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 ...
Classical First-Order Logic Introduction
... First-order logic (FOL) is a richer language than propositional logic. Its lexicon contains not only the symbols ∧, ∨, ¬, and → (and parentheses) from propositional logic, but also the symbols ∃ and ∀ for “there exists” and “for all”, along with various symbols to represent variables, constants, fun ...
... First-order logic (FOL) is a richer language than propositional logic. Its lexicon contains not only the symbols ∧, ∨, ¬, and → (and parentheses) from propositional logic, but also the symbols ∃ and ∀ for “there exists” and “for all”, along with various symbols to represent variables, constants, fun ...
Palo Alto 2016 - Stanford Introduction to Logic
... currently inefficient because it is first adding the elements to the array, then going back through the array again and evaluating the elements. Instead, the e.evaluate(i) command should be called inside the first for loop. In order to keep track of the index i properly, we can create another varia ...
... currently inefficient because it is first adding the elements to the array, then going back through the array again and evaluating the elements. Instead, the e.evaluate(i) command should be called inside the first for loop. In order to keep track of the index i properly, we can create another varia ...
First-Order Predicate Logic (2) - Department of Computer Science
... Theorem There exists a computer program that outputs exactly the tautologies of first-order predicate logic. ...
... Theorem There exists a computer program that outputs exactly the tautologies of first-order predicate logic. ...
slides
... Don’t get confused! The symbol |= is used in two different ways: I |= F F1 , . . . , Fn |= G In the first the left-hand-side is an interpretation, in the second it is a sequence (or set) of formulas. ...
... Don’t get confused! The symbol |= is used in two different ways: I |= F F1 , . . . , Fn |= G In the first the left-hand-side is an interpretation, in the second it is a sequence (or set) of formulas. ...
Understanding Intuitionism - the Princeton University Mathematics
... of the λ-calculus. The remaining rules are special to Arithmetic. If a is a variable-free term of L, the rules (R6)–(R9) reduce it to a numeral n, and then applications of (R10) and (R11) eliminate this occurrence of ρ by recursion. A code is irreducible in case none of these reduction rules applies ...
... of the λ-calculus. The remaining rules are special to Arithmetic. If a is a variable-free term of L, the rules (R6)–(R9) reduce it to a numeral n, and then applications of (R10) and (R11) eliminate this occurrence of ρ by recursion. A code is irreducible in case none of these reduction rules applies ...
lec26-first-order
... But there exists first order theories defined by axioms which are not sufficient for proving all T-valid formulas. ...
... But there exists first order theories defined by axioms which are not sufficient for proving all T-valid formulas. ...
Sequent calculus for predicate logic
... with ∈ {∧, ∨, →} and Q ∈ {∀, ∃}. Lemma 3.2. A cut free derivation π of a sequent σ in either the classical or intuitionistic sequent calculus only contains Gentzen subformulas of formulas occurring in σ. But note that the definition of Gentzen subformula is such that ∀x P x has infinitely many Gen ...
... with ∈ {∧, ∨, →} and Q ∈ {∀, ∃}. Lemma 3.2. A cut free derivation π of a sequent σ in either the classical or intuitionistic sequent calculus only contains Gentzen subformulas of formulas occurring in σ. But note that the definition of Gentzen subformula is such that ∀x P x has infinitely many Gen ...
The modal logic of equilibrium models
... (fullpast) for every w, for every finite P, Q ⊆ Vw such that P is nonempty, there is u such that: wRS u, Vu ∩ P = ∅ and Q ⊆ Vu ; (mtrans) for every w, u, wT , if wRS u and uRT wT then wRT wT ; (wconv) for every w, wT , if wRT wT then w = wT or wT RS w. The first two constraints are about RT , the ne ...
... (fullpast) for every w, for every finite P, Q ⊆ Vw such that P is nonempty, there is u such that: wRS u, Vu ∩ P = ∅ and Q ⊆ Vu ; (mtrans) for every w, u, wT , if wRS u and uRT wT then wRT wT ; (wconv) for every w, wT , if wRT wT then w = wT or wT RS w. The first two constraints are about RT , the ne ...
A Logic of Belief with the Complexity Measure
... B r = {ir , sr }, where ir is an initial belief set – beliefs that are initially actively hold by the agent, and sr = {α | c(α | ir ) ≤ r} is a potential belief set – beliefs for which an agent has a resource to infer them from his initial beliefs. In the logic of belief with complexity (lbc), the k ...
... B r = {ir , sr }, where ir is an initial belief set – beliefs that are initially actively hold by the agent, and sr = {α | c(α | ir ) ≤ r} is a potential belief set – beliefs for which an agent has a resource to infer them from his initial beliefs. In the logic of belief with complexity (lbc), the k ...
Propositional Logic
... • Express the syllogism as a conditional expression of the form P1 P2 ... Pn C • Create a table with one column for each variable and each subexpression occurring in the formula • Create one row for each possible assignment of T and F to the variables • Fill in the entries for variables with ...
... • Express the syllogism as a conditional expression of the form P1 P2 ... Pn C • Create a table with one column for each variable and each subexpression occurring in the formula • Create one row for each possible assignment of T and F to the variables • Fill in the entries for variables with ...
Document
... An interpretation gives meaning to the nonlogical symbols of the language. An assignment of facts to atomic wffs a fact is taken to be either true or false about the world thus, by providing an interpretation, we also provide the truth value of each of the atoms ...
... An interpretation gives meaning to the nonlogical symbols of the language. An assignment of facts to atomic wffs a fact is taken to be either true or false about the world thus, by providing an interpretation, we also provide the truth value of each of the atoms ...
Methods of Proofs Recall we discussed the following methods of
... An implication is trivially true when its conclusion is always true. A declared mathematical proposition whose truth value is unknown is called a conjecture. One of the main functions of a mathematician (and a computer scientist) is to decide the truth value of their claims (or someone else’s claims ...
... An implication is trivially true when its conclusion is always true. A declared mathematical proposition whose truth value is unknown is called a conjecture. One of the main functions of a mathematician (and a computer scientist) is to decide the truth value of their claims (or someone else’s claims ...
x - Agus Aan
... Proof by contradiction • We assume that all original facts are TRUE. • We add a new fact (the contradiction of sentence we are trying to prove is TRUE). • If we can infer that FALSE is TRUE we know the knowledgebase is corrupt. • The only thing that might not be TRUE is the negation of the goal tha ...
... Proof by contradiction • We assume that all original facts are TRUE. • We add a new fact (the contradiction of sentence we are trying to prove is TRUE). • If we can infer that FALSE is TRUE we know the knowledgebase is corrupt. • The only thing that might not be TRUE is the negation of the goal tha ...
Kurt Gödel and His Theorems
... as any attempt at such a formalism will omit some true mathematical statements • A theory such as Peano arithmetic cannot even prove its own consistency • There is no mechanical way to decide the truth (or provability) of statements in any consistent extension of Peano arithmetic ...
... as any attempt at such a formalism will omit some true mathematical statements • A theory such as Peano arithmetic cannot even prove its own consistency • There is no mechanical way to decide the truth (or provability) of statements in any consistent extension of Peano arithmetic ...
Chapter 1, Part I: Propositional Logic
... one of the prerequisites, but may have taken both. This is the meaning of disjunction. For p ∨q to be true, either one or both of p and q must be true. “Exclusive Or” - When reading the sentence “Soup or salad comes with this entrée,” we do not expect to be able to get both soup and salad. This is ...
... one of the prerequisites, but may have taken both. This is the meaning of disjunction. For p ∨q to be true, either one or both of p and q must be true. “Exclusive Or” - When reading the sentence “Soup or salad comes with this entrée,” we do not expect to be able to get both soup and salad. This is ...
Propositional Logic
... Definition 2.1 Let Op = {¬, ∧, ∨, ⇒, ⇔ , ( , )} be the set of logical operators and Σ a set of symbols. The sets Op, Σ and {t, f } are pairwise disjoint. Σ is called the signature and its elements are the proposition variables. The set of propositional logic formulas is now recursively defined: • t ...
... Definition 2.1 Let Op = {¬, ∧, ∨, ⇒, ⇔ , ( , )} be the set of logical operators and Σ a set of symbols. The sets Op, Σ and {t, f } are pairwise disjoint. Σ is called the signature and its elements are the proposition variables. The set of propositional logic formulas is now recursively defined: • t ...
proceedings version
... The next proposition says that when checking satisfaction it is enough to only consider models with finite valuations. Proposition 2. Let ϕ be a L[T],[S] formula. Let M = hW, T , S, Vi be a Kripke model satisfying (d), (alt), (heredity), (negatable), (mtrans), and (wconv). Let the valuation V ϕ be d ...
... The next proposition says that when checking satisfaction it is enough to only consider models with finite valuations. Proposition 2. Let ϕ be a L[T],[S] formula. Let M = hW, T , S, Vi be a Kripke model satisfying (d), (alt), (heredity), (negatable), (mtrans), and (wconv). Let the valuation V ϕ be d ...
Formal logic
... in which ϕ is true. A formula is said to be valid if it is true under all circumstances, that is, if every valuation is a model of ϕ: ϕ is valid if I V (ϕ) = 1 for all valuations V . For instance, it is easy to check that p → (q → p) is a valid formula. Similarly, if V is a model of all the formulas ...
... in which ϕ is true. A formula is said to be valid if it is true under all circumstances, that is, if every valuation is a model of ϕ: ϕ is valid if I V (ϕ) = 1 for all valuations V . For instance, it is easy to check that p → (q → p) is a valid formula. Similarly, if V is a model of all the formulas ...
Beginning Deductive Logic
... one has set aside enough time and space to craft at least a sustained essay, or perhaps even a book. Well, we fancy ourselves brave; here goes; all done in but one sentence: Logic is the science and engineering of reasoning. Maybe you don’t find this answer enlightening. If so, that could be because ...
... one has set aside enough time and space to craft at least a sustained essay, or perhaps even a book. Well, we fancy ourselves brave; here goes; all done in but one sentence: Logic is the science and engineering of reasoning. Maybe you don’t find this answer enlightening. If so, that could be because ...