Completeness in modal logic - Lund University Publications
... Scott-Montague semantics are then put in a different yet equivalent formulation, where the “N”-function from worlds to classes of subsets of W is replaced by the function “f”, simply from subsets of W to subsets of W. The idea is that when the evaluation on a Scott-Montague frame is set, all we need ...
... Scott-Montague semantics are then put in a different yet equivalent formulation, where the “N”-function from worlds to classes of subsets of W is replaced by the function “f”, simply from subsets of W to subsets of W. The idea is that when the evaluation on a Scott-Montague frame is set, all we need ...
210ch2 - Dr. Djamel Bouchaffra
... Note: f associates with each x in A one and only one y in B. A is called the domain and B is called the codomain. If f(x) = y y is called the image of x under f x is called a preimage of y (note there may be more than one preimage of y but there is only one image of x). The range of f is the set of ...
... Note: f associates with each x in A one and only one y in B. A is called the domain and B is called the codomain. If f(x) = y y is called the image of x under f x is called a preimage of y (note there may be more than one preimage of y but there is only one image of x). The range of f is the set of ...
(pdf)
... time. Similarly, if an irrational number is in the range of a function, it would be impossible to write the number in a finite amount of time. Because of this, it makes sense to restrict our questions about the computability of a function to functions from rational numbers to rational numbers. Becau ...
... time. Similarly, if an irrational number is in the range of a function, it would be impossible to write the number in a finite amount of time. Because of this, it makes sense to restrict our questions about the computability of a function to functions from rational numbers to rational numbers. Becau ...
BASIC COUNTING - Mathematical sciences
... • Set Theory: Informally we define a set as a collection of objects. The resulting theory of how one can operate on sets is known as naïve set theory. It is naïve because the informal definition leads to subtle paradoxes. A more careful definition of set removes these paradoxes and leaves the conclu ...
... • Set Theory: Informally we define a set as a collection of objects. The resulting theory of how one can operate on sets is known as naïve set theory. It is naïve because the informal definition leads to subtle paradoxes. A more careful definition of set removes these paradoxes and leaves the conclu ...
appendix-1
... Notice that the statement mentions that a card with an even number on one side has a vowel on the other. It does not state that a card with a vowel on one side must have an even number on the other side. That may or may not be so. The rule also does not state that a card with an odd number on one si ...
... Notice that the statement mentions that a card with an even number on one side has a vowel on the other. It does not state that a card with a vowel on one side must have an even number on the other side. That may or may not be so. The rule also does not state that a card with an odd number on one si ...
Sets with dependent elements: Elaborating on Castoriadis` notion of
... In his seminal work [2], C. Castoriadis1 is concerned, among many other things, with the way collections of things are presented to (or created by) human mind. In particular, he aims at making us rethink of the “self-evident” belief of western rationalistic tradition that a collection needs always t ...
... In his seminal work [2], C. Castoriadis1 is concerned, among many other things, with the way collections of things are presented to (or created by) human mind. In particular, he aims at making us rethink of the “self-evident” belief of western rationalistic tradition that a collection needs always t ...
Hilbert Calculus
... By induction on the structure of F (and using Lemma III): Atomic formulas: F = A. Easy. Negation: F = ¬G. We have: A(F ) = 1 iff A(G) = 0 iff G 6∈ S iff ¬G ∈ S iff F ∈ S. Implication: F = F1 → F2 . We have: A(F ) = 1 iff A(F1 → F2 ) = 1 iff (A(F1 ) = 0 or A(F2 ) = 1) iff (F1 6∈ S or F2 ∈ S) iff F1 → ...
... By induction on the structure of F (and using Lemma III): Atomic formulas: F = A. Easy. Negation: F = ¬G. We have: A(F ) = 1 iff A(G) = 0 iff G 6∈ S iff ¬G ∈ S iff F ∈ S. Implication: F = F1 → F2 . We have: A(F ) = 1 iff A(F1 → F2 ) = 1 iff (A(F1 ) = 0 or A(F2 ) = 1) iff (F1 6∈ S or F2 ∈ S) iff F1 → ...
Resources - CSE, IIT Bombay
... - Every interpretation I of S contains an interpretation I’ of S’ - So, if I’ falsifies S’, then I must also falsify S’ - Since S’ is falsified by every interpretation I’, it must also be falsified by every interpretation I of S - i.e. S is falsified by every interpretation of S - Hence S is unsatis ...
... - Every interpretation I of S contains an interpretation I’ of S’ - So, if I’ falsifies S’, then I must also falsify S’ - Since S’ is falsified by every interpretation I’, it must also be falsified by every interpretation I of S - i.e. S is falsified by every interpretation of S - Hence S is unsatis ...
pptx - CSE, IIT Bombay
... • We investigated the applicability of logic as a language for the representation of a number of medical reasoning models. • It was shown that the language of first-order predicate logic allowed for the precise, and compact, representation of these models. • Generally, in translating domain knowledg ...
... • We investigated the applicability of logic as a language for the representation of a number of medical reasoning models. • It was shown that the language of first-order predicate logic allowed for the precise, and compact, representation of these models. • Generally, in translating domain knowledg ...
FORMALIZATION OF HILBERT`S GEOMETRY OF INCIDENCE AND
... the following task 1: “To connect two points with a line, and to find the intersection point of two lines in case they are not parallel” (1899, 78). Now, the second part is quite problematic, as parallelism is only introduced in Axiom III: “In a plane with a point A outside a line a one and only o ...
... the following task 1: “To connect two points with a line, and to find the intersection point of two lines in case they are not parallel” (1899, 78). Now, the second part is quite problematic, as parallelism is only introduced in Axiom III: “In a plane with a point A outside a line a one and only o ...
The equational theory of N, 0, 1, +, ×, ↑ is decidable, but not finitely
... this research, namely the theory of integers under sum, product, exponentiation with two constants for zero and one. This paper closes this long standing open problem, by providing an elementary proof, relying on previous work of R. Gurevič, of the fact that Tarski’s original system is decidable, y ...
... this research, namely the theory of integers under sum, product, exponentiation with two constants for zero and one. This paper closes this long standing open problem, by providing an elementary proof, relying on previous work of R. Gurevič, of the fact that Tarski’s original system is decidable, y ...
duality of quantifiers ¬8xA(x) 9x¬A(x) ¬9xA(x) 8x¬A(x)
... There are problems that cannot be solved by computer programs (i.e. algorithms) even assuming unlimited time and space. What is an “algorithm”? The following are all equivalent: - C programs, scheme programs, Java programs . . . - Turing machines (Turing’s idea of an “algorithm”) ...
... There are problems that cannot be solved by computer programs (i.e. algorithms) even assuming unlimited time and space. What is an “algorithm”? The following are all equivalent: - C programs, scheme programs, Java programs . . . - Turing machines (Turing’s idea of an “algorithm”) ...
A Short Glossary of Metaphysics
... in or among which constitute the event. Some philosophers use ‘event’ very broadly for all occurrents. existence From ex-sistere, ex-stare, Latin for ‘to stand out’, a relatively late coinage. The fundamental notion of metaphysics. There are numerous theories as to what existence is, but it is prob ...
... in or among which constitute the event. Some philosophers use ‘event’ very broadly for all occurrents. existence From ex-sistere, ex-stare, Latin for ‘to stand out’, a relatively late coinage. The fundamental notion of metaphysics. There are numerous theories as to what existence is, but it is prob ...
Lecture Notes 2
... mathematicians, is just as rigorous. It consists of sentences describing the situation at hand, the inferences being made, and the justification of each inference. ...
... mathematicians, is just as rigorous. It consists of sentences describing the situation at hand, the inferences being made, and the justification of each inference. ...
Elementary Logic
... A (propositional) sequent is an expression of the form Γ ` ∆, where Γ = A1 , A2 , · · · , Am and ∆ = B1 , B2 , · · · , Bn are finite (possibly empty) sequences of (propositional) formulae. In a sequent Γ ` ∆, Γ is called the antecedent (also context) and ∆ the consequent. Note: many authors prefer t ...
... A (propositional) sequent is an expression of the form Γ ` ∆, where Γ = A1 , A2 , · · · , Am and ∆ = B1 , B2 , · · · , Bn are finite (possibly empty) sequences of (propositional) formulae. In a sequent Γ ` ∆, Γ is called the antecedent (also context) and ∆ the consequent. Note: many authors prefer t ...
1 The calculus of “predicates”
... adds to the language of the propositional calculus: names of individuals belonging to some domain or universe of discourse; variables standing for these names (ranging over the domain), predicate symbols, and quantifiers. In first-order logic there are also function symbols, but we concentrate for p ...
... adds to the language of the propositional calculus: names of individuals belonging to some domain or universe of discourse; variables standing for these names (ranging over the domain), predicate symbols, and quantifiers. In first-order logic there are also function symbols, but we concentrate for p ...
slides
... If there are infinitely many possible values for X the meaning of this expression cannot be represented using a propositional formula. In AG, the meaning of aggregate expressions is captured using an infinitary propositional formula. The definition is based on the semantics for propositional aggrega ...
... If there are infinitely many possible values for X the meaning of this expression cannot be represented using a propositional formula. In AG, the meaning of aggregate expressions is captured using an infinitary propositional formula. The definition is based on the semantics for propositional aggrega ...
Sam Sanders
... E-mail address: [email protected], http://cage.ugent.be/∼sasander. 1Note that the terms ‘weak’, ‘natural’ and ‘strong’ are used informally here, and relative to the context of RCA0 . ...
... E-mail address: [email protected], http://cage.ugent.be/∼sasander. 1Note that the terms ‘weak’, ‘natural’ and ‘strong’ are used informally here, and relative to the context of RCA0 . ...
