Propositional/First
... • A valid sentence is true in all worlds under all interpretations • If an implication sentence can be shown to be valid, then—given its premise—its consequent can be derived • Different logics make different commitments about what the world is made of and what kind of beliefs we can have regarding ...
... • A valid sentence is true in all worlds under all interpretations • If an implication sentence can be shown to be valid, then—given its premise—its consequent can be derived • Different logics make different commitments about what the world is made of and what kind of beliefs we can have regarding ...
chapter1p3 - WordPress.com
... Let us look at the examples considered earlier. In the first one there are two variables and the minterms occurring in the expression correspond to 0, 2 and 3 and this is written as 0, 2, 3. In the second example the minterms correspond to 7, 6, 3, 1 respectively and the expression is written as 1 ...
... Let us look at the examples considered earlier. In the first one there are two variables and the minterms occurring in the expression correspond to 0, 2 and 3 and this is written as 0, 2, 3. In the second example the minterms correspond to 7, 6, 3, 1 respectively and the expression is written as 1 ...
First-order logic syntax and semantics
... a ∈ A holds (A, %ax ) |= ϕ a i for all a ∈ A holds (B, h ◦ (%x )) |= ϕ h(a) i for all h (a) ∈ B holds (B, (h ◦ %)x ) |= ϕ b i for all b ∈ B holds (B, (h ◦ %)x ) |= ϕ i (B, h ◦ %) |= (∀x ϕ) i for all ...
... a ∈ A holds (A, %ax ) |= ϕ a i for all a ∈ A holds (B, h ◦ (%x )) |= ϕ h(a) i for all h (a) ∈ B holds (B, (h ◦ %)x ) |= ϕ b i for all b ∈ B holds (B, (h ◦ %)x ) |= ϕ i (B, h ◦ %) |= (∀x ϕ) i for all ...
Multi-Agent Only
... I For a single agent possibilities are just worlds. I For many agents possibilities include other agents beliefs. ...
... I For a single agent possibilities are just worlds. I For many agents possibilities include other agents beliefs. ...
Robot Morality and Review of classical logic.
... Analytic philosophy (like proving God’ Existence, free will, the problem of evil, etc) Many other… At this point I should ask all students to give another examples of similar problems that they want to solve ...
... Analytic philosophy (like proving God’ Existence, free will, the problem of evil, etc) Many other… At this point I should ask all students to give another examples of similar problems that they want to solve ...
1. Axioms and rules of inference for propositional logic. Suppose T
... For Ass, Ex, Contr and Cut this amounts to the so called “generalized rules of inference” on stated and proved on pp. 91-93 of the coursepack. The rest are a straightforward exercise for the reader making use of associativity. ...
... For Ass, Ex, Contr and Cut this amounts to the so called “generalized rules of inference” on stated and proved on pp. 91-93 of the coursepack. The rest are a straightforward exercise for the reader making use of associativity. ...
Lecturecise 19 Proofs and Resolution Compactness for
... Γ ` F means that there is a proof tree with leaves Γ that derives F ...
... Γ ` F means that there is a proof tree with leaves Γ that derives F ...
Section 3 - UCLA Department of Mathematics
... semantics, we will allow for the assignment of a truth-value to P12 v3 c given an assignment of v3 to some particular object. What are the objects over which our variables are to range? A natural answer would be that they range over all objects. If we made this choice, then we could interpret ∀v3 as ...
... semantics, we will allow for the assignment of a truth-value to P12 v3 c given an assignment of v3 to some particular object. What are the objects over which our variables are to range? A natural answer would be that they range over all objects. If we made this choice, then we could interpret ∀v3 as ...
3 The semantics of pure first
... semantics, we will allow for the assignment of a truth-value to P12 v3 c given an assignment of v3 to some particular object. What are the objects over which our variables are to range? A natural answer would be that they range over all objects. If we made this choice, then we could interpret ∀v3 as ...
... semantics, we will allow for the assignment of a truth-value to P12 v3 c given an assignment of v3 to some particular object. What are the objects over which our variables are to range? A natural answer would be that they range over all objects. If we made this choice, then we could interpret ∀v3 as ...
3 The semantics of pure first
... semantics, we will allow for the assignment of a truth-value to P12 v3 c given an assignment of v3 to some particular object. What are the objects over which our variables are to range? A natural answer would be that they range over all objects. If we made this choice, then we could interpret ∀v3 as ...
... semantics, we will allow for the assignment of a truth-value to P12 v3 c given an assignment of v3 to some particular object. What are the objects over which our variables are to range? A natural answer would be that they range over all objects. If we made this choice, then we could interpret ∀v3 as ...
Notes on Propositional and Predicate Logic
... • Simplify all subexpressions of the form (not (not p)) to p • Move all occurrences of or “inside” occurrences of and • Simplify all or- expressions for example by rewriting (or (or p q) r) as (or p q r) , and similarly for and Each premise is converted to conjunctive normal form in this way. Then t ...
... • Simplify all subexpressions of the form (not (not p)) to p • Move all occurrences of or “inside” occurrences of and • Simplify all or- expressions for example by rewriting (or (or p q) r) as (or p q r) , and similarly for and Each premise is converted to conjunctive normal form in this way. Then t ...
P Q
... It allows any expression to be substituted for every occurrence of a symbol in a proposition that is an axiom or theorem already known to be true For instance, (BB)B may have the expression A substituted for B to produce (AA)A ...
... It allows any expression to be substituted for every occurrence of a symbol in a proposition that is an axiom or theorem already known to be true For instance, (BB)B may have the expression A substituted for B to produce (AA)A ...
Lesson 2
... • Hence if we prove that the conclusion logically follows from the assumptions, then by virtue of it we do not prove that the conclusion is true • It is true, provided the premises are true • The argument the premises of which are true is called sound. • Truthfulness or Falseness of premises can be ...
... • Hence if we prove that the conclusion logically follows from the assumptions, then by virtue of it we do not prove that the conclusion is true • It is true, provided the premises are true • The argument the premises of which are true is called sound. • Truthfulness or Falseness of premises can be ...
Logic and Proof
... (1) It is not sunny this afternoon (¬ p) and it is colder than yesterday (q). (2) We will go swimming (r) only if it is sunny (p). (3) If we do not go swimming (¬ r), then we will take a canoe trip (s). (4) If we take a canoe trip (s), then we will be home by sunset (t). Therefore, we will be home b ...
... (1) It is not sunny this afternoon (¬ p) and it is colder than yesterday (q). (2) We will go swimming (r) only if it is sunny (p). (3) If we do not go swimming (¬ r), then we will take a canoe trip (s). (4) If we take a canoe trip (s), then we will be home by sunset (t). Therefore, we will be home b ...
A Simple Exposition of Gödel`s Theorem
... odd prime numbers, but 2 as well. Then a sequence of well-formed formulae can be expressed by an even number, and a putative proof of well-formed formula no.1729 would look something like ...
... odd prime numbers, but 2 as well. Then a sequence of well-formed formulae can be expressed by an even number, and a putative proof of well-formed formula no.1729 would look something like ...
Practice Problem Set 1
... P M1 (f −1 (b1 ), f −1 (b2 ), . . . f −1 (bk )). It can be shown that if M1 and M2 are isomorphic Σ-structures, then for every first-order logic sentence φ on the signature Σ, M1 |= φ iff M2 |= φ. Now consider Σ = {=}, i.e., the signature containing only the equality predicate. We wish to show that ...
... P M1 (f −1 (b1 ), f −1 (b2 ), . . . f −1 (bk )). It can be shown that if M1 and M2 are isomorphic Σ-structures, then for every first-order logic sentence φ on the signature Σ, M1 |= φ iff M2 |= φ. Now consider Σ = {=}, i.e., the signature containing only the equality predicate. We wish to show that ...
Discrete Structures & Algorithms Propositional Logic
... • Direct proof: Assume p is true, and prove q. • Indirect proof: Assume q, and prove p. • Vacuous proof: Prove p by itself. • Trivial proof: Prove q by itself. • Proof by cases: Show p(a b), and (aq) and (bq). ...
... • Direct proof: Assume p is true, and prove q. • Indirect proof: Assume q, and prove p. • Vacuous proof: Prove p by itself. • Trivial proof: Prove q by itself. • Proof by cases: Show p(a b), and (aq) and (bq). ...
Lesson 2
... • Hence if we prove that the conclusion logically follows from the assumptions, then by virtue of it we do not prove that the conclusion is true • It is true, provided the premises are true • The argument the premises of which are true is called sound. • Truthfulness or Falseness of premises can be ...
... • Hence if we prove that the conclusion logically follows from the assumptions, then by virtue of it we do not prove that the conclusion is true • It is true, provided the premises are true • The argument the premises of which are true is called sound. • Truthfulness or Falseness of premises can be ...
paper by David Pierce
... We make no new row for x4 , since 4 = 1 in Z/3Z, so x4 has already been defined. If we did try to make a row for x4 , using (2.4), then it would not agree with the row for x1 . Thus, although we can use equations (2.3) and (2.4) to give a definition, in Peano’s sense, of exponentiation in Z/3Z, those ...
... We make no new row for x4 , since 4 = 1 in Z/3Z, so x4 has already been defined. If we did try to make a row for x4 , using (2.4), then it would not agree with the row for x1 . Thus, although we can use equations (2.3) and (2.4) to give a definition, in Peano’s sense, of exponentiation in Z/3Z, those ...
On Correctness of Mathematical Texts from a Logical and Practical
... of the last century when first theorem proving programs were created [1]. It is worth noting how ambitious was the title of Wang’s article! Numerous attempts to “mechanize” mathematics led to less ambitious and more realistic idea of “computer aided” mathematics as well as to the notion of “proof ass ...
... of the last century when first theorem proving programs were created [1]. It is worth noting how ambitious was the title of Wang’s article! Numerous attempts to “mechanize” mathematics led to less ambitious and more realistic idea of “computer aided” mathematics as well as to the notion of “proof ass ...
Chapter 9: Quantified Formulas
... equivalent formula without quantifiers.4 The procedures that we present next require that all the quantifiers are eliminated in order to check for validity. It is sufficient to show that there exists a procedure for eliminating an existential quantifier. Universal quantifiers can be eliminated by ma ...
... equivalent formula without quantifiers.4 The procedures that we present next require that all the quantifiers are eliminated in order to check for validity. It is sufficient to show that there exists a procedure for eliminating an existential quantifier. Universal quantifiers can be eliminated by ma ...
pdf file
... proving T.b.x̂ . An instantiation of P 0 looks promising, because the goal T.b.x̂ can be reached easily from at least the first disjunct of its body. This would yield a proof that started from a premise and ended with the goal. On the other hand, we could begin with the goal T.b.x̂ and immediately us ...
... proving T.b.x̂ . An instantiation of P 0 looks promising, because the goal T.b.x̂ can be reached easily from at least the first disjunct of its body. This would yield a proof that started from a premise and ended with the goal. On the other hand, we could begin with the goal T.b.x̂ and immediately us ...
Lindenbaum lemma for infinitary logics
... / T , enumerate all the rules, and in each step expand T in a way that makes sure that either it already contains the conclusion of the given rule or the rule is not applicable to any of its extensions not proving certain special formula. We use the existing disjunction connective to combine and ‘re ...
... / T , enumerate all the rules, and in each step expand T in a way that makes sure that either it already contains the conclusion of the given rule or the rule is not applicable to any of its extensions not proving certain special formula. We use the existing disjunction connective to combine and ‘re ...
Slide 1
... pq Converse: q p Contrapositive: q p Inverse: p q Two compound propositions are equivalent if they always have the same truth value – The contrapositive is equivalent to the original statement – The converse is equivalent to the inverse ...
... pq Converse: q p Contrapositive: q p Inverse: p q Two compound propositions are equivalent if they always have the same truth value – The contrapositive is equivalent to the original statement – The converse is equivalent to the inverse ...