INDUCTION
... “The next piece of snow that I will examine will be cold. “ All beliefs about unobserved matters of fact are derived from experience by induction. ...
... “The next piece of snow that I will examine will be cold. “ All beliefs about unobserved matters of fact are derived from experience by induction. ...
Proof Theory in Type Theory
... The starting motivation of this work was a comparison between the work of Tait/Schütte and the work of Lorenzen/Novikov on cut-elimination. From a constructive perspective, the analysis of sequent calculus is most elegantly expressed in Lorenzen/Novikov’s way, as the admissibility of the cut rule [ ...
... The starting motivation of this work was a comparison between the work of Tait/Schütte and the work of Lorenzen/Novikov on cut-elimination. From a constructive perspective, the analysis of sequent calculus is most elegantly expressed in Lorenzen/Novikov’s way, as the admissibility of the cut rule [ ...
4 slides/page
... Theorem: An argument is valid iff the conjunction of its premises logically implies the conclusion. Proof: Suppose the argument is valid. We want to show (A1 ∧ . . . ∧ An) ⇒ B is a tautology. • Do we have to try all 2k truth assignments (where k = #primitive propositions in A1, . . . , An, B). It’s ...
... Theorem: An argument is valid iff the conjunction of its premises logically implies the conclusion. Proof: Suppose the argument is valid. We want to show (A1 ∧ . . . ∧ An) ⇒ B is a tautology. • Do we have to try all 2k truth assignments (where k = #primitive propositions in A1, . . . , An, B). It’s ...
Logic and Reasoning
... presented as fact. However, not all information is common sense to the audience so the speaker has to connect the dots. To do this, the speaker can use: – Deductive reasoning – Inductive reasoning ...
... presented as fact. However, not all information is common sense to the audience so the speaker has to connect the dots. To do this, the speaker can use: – Deductive reasoning – Inductive reasoning ...
Slides - UCSD CSE
... For the formula to be true, if p is true, then q can’t be false. If p is false, the formula is true ...
... For the formula to be true, if p is true, then q can’t be false. If p is false, the formula is true ...
Ch1 - COW :: Ceng
... Propositional logic is one of the simplest logics Propositional logic has direct applications e.g. 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 is one of the simplest logics Propositional logic has direct applications e.g. 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
... 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 ...
(P Q). - Snistnote
... A statement formula which is true regardless of the truth values of the statements which replace the variables in it is called a universally valid formula or a tautology or a logical truth A statement formula which is false regardless of the truth values of the statements which replace the varia ...
... A statement formula which is true regardless of the truth values of the statements which replace the variables in it is called a universally valid formula or a tautology or a logical truth A statement formula which is false regardless of the truth values of the statements which replace the varia ...
Programming and Problem Solving with Java: Chapter 14
... proof be made invalid by adding additional premises or assumptions? ...
... proof be made invalid by adding additional premises or assumptions? ...
Logical Fallacies Chart APLAC TERM DEFINITION EXAMPLE 1
... therefore, the logic is faulty; This fallacy's most popular appearance is in the form of a challenging question, because questions with contradictory premises are such brain teasers. Someone tries to win support for their argument or idea by exploiting her or his opponent's feelings of pity or guilt ...
... therefore, the logic is faulty; This fallacy's most popular appearance is in the form of a challenging question, because questions with contradictory premises are such brain teasers. Someone tries to win support for their argument or idea by exploiting her or his opponent's feelings of pity or guilt ...
Comments on predicative logic
... This is a nice alternative, but we discuss another one. Namely, restrict the range of the ∀-elimination rule to atomic formulas. For lack of a better name, let us call this restricted calculus atomic PSOLi . Observe that Theorem 1 still goes through with atomic PSOLi (instead of predicative PSOLi ). ...
... This is a nice alternative, but we discuss another one. Namely, restrict the range of the ∀-elimination rule to atomic formulas. For lack of a better name, let us call this restricted calculus atomic PSOLi . Observe that Theorem 1 still goes through with atomic PSOLi (instead of predicative PSOLi ). ...
PDF
... together with the assumption A, then the formula A → B is deducible from ∆ alone. Conversely, if we can deduce A → B from ∆, and if in addition we assume A, then B can be deduced. The deduction theorem conforms with our intuitive understanding of how mathematical proofs work: if we want to prove the ...
... together with the assumption A, then the formula A → B is deducible from ∆ alone. Conversely, if we can deduce A → B from ∆, and if in addition we assume A, then B can be deduced. The deduction theorem conforms with our intuitive understanding of how mathematical proofs work: if we want to prove the ...
Homework 5
... (3) Construct an example of a formula that is satisfiable in a denumerable universe but not in any finite one (exercise 3, page 50 of Smullyan). (4) Show that a first-order formula A is valid if and only if ∼A is satisfiable. Show that A is satisfiable if and only if ∼A is valid (exercise 4, page 50 ...
... (3) Construct an example of a formula that is satisfiable in a denumerable universe but not in any finite one (exercise 3, page 50 of Smullyan). (4) Show that a first-order formula A is valid if and only if ∼A is satisfiable. Show that A is satisfiable if and only if ∼A is valid (exercise 4, page 50 ...
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 ...
Answers - stevewatson.info
... but an interpretation is a model of Th() iff it is a model of by the construction of Th(), so every model of satisfies . ie. but then Th(). QED. ...
... but an interpretation is a model of Th() iff it is a model of by the construction of Th(), so every model of satisfies . ie. but then Th(). QED. ...
Logic in Proofs (Valid arguments) A theorem is a hypothetical
... hypothesis, a tautology, or a consequence of previous members of the chain by using an allowable rule of inference. In creating a formal proof we use Substitution Rules Names don’t matter in a tautology (only the form)! Equivalences do not change truth value! Consider a proof of [(p 6 q) v (q 6 r) v ...
... hypothesis, a tautology, or a consequence of previous members of the chain by using an allowable rule of inference. In creating a formal proof we use Substitution Rules Names don’t matter in a tautology (only the form)! Equivalences do not change truth value! Consider a proof of [(p 6 q) v (q 6 r) v ...
Part 1: Propositional Logic
... Obviously, A(F ) depends only on the values of those finitely many variables in F under A. If F contains n distinct propositional variables, then it is sufficient to check 2n valuations to see whether F is satisfiable or not. ⇒ truth table. So the satisfiability problem is clearly decidable (but, by ...
... Obviously, A(F ) depends only on the values of those finitely many variables in F under A. If F contains n distinct propositional variables, then it is sufficient to check 2n valuations to see whether F is satisfiable or not. ⇒ truth table. So the satisfiability problem is clearly decidable (but, by ...
Propositional Logic Predicate Logic
... “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 specify the set that x ranges over. Note. Nullary predicates (or, predicates wi ...
... “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 specify the set that x ranges over. Note. Nullary predicates (or, predicates wi ...
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, ...
POSSIBLE WORLDS AND MANY TRUTH VALUES
... constructed, via ¬, ∨, , from formulas τai pj , and again α00 ⇔ α000 is valid on every frame. Finally, let β be obtained from α000 by replacing each τai pj by a new variable qij , let γ be a formula which “says” that, necessarily, for each j exactly one qij holds, and let α∗ be γ ⇒ β. Then α∗ is a ...
... constructed, via ¬, ∨, , from formulas τai pj , and again α00 ⇔ α000 is valid on every frame. Finally, let β be obtained from α000 by replacing each τai pj by a new variable qij , let γ be a formula which “says” that, necessarily, for each j exactly one qij holds, and let α∗ be γ ⇒ β. Then α∗ is a ...
Logic Logical Concepts Deduction Concepts Resolution
... Intuitively, this is how resolution works in first-order logic, All Humans are Mortals Socrates is Human Therefore, Socrates is Mortal ...
... Intuitively, this is how resolution works in first-order logic, All Humans are Mortals Socrates is Human Therefore, Socrates is Mortal ...
Exam 2 study guide
... Proving a formula of the form φ→ ψ, where the conditional φ→ψ is provable: first prove the conditional φ→ψ, then Necessitate, then distribute the over the → using K . Proving a formula of the form ◊φ→◊ψ, where the conditional φ→ψ is provable. As above, but use K◊. Proving a formula of the form φ ...
... Proving a formula of the form φ→ ψ, where the conditional φ→ψ is provable: first prove the conditional φ→ψ, then Necessitate, then distribute the over the → using K . Proving a formula of the form ◊φ→◊ψ, where the conditional φ→ψ is provable. As above, but use K◊. Proving a formula of the form φ ...
L11
... It is minimal in structure but as powerful as the truth table and natural deduction approaches. − The proofs of the theorems are often difficult and require a guess in selection of appropriate axiom(s) and rules. − These methods basically require forward chaining strategy where we start with the giv ...
... It is minimal in structure but as powerful as the truth table and natural deduction approaches. − The proofs of the theorems are often difficult and require a guess in selection of appropriate axiom(s) and rules. − These methods basically require forward chaining strategy where we start with the giv ...