Glossary of Terms

Algebra
Algebra is a branch of mathematics that deals with relations between quantities represented as symbols or letters.
 
Bijective
A bijective function is both injective and surjective.
 
Disjoint Unions
A disjoint union is formed by taking mutually disjoint copies of each item to be included and combining them into one set. If two items are not mutually disjoint, the domains are first indexed to achieve disjoint sets.
Epistemic
The word itself means the following:
Of, relating to, or involving knowledge; cognitive.
In the modal sense, epistemic logic is used to describe the logic of systems and agents concerned with knowledge. The most common forms of epistemic statements are:

  • "It is certainly true that..."
  • "It may (given the available information) be true that..."

    For examples and more details refer to WordIQ
     
    • Extensional vs. Intensional
    While reading from Section 2.3 - The Lewis Systems - in Goldblatt's "Mathematical Modal Logic: A view of its evolution" (large file) we encountered the notion of extensional and intensional meanings of disjunction. What we learned is the following: From the paper, taking "a implies b" to mean that "either not-a or b", we have different phrases to describe the same thing. The intensional meaning of disjunction is that "at least one of the disjoined propositions is necessarily true." On the other hand, the extensional meaning states that "it is false that a is true and b is false". The subtle difference is that the intensional definition does not explicitly enumerate elements belonging to the set whereas the extensional definition does. Something significant about the difference between intensional and extensional when defining sets is that an extensional definition explicitly enumerates the elements and the set must therefore be finite. Using an intensional definition, on the other hand, allows for infinite sets. In 1932, C. I. Lewis made this observation:
    the expositors of the algebra of logic have not always taken pains to indicate that there is a difference between the algebraic and orinary meanings implication
    -- From Goldblatt quoting Lewis' book Symbolic logic

     
    Falsifying Model
    Falsifying models are models such that given a formula F, a model M in a world w can be created such that the formula F is invalid. If this is the case, M is the falsifying model for the given formula F.
     
    Finite Model Property
    The finite model property states that if a modal formula is satisfiable on an arbitrary model, then it is satisfiable on a finite model.
     
    Frame
    A frame is a pair (W, R) where W is a non-empty set and R is a binary relation on W.
     
    Homomorphism
    A homomorphism is a function f that maps from one mathematical structure (the source) to another (the target) and preserves the structure of the first. Homomorphisms allow us to conclude that all the relations of the source are present in the target, but not necessarily vice-versa.
     
    Image Finite
    Let T be a modal similarity type and M a T-model. M is image-finite if for each state u in M and each relation R in M, the set {(v1,....vn)|Ruv1...vn) is finited. There are no restrictions on the total number of relations in R in the model M just that each of them is image finite.
    Injective
    An injective function f:A -> B is one that maps elements of the set A to the set B such that for every x in A, f(x) is in B. There may be elements in B that are not reachable by f(x). This is also called a one-to-one mapping.
     
    Isomorphism
    An isomorphism is a bijective strong homomorphism.
     
    Labeled Transition System
    A labeled transition system or simply a transition system is a relational structure which consists of a pair of W non empty state set and a set of A labels which is non-empty as well. They can be viewed as an abstract way of representing computation. The states are the possible states of the computer and the labels are the programs that can be executed to go from one state to the other.
    Logic
    Logic is a branch of philosophy concerned with classifying the structure of arguments.
     
    Modality
    Modality refers to the sentential operator associated with a propositional statement that classifies the statement as "necessary" or "possibly".
     
    Modally Equivalent
    Modal equivalence is satisified for two particular models when for all modal formulas F, M, w satisifies F i
    Modal Logic - Another definition
    Modal logic is the bisimulation invariant fragment of first-order-logic.
    Model
    A model in modal logic is a pair (F, V), where F is a frame and V is a valuation function that labels the points in the model where some p is true.
     
    Morphism
    Same as a homomorphism.
     
    n-Bismulations
    We say that w and w' are n-bimilar for two models M and M' if there exists a sequence of binary relations Zn superset of Zn-1....superset of Z0 with the following properties:
    1. w Znw'
    2. if vZ0v' then v and v' agree on all propositional letters
    3. if VZi+1v' and Rvu, then there exists u' with R'v'u' and uZiu'
    4. if vZi+1v' and R'v'u', then there exists u with Rvu and uZiu'
    Intuitively, if w is b-similar to w', then w and w' bi-bimulate to a depth of n. The converse is not necessarily true.
    Nabla
  • The upside-down capital delta symbol , also called "del" used to denote the gradient and other vector derivatives
  • The dual of a modal operator of arity 2
  • In the modal logic sense, the dual of a triangle <> is the [] and vice versa.
     
    • Necessitation
    The rule of necessitation in modal logic basically means that anything that can be derived from necessarily true statements must also be a necessarily true statement.
     
    Normal Logic
    In logic, normal modal logic is a set L of modal formulas such that L contains
  • all propositional tautologies,
  • Kripke's schema: []A \to B) \to (\Box A \to \Box B), and L is closed under
  • substitution,
  • detachment rule: from A and A¢ªB infer B,
  • necessitation rule: from A infer \Box A. The minimal normal modal logic is known as K.
     
    • Relational Structure
    A relational structure is a tuple T whose first component is a non-empty set W called the universe or domain. The remaining components are relations on W. The elements of W can have a variety of names such as points, states, nodes, worlds, times, instants and situations.
    Sentential Operators
    A sentential operator is a grammatical operator that is applied to sentences to form more complex sentences. These are divided into the following types:

  • the negative operator "not,"
  • the conjunctive operator "and,"
  • the disjunctive operator "or,"
  • the conditional operator "if"

    Additional sentential operators may also be studied in Propositional Logic, such as the modal operators "necessary" and "possible". Sentential operators are also called connectives.

    courtesy: Link
     
    • Surjective
    A surjective function f:A -> B is one that maps elements of the set A to the set B such that for every f(x) in B, there is a x in A. There may be more elements in A that do not map to an element of B. This is also called an onto mapping.
     
    Strong Homomorphism
    Strong homomorphisms entail stronger versions of the morphism which include relationships which have to be present in the original as well as the homomorphism of the model. The definition of an isomorphism is modified to read the following:
    The strong homomorphism is from M into M' and f:M->M'
    1. For each proposition p and element w from M, w belongs to  V(p) iff f(w) belongs to V'(p)
    2. For each n >= 0 and each n-ary operator in T, and (n+1)-tuple w from M, (w0, w1....wn) belong to R iff (f(w0), f(w1)....f(wn)) belong to R'. (strong homomorphic condition)