Conditional proof can only be used to deduce a conditional claim
... upon the formula assumed. Or, in other words, what is deduced is always the consequent of the antecedent that was assumed. For example, if you assume p and deduce t you have proven p t, not t by itself. When using conditional proof, any formula derived within the conditional proof lines, that is, ...
... upon the formula assumed. Or, in other words, what is deduced is always the consequent of the antecedent that was assumed. For example, if you assume p and deduce t you have proven p t, not t by itself. When using conditional proof, any formula derived within the conditional proof lines, that is, ...
THE FEFERMAN-VAUGHT THEOREM We give a self
... many disjoint, stationary sets hXn : n < ωi. Define a function f ∈ Gi by, if i ∈ Xn , picking f (i) ∈ Gi to have order larger than n. In particular, n · f (i) 6= 0 for all i ∈ Xn . By the Feferman-Vaught Theorem, for each n < ω, Y Gi /F n · [f ]F = 0 iff {i < ω1 : Gi n · f (i) = 0}Q∈ F . Since t ...
... many disjoint, stationary sets hXn : n < ωi. Define a function f ∈ Gi by, if i ∈ Xn , picking f (i) ∈ Gi to have order larger than n. In particular, n · f (i) 6= 0 for all i ∈ Xn . By the Feferman-Vaught Theorem, for each n < ω, Y Gi /F n · [f ]F = 0 iff {i < ω1 : Gi n · f (i) = 0}Q∈ F . Since t ...
Propositional logic
... Definition: an assignment to a set V of variables is a function s: V Æ {T,F}. Each assignment is inductively extended to apply to wffs. For wffs a and b • s(ÿa) = ÿs(a), • s(aŸb) = s(a) Ÿ s(b), • s(a⁄b) = s(a) ⁄ s(b), • s(afib) = s(a) fi s(b), • s(aÛb) = s(a) Û s(b), and • s(T) = T, s(F) = F, Defini ...
... Definition: an assignment to a set V of variables is a function s: V Æ {T,F}. Each assignment is inductively extended to apply to wffs. For wffs a and b • s(ÿa) = ÿs(a), • s(aŸb) = s(a) Ÿ s(b), • s(a⁄b) = s(a) ⁄ s(b), • s(afib) = s(a) fi s(b), • s(aÛb) = s(a) Û s(b), and • s(T) = T, s(F) = F, Defini ...
Logic Logical Concepts Deduction Concepts Resolution
... Let D be the domain of natural numbers. Consider the formula ∀x∃yP (x, y) In order to evaluate if this formula is true or false, we need to give the predicate symbol P an interpretation Suppose we interpret P as the < relation, i.e., P (x, y) means "x is less than y" Under this interpretation, the f ...
... Let D be the domain of natural numbers. Consider the formula ∀x∃yP (x, y) In order to evaluate if this formula is true or false, we need to give the predicate symbol P an interpretation Suppose we interpret P as the < relation, i.e., P (x, y) means "x is less than y" Under this interpretation, the f ...
Resources - CSE, IIT Bombay
... => and ¬ form a minimal set (can express other operations) - Prove it. Tautologies are formulae whose truth value is always T, whatever the assignment is ...
... => and ¬ form a minimal set (can express other operations) - Prove it. Tautologies are formulae whose truth value is always T, whatever the assignment is ...
1. What is propositional logic? With respect to AI, what is it good for
... b. ( A AND B ), where A and B are a propositional variable ( A AND B ) is a conjunction of A and B. c. ( A OR B ), where A and B are a propositional variable ( A OR B ) is a disjunction of A and B. d. ( A IMPLIES B ), where A and B are a propositional variable ( A IMPLIES B ) is an implication when, ...
... b. ( A AND B ), where A and B are a propositional variable ( A AND B ) is a conjunction of A and B. c. ( A OR B ), where A and B are a propositional variable ( A OR B ) is a disjunction of A and B. d. ( A IMPLIES B ), where A and B are a propositional variable ( A IMPLIES B ) is an implication when, ...
Lecture Notes 2
... Observe that De Morgan’s laws combined with (ii) allow to express ∧ via ∨ and vice versa. For example, (ii) implies that a disjunction U ∨ V , where U , V are variables, can be replaced by an equivalent formula ¬¬U ∨ ¬¬V , which in turn, by the first of De Morgan’s laws is equivalent to ¬(¬U ∧ ¬V ). ...
... Observe that De Morgan’s laws combined with (ii) allow to express ∧ via ∨ and vice versa. For example, (ii) implies that a disjunction U ∨ V , where U , V are variables, can be replaced by an equivalent formula ¬¬U ∨ ¬¬V , which in turn, by the first of De Morgan’s laws is equivalent to ¬(¬U ∧ ¬V ). ...
Natural Deduction Calculus for Quantified Propositional Linear
... of time [Wolper (1981)]. Nevertheless, each of these logics uses its own specific syntax and it makes sense to consider how easy these logics can be used in specification. We believe that in this list QPTL indeed occupies a special place. For example, ETL and linear time µ − calculus formulae are ve ...
... of time [Wolper (1981)]. Nevertheless, each of these logics uses its own specific syntax and it makes sense to consider how easy these logics can be used in specification. We believe that in this list QPTL indeed occupies a special place. For example, ETL and linear time µ − calculus formulae are ve ...
Propositional Logic
... us always true conclusions. The validity of an argument do not depend on the truth of the premises but with the fact that if someone accepts the truth of the premises he/she must accept the conclusion. If someone does not accept the premises, he/she wont accept the conclusions but this does not inva ...
... us always true conclusions. The validity of an argument do not depend on the truth of the premises but with the fact that if someone accepts the truth of the premises he/she must accept the conclusion. If someone does not accept the premises, he/she wont accept the conclusions but this does not inva ...
Ch1 - COW :: Ceng
... circuit design There are efficient algorithms for reasoning in propositional logic Propositional logic is a foundation for most of the more expressive logics ...
... circuit design There are efficient algorithms for reasoning in propositional logic Propositional logic is a foundation for most of the more expressive logics ...
Propositional Logic Predicate Logic
... A variable that represents a predicate with variables x1 , . . . , xn . ∀x.A “For any x, A” A is true for all individuals x. ∃x.A “There exists x s.t. A” B is true for some individual x. We also use individual constant a, b, c, etc. For some specific theories, we may write ∀x ∈ X.A or ∃x ∈ X.A to sp ...
... A variable that represents a predicate with variables x1 , . . . , xn . ∀x.A “For any x, A” A is true for all individuals x. ∃x.A “There exists x s.t. A” B is true for some individual x. We also use individual constant a, b, c, etc. For some specific theories, we may write ∀x ∈ X.A or ∃x ∈ X.A to sp ...
x, y, x
... Logical Equivalences for Predicates We can define valid, satisfiable and unsatisfiable in the same manner as with propositional logic. Let F be a formula of a first order language L. We say F is: 1. valid or a tautology if it is satisfied by every interpretation of L 2. satisfiable if some interpre ...
... Logical Equivalences for Predicates We can define valid, satisfiable and unsatisfiable in the same manner as with propositional logic. Let F be a formula of a first order language L. We say F is: 1. valid or a tautology if it is satisfied by every interpretation of L 2. satisfiable if some interpre ...
Predicate Logic
... Logical Equivalences for Predicates We can define valid, satisfiable and unsatisfiable in the same manner as with propositional logic. Let F be a formula of a first order language L. We say F is: 1. valid or a tautology if it is satisfied by every interpretation of L 2. satisfiable if some interpre ...
... Logical Equivalences for Predicates We can define valid, satisfiable and unsatisfiable in the same manner as with propositional logic. Let F be a formula of a first order language L. We say F is: 1. valid or a tautology if it is satisfied by every interpretation of L 2. satisfiable if some interpre ...
coppin chapter 07e
... Allows us to reason about certainties, and possible worlds. If a statement A is contingent then we say that A is possibly true, which is written: ◊A If A is non-contingent, then it is necessarily true, which is written: A ...
... Allows us to reason about certainties, and possible worlds. If a statement A is contingent then we say that A is possibly true, which is written: ◊A If A is non-contingent, then it is necessarily true, which is written: A ...
Programming and Problem Solving with Java: Chapter 14
... Allows us to reason about certainties, and possible worlds. If a statement A is contingent then we say that A is possibly true, which is written: ◊A If A is non-contingent, then it is necessarily true, which is written: A ...
... Allows us to reason about certainties, and possible worlds. If a statement A is contingent then we say that A is possibly true, which is written: ◊A If A is non-contingent, then it is necessarily true, which is written: A ...
4 slides/page
... • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the following: Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence thi ...
... • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the following: Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence thi ...
Friedman`s Translation
... Theorem 1.5. If ` A is derivable in classical predicate logic and if no free variable of R occurs in the derivation, then ` A¬R is derivable in intuitionistic predicate logic. In order to obtain Theorem 1.5 for arithmetic, it remains to show that HA∗ proves the ¬R -translation of all its axioms. The ...
... Theorem 1.5. If ` A is derivable in classical predicate logic and if no free variable of R occurs in the derivation, then ` A¬R is derivable in intuitionistic predicate logic. In order to obtain Theorem 1.5 for arithmetic, it remains to show that HA∗ proves the ¬R -translation of all its axioms. The ...
1
... 1. (a) Identify the free and bound variable occurrences in the following logical formulas: • ∀x∃y(Rxz ∧ ∃zQyxz), • ∀x((∃yRxy→Ax)→Bxy), • ∀x(Ax→∃yBy) ∧ ∃z(Cxz→∃xDxyz). (b) Give the definition of a atomic formula of predicate logic and of a valuation of terms s based on a variable assignment s. (c) Pr ...
... 1. (a) Identify the free and bound variable occurrences in the following logical formulas: • ∀x∃y(Rxz ∧ ∃zQyxz), • ∀x((∃yRxy→Ax)→Bxy), • ∀x(Ax→∃yBy) ∧ ∃z(Cxz→∃xDxyz). (b) Give the definition of a atomic formula of predicate logic and of a valuation of terms s based on a variable assignment s. (c) Pr ...
Slides
... Instantiating the framework for a specific logic L, requires a deductive system for L that meets several criteria. Linear arithmetic, EUF, arrays etc all have it. ...
... Instantiating the framework for a specific logic L, requires a deductive system for L that meets several criteria. Linear arithmetic, EUF, arrays etc all have it. ...
Section 6.1 How Do We Reason? We make arguments, where an
... followed by a single statement, called the conclusion. The hope is that we make valid arguments, where an argument is valid if the truth of the premises implies the truth of the conclusion. We can use rules of logic to make valid arguments. The most common rule of logic is modus ponens (mode that af ...
... followed by a single statement, called the conclusion. The hope is that we make valid arguments, where an argument is valid if the truth of the premises implies the truth of the conclusion. We can use rules of logic to make valid arguments. The most common rule of logic is modus ponens (mode that af ...
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) ...
A Subatomic Proof System - Department of Computer Science
... Φ ∨ Ψ cannot. This is basically the definition of deep inference and it holds for every language, not just propositional classical logic. A further advantage of deep inference is that, contrary to Gentzen theory [10], self-dual noncommutative connectives such as the ones that we use for atoms here c ...
... Φ ∨ Ψ cannot. This is basically the definition of deep inference and it holds for every language, not just propositional classical logic. A further advantage of deep inference is that, contrary to Gentzen theory [10], self-dual noncommutative connectives such as the ones that we use for atoms here c ...
Bound and Free Variables Theorems and Proofs
... Suppose we have a random graph with n vertices. How likely is it to be connected? • What is a random graph? ◦ If it has n vertices, there are C(n, 2) possible edges, and 2C(n,2) possible graphs. What fraction of them is connected? ◦ One way of thinking about this. Build a graph using a random proces ...
... Suppose we have a random graph with n vertices. How likely is it to be connected? • What is a random graph? ◦ If it has n vertices, there are C(n, 2) possible edges, and 2C(n,2) possible graphs. What fraction of them is connected? ◦ One way of thinking about this. Build a graph using a random proces ...
Notes
... exactly to the proof rules of propositional intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponent ...
... exactly to the proof rules of propositional intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponent ...
hilbert systems - CSA
... S U {~X} is also Consistent If not, S U {~X} |- X S |- (~X > X) { Deduction Theorem } (~X > X) > X { Theorem } S |- X { Modus Ponens } S U {~X} is satisfiable { Model Existence Theorem } S U {X} is not valid. ...
... S U {~X} is also Consistent If not, S U {~X} |- X S |- (~X > X) { Deduction Theorem } (~X > X) > X { Theorem } S |- X { Modus Ponens } S U {~X} is satisfiable { Model Existence Theorem } S U {X} is not valid. ...