Glossary

10.2 Kripke frames and models

4 min read•august 7, 2024

Kripke frames and models are the backbone of , providing a way to visualize and reason about possibility and necessity. They consist of possible worlds connected by accessibility relations, allowing us to interpret modal statements across different scenarios.

These structures help us understand various modal logics by defining different properties of accessibility relations. Reflexivity, symmetry, transitivity, and other characteristics correspond to specific axioms, shaping how we interpret necessity and possibility in different contexts.

Kripke Frames and Models

Defining Kripke Frames

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  • FF is a pair (W,R)(W,R) where WW is a non-empty set of possible worlds and RR is a binary relation on WW called the
  • WW represents the set of all conceivable scenarios or states of affairs in the context of modal logic
  • Each element w∈Ww \in W is a possible world that captures a specific configuration of propositions being true or false
  • R⊆W×WR \subseteq W \times W determines which worlds are considered accessible from each world in the frame
  • If (w,v)∈R(w,v) \in R, then world vv is accessible from world ww, meaning that if a proposition is necessary at ww, it must be true at vv

Extending Frames to Models

  • MM is a triple (W,R,V)(W,R,V) where (W,R)(W,R) is a Kripke frame and VV is a valuation function
  • V:Prop→2WV: Prop \to 2^W assigns to each propositional variable p∈Propp \in Prop the set of worlds where pp is true
  • For each p∈Propp \in Prop and w∈Ww \in W, w∈V(p)w \in V(p) means that pp is true at world ww in the model MM
  • Truth of modal formulas at a world in a Kripke model depends on both the valuation and the accessibility relation
  • Example: M,w⊨□φM,w \models \square \varphi iff for all vv such that (w,v)∈R(w,v) \in R, M,v⊨φM,v \models \varphi, meaning φ\varphi is necessary at ww if it is true at all accessible worlds

Interpreting Accessibility

  • Accessibility relation RR captures the notion of relative possibility between worlds in a Kripke frame or model
  • Different interpretations of RR correspond to different modal logics and their associated axioms
  • In epistemic logic, (w,v)∈R(w,v) \in R means that world vv is considered possible from the perspective of an agent's knowledge at world ww
  • In deontic logic, (w,v)∈R(w,v) \in R means that world vv is considered ideal or permissible according to the norms or obligations at world ww
  • In temporal logic, (w,v)∈R(w,v) \in R means that world vv is a future state reachable from world ww, capturing the flow of time

Properties of Accessibility Relations

Reflexivity and Symmetry

  • Reflexivity: ∀w∈W:(w,w)∈R\forall w \in W: (w,w) \in R, meaning each world is accessible from itself
  • Reflexivity corresponds to the modal axiom □φ→φ\square \varphi \to \varphi (axiom T), capturing the idea that necessity implies truth
  • Example: In epistemic logic, reflexivity means that if an agent knows φ\varphi, then φ\varphi is actually true
  • Symmetry: ∀w,v∈W:(w,v)∈R  ⟹  (v,w)∈R\forall w,v \in W: (w,v) \in R \implies (v,w) \in R, meaning accessibility is bidirectional
  • Symmetry corresponds to the modal axiom φ→□◊φ\varphi \to \square \Diamond \varphi (axiom B), capturing the idea that truth implies possibility of necessity
  • Example: In epistemic logic, symmetry means that if φ\varphi is considered possible by an agent, then the agent considers it possible to know φ\varphi

Transitivity and Euclideanness

  • Transitivity: ∀w,v,u∈W:(w,v)∈R∧(v,u)∈R  ⟹  (w,u)∈R\forall w,v,u \in W: (w,v) \in R \wedge (v,u) \in R \implies (w,u) \in R, meaning accessibility is transitive
  • Transitivity corresponds to the modal axiom □φ→□□φ\square \varphi \to \square \square \varphi (axiom 4), capturing the idea that necessity is idempotent
  • Example: In epistemic logic, transitivity means that if an agent knows that they know φ\varphi, then they know φ\varphi
  • : ∀w,v,u∈W:(w,v)∈R∧(w,u)∈R  ⟹  (v,u)∈R\forall w,v,u \in W: (w,v) \in R \wedge (w,u) \in R \implies (v,u) \in R, meaning worlds accessible from the same world are accessible from each other
  • Euclideanness corresponds to the modal axiom ◊φ→□◊φ\Diamond \varphi \to \square \Diamond \varphi (axiom 5), capturing the idea that possibility of necessity implies necessity of possibility
  • Example: In epistemic logic, Euclideanness means that if an agent considers φ\varphi possible and ψ\psi possible, then in situations where φ\varphi is true, the agent must also consider ψ\psi possible

Seriality and Combinations

  • : ∀w∈W,∃v∈W:(w,v)∈R\forall w \in W, \exists v \in W: (w,v) \in R, meaning each world has at least one accessible world
  • Seriality corresponds to the modal axiom □φ→◊φ\square \varphi \to \Diamond \varphi (axiom D), capturing the idea that necessity implies possibility
  • Example: In deontic logic, seriality means that for each situation, there is at least one ideal or permissible situation
  • Combining properties of accessibility relations leads to different modal logics with specific axiomatizations
  • Example: Modal logic S4 has a reflexive and transitive accessibility relation, characterized by axioms K, T, and 4
  • Example: Modal logic S5 has a reflexive, symmetric, and transitive accessibility relation, characterized by axioms K, T, B, 4, and 5

Key Terms to Review (19)

Accessibility Relation: An accessibility relation is a binary relation between possible worlds that indicates how one world can 'access' another in modal logic. This concept is crucial for understanding how modal operators, like necessity and possibility, work within a given framework. The nature of the accessibility relation helps to define the semantics of modal logic, influencing what statements are true or false across different worlds.
Computational linguistics: Computational linguistics is an interdisciplinary field that focuses on the computational aspects of the human language capacity. It combines linguistics, computer science, and artificial intelligence to analyze and model language in a way that machines can understand. This area is crucial for the development of natural language processing systems, which rely on structured models, such as Kripke frames, to interpret meaning and semantics.
David Lewis: David Lewis was a prominent 20th-century philosopher known for his work in modal logic, metaphysics, and philosophy of language. His contributions include the development of modal realism, which posits that possible worlds are just as real as the actual world, a concept that has significant implications for understanding Kripke frames and models in logic.
Euclidean Property: The Euclidean property refers to a specific characteristic of relations in Kripke frames where, if a world can access itself, then it can access any other world that it can access. This property plays a crucial role in modal logic, particularly in the context of defining certain types of logical systems and their semantics. It highlights the nature of accessibility relations between possible worlds, impacting how we interpret modal formulas and the truth conditions associated with them.
Intuitionistic logic: Intuitionistic logic is a form of logic that emphasizes the constructive nature of mathematical proofs, where a statement is only considered true if there is a method to construct an example demonstrating its truth. This approach leads to different interpretations of logical connectives and quantifiers compared to classical logic, making it essential for understanding various proof systems, the foundations of logic, and connections between different logical frameworks.
Kripke frame: A Kripke frame is a mathematical structure used in modal logic to represent the relationships between possible worlds. It consists of a set of worlds and a relation that indicates how these worlds are connected, allowing for the evaluation of modal statements such as necessity and possibility. This structure is fundamental in understanding how different interpretations of truth can be modeled across various contexts.
Kripke Model: A Kripke model is a mathematical structure used in modal logic to evaluate the truth of modal propositions across possible worlds. It consists of a set of possible worlds, a relation between those worlds, and an interpretation function that assigns truth values to propositions in each world. This framework helps analyze notions like necessity and possibility, providing a way to understand how statements can be true in some contexts but not others.
Modal logic: Modal logic is a type of formal logic that extends classical logic to include operators expressing modality, such as necessity and possibility. This allows for reasoning about statements that are not strictly true or false, enabling discussions about what could be, must be, or might have been. Modal logic connects to various areas, enhancing our understanding of semantics, proof structures, and computational applications.
Necessity operator: The necessity operator is a modal logic operator that expresses that a proposition is necessarily true in all possible worlds or contexts. It is typically denoted by the symbol '□' and indicates that if something is necessary, it cannot be false in any conceivable scenario. This concept is crucial for understanding how modal logic interprets statements about possibility and necessity, as it provides a formal way to differentiate between what could be the case versus what must be the case.
Philosophical logic: Philosophical logic is a branch of philosophy that focuses on the application of formal logic to philosophical problems and questions. It seeks to analyze the principles of valid reasoning, truth, and meaning, often intersecting with other areas such as epistemology and metaphysics. This field not only explores traditional logical systems but also investigates non-classical logics that challenge conventional understandings of validity and inference.
Possibility Operator: The possibility operator is a modal operator that expresses the idea that something is possible, typically denoted as \textbf{◇} (diamond). It plays a key role in modal logic by allowing statements about what could be the case, in contrast to what must be the case. This operator is crucial for exploring the nuances of necessity and possibility, especially when combined with other logical expressions.
Possible world semantics: Possible world semantics is a framework used in modal logic to evaluate the truth of propositions in different hypothetical situations, or 'possible worlds.' This approach allows for a richer understanding of modalities such as necessity and possibility by considering how statements hold true across various scenarios, rather than being limited to just the actual world. It connects deeply with Kripke frames and models, where relationships between possible worlds are represented, allowing for a formal analysis of logical systems.
Reflexive frame: A reflexive frame is a specific type of Kripke frame where every world in the frame can access itself. This means that for any world 'w', the relation 'R' satisfies 'w R w'. Reflexive frames are significant because they establish a foundational property for modal logic, particularly in the context of expressing necessity and possibility in a way that reflects self-reference.
Saul Kripke: Saul Kripke is a prominent philosopher and logician known for his influential work in modal logic, semantics, and the philosophy of language. His contributions to the understanding of possible worlds semantics and the formalization of modal logics have reshaped how we interpret necessity and possibility in logical systems. By introducing Kripke frames and models, he provided a systematic approach to semantics that bridged the gap between syntactic formulations and their interpretations in logic.
Seriality: Seriality refers to a property of a relation in Kripke frames that ensures every world in the frame has at least one accessible successor world. This characteristic is crucial for certain modal logics, enabling the representation of various modalities such as necessity and possibility. In the context of Kripke models, seriality ensures that if a world can access another, there is always at least one option for further exploration.
Symmetric relation: A symmetric relation is a type of binary relation in which, for any two elements, if one element is related to the other, then the second element is also related to the first. This property is significant in various contexts, especially when discussing Kripke frames and models, as it influences the accessibility relations between possible worlds. In Kripke semantics, a symmetric relation implies that if a world can access another, then that world can also be accessed back, which affects the truth conditions of modal statements.
Transitive Frame: A transitive frame is a type of Kripke frame where the accessibility relation is transitive, meaning that if a world A accesses world B and world B accesses world C, then world A must also access world C. This property is crucial because it influences the semantics of modal logic, particularly in relation to the validity of certain modal formulas and their interpretations across different possible worlds.
Truth Assignment: A truth assignment is a mapping that assigns a truth value, typically 'true' or 'false', to each propositional variable within a logical formula. This concept is crucial for evaluating the semantics of propositional logic, allowing us to determine the overall truth value of complex expressions based on the truth values of their constituent variables. Understanding truth assignments is key to interpreting models and establishing whether certain interpretations satisfy specific logical formulas.
Validity: Validity refers to the property of an argument whereby, if the premises are true, the conclusion must also be true. This concept is crucial in assessing logical reasoning and its implications across various logical systems, ensuring that arguments lead to accurate conclusions based on their premises.
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