued that IF and extended IF logic should be used as a basis for the foundations of mathematics; this proposal has been met in some cases with skepticism.

Syntax

A number of slightly different presentations of IF logic have appeared in the literature; here we follow.

Terms and atomic formulas

Terms and atomic formulas are defined exactly as in first-order logic with equality.

IF formulas

For a fixed signature σ, formulas of IF logic are defined as follows:
  1. Any atomic formula is an IF formula.
  2. If is an IF formula, then is an IF formula.
  3. If and are IF formulas, then and are IF formulas.
  4. If is a formula, is a variable, and is a finite set of variables, then and are also IF formulas.

    Free variables

The set of the free variables of an IF formula is defined inductively as follows:
  1. If is an atomic formula, then is the set of all variables occurring in it.
  2. ;
  3. ;
  4. .
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 are counted as free variables.

IF Sentences

An IF formula 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.
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 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, a structure, and an assignment.

Players

The semantic game has two players, called Eloise and Abelard.

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 is in negation normal form, with negations symbols occurring only in front of atomic subformulas.
  1. If is a literal, the game ends, and, if is true in , then Eloise wins; otherwise, Abelard wins.
  2. If, then Abelard chooses one of the subformulas, and the corresponding game is played.
  3. If, then Eloises chooses one of the subformulas, and the corresponding game is played.
  4. If, then Abelard chooses an element of, and game is played.
  5. If, then Eloise chooses an element of, and game is played.
More generally, if 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, some subgame is played; we call the assignment associated to, and the subformula occurrence associated to. The player associated to is Eloise in case the most external logical operator in is or, and Abelard in case it is or.
The set of allowed moves in a history is if the most external operator of is or ; it is in case the most external operator of is or.
Given two assignments of same domain, and we write if 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 is in the information set, the player associated to does not know whether he is in or in some other history of.
Consider two histories such that the associated are identical subformula occurrences of the form ; if furthermore, we write or , 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 for the set of histories to which Eloise is associated, and similarly 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 for Eloise is winning if Eloise wins in each terminal history that can be reached by playing according to. Similarly for Abelard.

Truth, falsity, indeterminacy

An IF sentence is true in a structure 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 is an IF sentence with empty slash sets, associate to it the first-order formula which is identical to it, except in that each IF quantifier is replaced by the corresponding first-order quantifier. 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 IF formulas; more exactly, it is possible to define what it means for an IF formula to be satisfied, on a structure, by a team .
The associated games begin with the random choice of an assignment ; 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.

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, we first define its skolemization relativized to a finite set of variables. For every existential quantifier occurring in, let be a new function symbol. We write for the formula which is obtained substituting, in, all free occurrences of the variable with the term. The Skolemization of relative to,, is defined by the following inductive clauses:
  1. if is a literal.
  2. if.
  3. .
  4. , where is a list of the variables in.
If is an IF sentence, its Skolemization is defined as.

Kreiselization

Given an IF formula, associate, to each universal quantifier occurring in it, a new function symbol . Then, the Kreiselization of relative to a finite set of variables, is defined by the following inductive clauses:
  1. if is a literal.
  2. .
  3. .
  4. , where is a list of the variables in.
If is an IF sentence, its Kreiselization is defined as.

Truth, falsity, indeterminacy

Given an IF sentence with existential quantifiers, a structure, and a list of functions of appropriate arities, we denote as the expansion of which assigns the functions as interpretations for the Skolem functions of.
An IF sentence is true on a structure iff there is a tuple of functions such that.
Similarly, iff there is a tuple 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 be a structure and let be a finite set of variables. Then a team over with domain is a set of assignments over with domain, that is, a set of functions from to.

Duplicating and supplementing teams

Duplicating and supplementing are two operations on teams which are related to the semantics of universal and existential quantification.
  1. Given a team over a structure and a variable, the duplicating team is the team.
  2. Given a team over a structure, a function and a variable, the supplementing team is the team.
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 with same variable domain, we write if for every variable.
Given a team on a structure and a finite set of variables, we say that a function is -uniform if whenever.

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:
  1. if and only if, for every assignment, in the sense of first-order logic.
  2. if and only if, for every assignment, in the sense of first-order logic.
  3. if and only if.
  4. if and only if and.
  5. if and only if there exist teams and such that and and .
  6. if and only if.
  7. if and only if there exists a -uniform function such that.
Negative satisfaction:
  1. if and only if, for every assignment, the tuple is not in the interpretation of.
  2. if and only if, for every assignment, .
  3. if and only if.
  4. if and only if there exist teams and such that and and .
  5. if and only if and.
  6. if and only if there exists a -uniform function such that.
  7. if and only if.

    Truth, falsity, indeterminacy

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

Relationship with Game-Theoretical Semantics

For any team on a structure, and any IF formula, we have:
iff
and
iff.
From this it immediately follows that, for sentences,, 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.
if for any structure and any team such that.
if and.
if for any structure and any team such that.
if and.
if and.
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, while truth equivalence treats IF sentences as if they were 2-valued.

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 is a finite set of variables and, one can state that is truth equivalent to relative to in case for any structure and any team of domain.

Model-theoretic properties

Sentence level

IF sentences can be translated in a truth-preserving fashion into sentences of existential second-order logic by means of the Skolemization procedure. Vice versa, every 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 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 and in many cases similar to properties of FOL.
We denote by a set of IF sentences.
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 . In this kind of translation, an extra n-ary predicate symbol is used to represent an n-variable team. This is motivated by the fact that, once an ordering of the variables of has been fixed, it is possible to associate a relation to the team. With this conventions, an IF formula is related to its translation thus:
where is the expansion of that assigns as interpretation for the predicate.
Through this correlation, it is possible to say that, on a structure, an IF formula of n free variables defines a family of n-ary relations over .
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 with an extra predicate .

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 . Hintikka 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 and bring it closer to first-order logic including a compactness theorem, a Löwenheim–Skolem theorem, and a Craig interpolation theorem.. However, Väänänen proved that the set of Gödel numbers of valid sentences of IF logic with at least one binary
predicate symbol is recursively isomorphic with the corresponding set of Gödel numbers of valid second-order sentences in a vocabulary that contains one binary predicate symbol. Furthermore, Väänänen showed that Val2 is the complete Π2-definable set of integers, and that it is Val2 not in for any finite m and n. Väänänen summarizes the complexity results as follows:
Problemfirst-order logicIF/dependence/ESO logic
Decision
Non-validity
Consistency
Inconsistency

Feferman cites Väänänen's 2001 result to argue 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". Feferman also attacked the claimed usefulness of the extended IF logic, because the sentences in do not admit a game-theoretic interpretation.