# Foundational principles of team semantics

## Fredrik Engström, FLoV

Team semantics is, when compared to standard Tarskian semantics, a more expressive framework that can be used to express logical connectives, operations and atoms that can’t be expressed in Tarskian semantics. This includes branching quantifiers, notions of dependence and independence, trace quantification in linear-time temporal logic (LTL), and probabilistic notions from quantum mechanics.

Team semantics is based on the same notion of structure as Tarskian semantics, but instead of a single assignment satisfying a formula (or not), in team semantics a set, or a *team*, of assignments satisfies a formula (or not). In other words, the semantic value of a formula is lifted from a set of assignments (those that satisfy the formula) to a set of teams of assignments.

In almost all (with only one exception that I’m aware of) logical systems based on team semantics this lifting operation is the power set operation, and as a result the set of teams satisfying a formula is closed downwards. This is often taken as a basic and foundational principle of team semantics.

In this talk I will discuss this principle and present some ideas on why, or why not, the power set operation is the most natural lift operation. By using other lift operations we can get a more powerful semantics, but, it seems, also a more complicated one.

References:

- Engström, F. (2012) “Generalized quantifiers in dependence logic”
- Nurmi, V. (2009) “Dependence Logic: Investigations into Higher-Order Semantics Defined on Teams”
- Väänänen, J. (2007) “Dependence logic: A new approach to independence friendly logic”