Slides - UCSD CSE
... Prove that there is no largest integer (p) Assume, to the contrary that ______________________ (~p) Then, __________________________________ (formula that follows from p) Now, _________________________ (p " ~p) ...
... Prove that there is no largest integer (p) Assume, to the contrary that ______________________ (~p) Then, __________________________________ (formula that follows from p) Now, _________________________ (p " ~p) ...
HISTORY OF LOGIC
... The Logicist Period Gottlob Frege (1848 – 1925): – Considered to be the father of Analytic Philosophy. – His Objective was demonstrating that arithmetic is identical with logic. – He invented axiomatic predicate logic and quantified variables, which solved the problem of multiple generality. ...
... The Logicist Period Gottlob Frege (1848 – 1925): – Considered to be the father of Analytic Philosophy. – His Objective was demonstrating that arithmetic is identical with logic. – He invented axiomatic predicate logic and quantified variables, which solved the problem of multiple generality. ...
Negative translation - Homepages of UvA/FNWI staff
... Definition 1.1. Let ∇ be a function mapping formulas in propositional logic to formulas in propositional logic. Such a mapping is called a nucleus if the following statements are provable in intuitionistic logic: `IL ϕ → ∇ϕ `IL ∇(ϕ ∧ ψ) ↔ ( ∇ϕ ∧ ∇ψ ) `IL (ϕ → ∇ψ) → (∇ϕ → ∇ψ) Exercise 1. (a) Show tha ...
... Definition 1.1. Let ∇ be a function mapping formulas in propositional logic to formulas in propositional logic. Such a mapping is called a nucleus if the following statements are provable in intuitionistic logic: `IL ϕ → ∇ϕ `IL ∇(ϕ ∧ ψ) ↔ ( ∇ϕ ∧ ∇ψ ) `IL (ϕ → ∇ψ) → (∇ϕ → ∇ψ) Exercise 1. (a) Show tha ...
Exercises: Sufficiently expressive/strong
... (a) Suppose G is a sentence of T ’s language which is true iff G is not provable in T : can T decide G? (b) Suppose H is a sentence of T ’s language which is true iff H is provable in T : can T decide H? (c) (Looking ahead, but try thinking about it!) Suppose M is a sentence of T ’s language which i ...
... (a) Suppose G is a sentence of T ’s language which is true iff G is not provable in T : can T decide G? (b) Suppose H is a sentence of T ’s language which is true iff H is provable in T : can T decide H? (c) (Looking ahead, but try thinking about it!) Suppose M is a sentence of T ’s language which i ...
Plural Quantifiers
... that domain. A model of a set of axioms is a model that makes these axioms true. Now consider a set of first-order axioms for arithmetic (such as the standard Peano axioms). These axioms will contain some arithmetical expressions, like ‘0’, ‘S’, ‘+’, and ‘<’. The standard model of arithmetic interpr ...
... that domain. A model of a set of axioms is a model that makes these axioms true. Now consider a set of first-order axioms for arithmetic (such as the standard Peano axioms). These axioms will contain some arithmetical expressions, like ‘0’, ‘S’, ‘+’, and ‘<’. The standard model of arithmetic interpr ...
Decidable fragments of first-order logic Decidable fragments of first
... Fix any subset {b1 , b2 } of Bn of size r ∈ {1, 2} and recall that some r -table T{b1 ,b2 } from Tr is assiged to this subset. For any subset {b3 , . . . , bl+2 } of pairwise distinct elements of Bn that differ from b1 and b2 , consider the event that the table induced by b1 , . . . , bl+2 is equal ...
... Fix any subset {b1 , b2 } of Bn of size r ∈ {1, 2} and recall that some r -table T{b1 ,b2 } from Tr is assiged to this subset. For any subset {b3 , . . . , bl+2 } of pairwise distinct elements of Bn that differ from b1 and b2 , consider the event that the table induced by b1 , . . . , bl+2 is equal ...
Intro to First
... R mean, what “universe” of objects are the quantifiers ranging over. Note since there are no free variables, we need not be ask for the variable assignment. But for now we will restrict our attention to propositions only, i.e. formulas with no free variables. A theory constrains the meaning of const ...
... R mean, what “universe” of objects are the quantifiers ranging over. Note since there are no free variables, we need not be ask for the variable assignment. But for now we will restrict our attention to propositions only, i.e. formulas with no free variables. A theory constrains the meaning of const ...
Full version - Villanova Computer Science
... • The Identity rule of G1 can be called an axiom, as it takes no premises. • The rules -Introduction and -Introduction are called logical rules as they “explain” the logical meanings of connectives. • The remaining rules of Exchange, Weakening and Contraction are called structural rules, as they o ...
... • The Identity rule of G1 can be called an axiom, as it takes no premises. • The rules -Introduction and -Introduction are called logical rules as they “explain” the logical meanings of connectives. • The remaining rules of Exchange, Weakening and Contraction are called structural rules, as they o ...
lec5 - Indian Institute of Technology Kharagpur
... The agent can shoot the wumpus along a straight line The agent has only one arrow ...
... The agent can shoot the wumpus along a straight line The agent has only one arrow ...
Exam-Computational_Logic-Subjects_2016
... The theorem of deduction and its reverse. 7. Definitions: tautology, theorem, logical consequence, syntactic consequence, logical equivalence, consistent/contingent/valid/inconsistent formula, interpretation, model, anti-model. The axiomatic system of propositional logic. The axiomatic system of pro ...
... The theorem of deduction and its reverse. 7. Definitions: tautology, theorem, logical consequence, syntactic consequence, logical equivalence, consistent/contingent/valid/inconsistent formula, interpretation, model, anti-model. The axiomatic system of propositional logic. The axiomatic system of pro ...
Predicate Logic - Teaching-WIKI
... • Apply modus ponens reasoning to generalized rules • Combines And-Introduction, Universal-Elimination, and Modus Ponens – E.g, from P(c) and Q(c) and x (P(x) Q(x)) → R(x) derive R(c) ...
... • Apply modus ponens reasoning to generalized rules • Combines And-Introduction, Universal-Elimination, and Modus Ponens – E.g, from P(c) and Q(c) and x (P(x) Q(x)) → R(x) derive R(c) ...
pdf
... argument expressed in a syllogism. Leibniz (1666) explained the potential of logic to create a universal language for codifying human knowledge and mechanically resolving scientific and legal questions. He accepted the idea that the mind works by computation. Boole built on those ideas and created a ...
... argument expressed in a syllogism. Leibniz (1666) explained the potential of logic to create a universal language for codifying human knowledge and mechanically resolving scientific and legal questions. He accepted the idea that the mind works by computation. Boole built on those ideas and created a ...
Examples of Ground Resolution Proofs 1 Ground Resolution
... The following is a slight generalisation of the version of the Ground Resolution Theorem proved in the last lecture. Before we considered only a single formula in Skolem form. Here we consider a conjunction of such formulas, which is more convenient for the applications below. Theorem 2 (Ground Reso ...
... The following is a slight generalisation of the version of the Ground Resolution Theorem proved in the last lecture. Before we considered only a single formula in Skolem form. Here we consider a conjunction of such formulas, which is more convenient for the applications below. Theorem 2 (Ground Reso ...
Lecture 10: A Digression on Absoluteness
... One was shown by some guy using structures with constants, or something like that. One was shown by some other guy using ultraproducts, whatever those are (we might see this later in the course). Finally, there are topological methods involving scary things named for people. Proof. We are now in a p ...
... One was shown by some guy using structures with constants, or something like that. One was shown by some other guy using ultraproducts, whatever those are (we might see this later in the course). Finally, there are topological methods involving scary things named for people. Proof. We are now in a p ...
Notes Predicate Logic II
... φ1 , . . . , φ n ` ψ The theorem states that every valid sequent can be proven, and every sequent that can be proven is valid. This theorem was proven by Kurt Gödel in 1929 in his doctoral dissertation. A description of his proof, as well as the proofs of the following theorems, is beyond the scope ...
... φ1 , . . . , φ n ` ψ The theorem states that every valid sequent can be proven, and every sequent that can be proven is valid. This theorem was proven by Kurt Gödel in 1929 in his doctoral dissertation. A description of his proof, as well as the proofs of the following theorems, is beyond the scope ...
Tautologies Arguments Logical Implication
... Socrates is a man Therefore Socrates is mortal There is two unary predicates: Mortal and Man There is one constant: Socrates The domain is the set of all people ∀x(M an(x) ⇒ M ortal(x)) M an(Socrates) ...
... Socrates is a man Therefore Socrates is mortal There is two unary predicates: Mortal and Man There is one constant: Socrates The domain is the set of all people ∀x(M an(x) ⇒ M ortal(x)) M an(Socrates) ...
Exam #2 Wednesday, April 6
... There are no further clauses to be obtained from these by resolution. If we use the Davis-Putnam Procedure, first eliminating P to get {Q} and then Q to get no clauses, we also see that the original formula is not valid. 3. (P -> Q) -> ( (P -> R ) -> (Q -> R)) The negation of the formula in CNF is: ...
... There are no further clauses to be obtained from these by resolution. If we use the Davis-Putnam Procedure, first eliminating P to get {Q} and then Q to get no clauses, we also see that the original formula is not valid. 3. (P -> Q) -> ( (P -> R ) -> (Q -> R)) The negation of the formula in CNF is: ...
coppin chapter 07e
... Propositional Calculus is the language we use to reason about propositional logic. A sentence in propositional logic is called a well-formed formula (wff). ...
... Propositional Calculus is the language we use to reason about propositional logic. A sentence in propositional logic is called a well-formed formula (wff). ...
IntroToLogic - Department of Computer Science
... In any logical language expressive enough to describe the properties of the natural numbers, there are true statements that are undecidable -- their truth cannot be established by any algorithm. ...
... In any logical language expressive enough to describe the properties of the natural numbers, there are true statements that are undecidable -- their truth cannot be established by any algorithm. ...
POSSIBLE WORLDS AND MANY TRUTH VALUES
... using only the connectives ¬, ∨, – if β is not a variable, then τa β can be replaced by a compound (using just ¬, ∨, ) of formulas τb γ with γ shorter than β. Finitely many replacements of this sort lead to a formula α00 having only standard connectives and , in which τa ’s apply only to variabl ...
... using only the connectives ¬, ∨, – if β is not a variable, then τa β can be replaced by a compound (using just ¬, ∨, ) of formulas τb γ with γ shorter than β. Finitely many replacements of this sort lead to a formula α00 having only standard connectives and , in which τa ’s apply only to variabl ...
Lecture 14 Notes
... calculate the value of a composed formula from values of the subformulas. In first-order logic, our starting point has to be atomic formulas instead of propositional variables and we have to explain how to calculate the value of quantified formulas that may have infinitely many subformulas. The stan ...
... calculate the value of a composed formula from values of the subformulas. In first-order logic, our starting point has to be atomic formulas instead of propositional variables and we have to explain how to calculate the value of quantified formulas that may have infinitely many subformulas. The stan ...
logical axiom
... ponens”, which states that from formulas A and A → B, one my deduce B. It is easy to see that this rule preserves logical validity. The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In addition, it is also complete. Note that in the above se ...
... ponens”, which states that from formulas A and A → B, one my deduce B. It is easy to see that this rule preserves logical validity. The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In addition, it is also complete. Note that in the above se ...
First-order logic;
... Bottom line: If n is large, then it is almost certain that a random graph will be connected. In fact, with probability approaching 1, all nodes are connected by a path of length at most 2. ...
... Bottom line: If n is large, then it is almost certain that a random graph will be connected. In fact, with probability approaching 1, all nodes are connected by a path of length at most 2. ...
Predicate logic
... • Interpretation – Maps symbols of the formal language (predicates, functions, variables, constants) onto objects, relations, and functions of the “world” (formally: Domain, relational Structure, or Universe) • Valuation – Assigns domain objects to variables – The Valuation function can be used for ...
... • Interpretation – Maps symbols of the formal language (predicates, functions, variables, constants) onto objects, relations, and functions of the “world” (formally: Domain, relational Structure, or Universe) • Valuation – Assigns domain objects to variables – The Valuation function can be used for ...