Modal Logic
Matt Iverson
Sachin Shinde
Overview
 Introduction
 Description
 Applications
 Proof System
 Syntax
 Semantics
 Inference Rules
 Soundness & Completeness
 Assignment
Introduction
Description
 Modal Logics are logics that include the use of modalities
 Modalities qualify the truth of a judgment
 “John is very happy” instead of “John is happy.”
 Three examples of modalities
 Possibility, probability, and necessity
 Only two are used in traditional modal logic
 ◊ is the symbol for possibility
 □ is the symbol for necessity
 Example
 “If it is possible that it will rain today, then Tom will bring an umbrella”
 ◊ Rain(today) → Bring(tom, umbrella)
Differences from First-Order Logic
 Modal logics allow the qualification of statements
 For most versions of modal logic, there is no way to
represent such concepts formally in FOL.
Applications
 Linguistics
 Gives the ability to translate qualified statements into sentences
in a formal logical system.
 Algebras and Topologies
 Most initial research on modal logic studied the relationship it
had with algebras and topologies.
 Temporal Logic
 An extension of modal logic.
Proof System
Syntax
 CN-calculus
 Symbols
 Conditional (Implication) – “→”
 Negation – “¬”
 Modal CN-calculus
 Extra symbols
 Necessity – “□”
 Possibility – “◊” – defined in terms of necessity.
 ◊φ =df ¬□¬φ
 For the purposes of time and simplicity, quantifiers will not
be used.
Syntax (cont.)
 Different types of modal CN-calculi
 These types have different properties (different from inference
rules).
 If the Rule of Interchange of Equivalents (IE Rule) is valid for a modal
CN-calculus Σ, suppose ψ is an expression of our language that contains
an expression φ, and let ψ’ be the expression obtained from substituting
every occurrence of φ in ψ with an expression φ’. If ⊢Σ (φ ⟷ φ’),
then ⊢Σ (ψ ⟷ ψ’).
 This rule will be true for all modal CN-calculi that we will consider.
 The IE Rule can be proven to be equivalent to the Rule of Replacement
of Equivalents (RE Rule), stated below:
 For all expressions φ, ψ in our language, if ⊢Σ (φ ⟷ ψ), then
⊢Σ (□φ ⟷ □ψ).
Syntax (cont.)
 Rule of Uniform Substitution (US Rule):
For all expressions φ, ψ in our language, if ⊢Σ φ, and there is a logical
constant P in the expression φ, then we can prove the statement
formed by substituting ψ for P in φ, i.e. ⊢Σ φ[P/ ψ]
 Rule of Modal Necessitation (RN Rule):
 For all expressions φ in our language, if ⊢Σ φ, then ⊢Σ □φ

 Different modal CN-calculi use different rules.
 Quasi-Classical CN-calculi
 A modal CN-calculus Σ is quasi-classical if the IE Rule is valid.
 Classical CN-calculi
 A modal CN-calculus Σ is classical if the IE Rule and the US Rule are
valid.
Syntax (cont.)
 Quasi-Regular CN-calculi
 A modal CN-calculus Σ is quasi-regular if it is a quasi-classical modal
CN-calculus and for all expressions φ, ψ, we have
 ⊢Σ □(φ → ψ) → (□φ → □ψ)
 If ⊢Σ (φ → ψ), then ⊢Σ (□φ → □ψ)
 Regular CN-calculi
 A modal CN-calculus Σ is regular if it is a classical modal CN-calculus
and for all expressions φ, ψ, we have
 ⊢Σ □(φ → ψ) → (□φ → □ψ)
 If ⊢Σ (φ → ψ), then ⊢Σ (□φ → □ψ)
Syntax (cont.)
 Quasi-Normal CN-calculi
 A modal CN-calculus Σ is quasi-normal if it is a quasi-regular modal
CN-calculus and the RN rule is valid.
 Normal CN-calculi
 A modal CN-calculus Σ is normal if it is a regular modal CN-calculus and
the RN rule is valid.
 Some Standard Normal Modal CN-calculi
 The Modal System Kr
 Contains the following axioms for all φ, ψ, χ :
 □n(φ → (ψ → φ))
 □n((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)))
 □n((¬φ → ¬ψ) → (ψ → φ))
 □n(□(ψ → φ) → (□ψ → □φ))
Syntax (cont.)
 The Modal System M
 The Modal System Kr with the axiom □n(□φ → φ)
 “What is necessarily the case, simply is the case”
 The Modal System Br
 The Modal System M with the axiom □n(φ → □◊φ)
 “What is the case, is necessarily possibly the case”
 The Modal System S4
 The Modal System M with the axiom □n(□φ → □□φ)
 “What is necessary is not contingently necessary but necessarily
necessary”
Syntax (cont.)
 The Modal System S4.2
 The Modal System S4 with the axiom □n(◊□φ → □◊φ)
 “What is possibly necessary, is necessarily possible”
 The Modal System S4.3
 The Modal System S4 with the axiom:
 □n((◊φ ⋀ ◊ψ) → ◊((φ ⋀ ◊ψ) ⋁ (◊φ ⋀ ψ)))
 “It is a possibility of two possibilities that one is possible relative to the
other”
 The Modal System S5
 The Modal System M with the axiom □n(◊φ → □◊φ)
 “What is possible is not just possible, but necessarily possible”
Semantics
 Two possible interpretations of semantics for S5
 Possible Worlds Semantics
 Use possible worlds to express modal claims.
 For each distinct way the world could have been, there is said to be a
distinct possible world (we live in the actual world).
 True Propositions: true in the actual world.
 False Propositions: false in the actual world.
 Possible Propositions: true in at least one possible world.
 Necessary Propositions: true in all possible worlds.
 Contingent Propositions: true in some possible worlds, and false in
others.
 Impossible Propositions: true in no possible world.
Semantics (cont.)
 Relational (Frame) Semantics
 A related world system (modal frame) 〈W, R〉 consists of two objects
 A non-empty set W of possible worlds
 A binary relation R that holds between those worlds
o Also called the “accessibility relation”
 For a relational world system 〈W, R〉 , define ⊨i P for a logical character
P and a world i ∈ W to be the truth-value assigned to P in world i.
 Inductively define ⊨i φ for all expressions φ as follows:
 ⊨i ¬φ if and only if ⊭i φ
 ⊨i (φ → ψ) if and only if either ⊭i φ or ⊨i ψ
 ⊨i □φ if and only if for all j ∈ W, if iRj then ⊨i φ
 This is an extension of possible worlds semantics, and it can be modified
to be used with other non-classical logics.
Inference Rule
 One Rule!
 Modus Ponens (MP Inference Rule)
 If Γ is a set of sentences in modal CN-calculus, and φ is a sentence in
modal CN-calculus, then φ is a modus ponens consequence of Γ if and
only there exists a sequence Δ1, Δ2, … , Δn such that Δn = φ and for all
i ≤ n, either Δi ∈ Γ or, for some j, k < i, Δk = (Δj → Δi).
 The MP Inference Rule allows us to justify a step from a set of sentences
to another sentence in a proof as long as this other sentence is a modus
ponens consequence of the set of sentences.
Examples
 Show that □φ → □◊φ in modal system M
 “If something is necessary, then it is necessarily possible”
 Show that □◊□φ → □φ in modal system Br.
 “If something is necessarily possibly necessary, then it is
necessary”
 Is this provable in M?
Completeness & Soundness
 Soundness and Completeness for modal system S5 can be
proved using either the formalization of the possible worlds
semantics or the formalization of the Relational Semantics.
 The concept of tautological consequence from FOL gets
replaced in these proofs with a similar concept of L-truth and
L-truth2.
 The Completeness proofs both utilize ideas seen in the proof
of the First-Order Logic, mainly the formation of a formally
consistent and formally complete set (in modal logic, this
becomes maximally complete).
Bibliography
 Cocchiarella, Nino B. Modal Logic – An Introduction to Its
Syntax and Semantics. New York: Oxford University
Press, 2008. Print.
 “Modal Logic”. www.wikipedia.com. n.p.,n.d. Web. 6 Nov.
2009. <http://en.wikipedia.org/wiki/Modal_logic>
 “Modal Logic”. plato.stanford.edu. n.p., 2 Oct. 2009. Web. 6
Nov. 2009. <http://plato.stanford.edu/entries/logicmodal/>