OF CALGARY Modelling Agent Conversations for Action

propositions can be defined by using the universal and existential quantifiers. Theuniversal quantifier (represented by ∀) specifies the properties of all the objects that abideto certain characteristics. For example, to indicate that all natural numbers smaller than 50are also smaller than 100, one could use the expression “∀x: N | x < 50 • x < 100”, whichcan be read as “for all natural numbers x, such as x is smaller than 50, then x is smaller than100.” On the other hand, the existential quantifier (denoted as ∃) is used to indicate thatcertain properties are true for at least one object that abides to certain characteristics. Forexample, to indicate that there is at least one even number smaller than 50, one could usethe expression “∃x: N | x < 50 • x mod 2 = 0”, which can be read as “there exists at leastone natural number x, such that x is smaller than 50, such that the remainder of dividing xby 2 is zero.” Lastly, the unique existential quantifier (denoted as ∃1) is a specialization ofthe existential quantifier that does not only indicate that there exists an element with thespecified characteristics but also that there exists exactly one such element.