Introduction to logic
... not only deductively valid. They also have true premises. These arguments are called sound. Argument (2.4) above is valid, but not sound. Another example: (2.5) All fish fly. Anything which flies talks. So, all fish talk. This is a deductively valid argument but unlikely to persuade anyone. ...
... not only deductively valid. They also have true premises. These arguments are called sound. Argument (2.4) above is valid, but not sound. Another example: (2.5) All fish fly. Anything which flies talks. So, all fish talk. This is a deductively valid argument but unlikely to persuade anyone. ...
The semantics of propositional logic
... n ≥ 1, if M (n) is true, then M (n + 1) is true (this is the inductive step), then we can conclude that M (n) is true for all natural numbers n. (We don’t actually know how to do this for the specific statement M (n) above, which is called “Goldbach’s conjecture”.) Induction is a way of proving (in ...
... n ≥ 1, if M (n) is true, then M (n + 1) is true (this is the inductive step), then we can conclude that M (n) is true for all natural numbers n. (We don’t actually know how to do this for the specific statement M (n) above, which is called “Goldbach’s conjecture”.) Induction is a way of proving (in ...
WhichQuantifiersLogical
... relational structures of signature ⟨ k1,…,kn⟩ closed under isomorphism. A typical member of Q is of the form ⟨ U,P1,…,Pn⟩ where U is non-empty and Pi is a ki-ary relation on U. Given Q, with each U is associated the (local) quantifier QU on U which is the relation QU(P1,…,Pn) that holds between P1,… ...
... relational structures of signature ⟨ k1,…,kn⟩ closed under isomorphism. A typical member of Q is of the form ⟨ U,P1,…,Pn⟩ where U is non-empty and Pi is a ki-ary relation on U. Given Q, with each U is associated the (local) quantifier QU on U which is the relation QU(P1,…,Pn) that holds between P1,… ...
First Order Predicate Logic
... – Basic rules for formula in Predicate Calculus are same as those of Propositional Calculus. – A wide variety of statements are expressed in contrast to Propositional Calculus ...
... – Basic rules for formula in Predicate Calculus are same as those of Propositional Calculus. – A wide variety of statements are expressed in contrast to Propositional Calculus ...
Deciding Global Partial-Order Properties
... partially ordered events, or local states of processes. In local partial order logics, the truth of a formula is evaluated at a local state, and the temporal modalities relate causal precedences among local states. Examples of such logics include TRPTL[13] and TLC [I]. In global partial order logics ...
... partially ordered events, or local states of processes. In local partial order logics, the truth of a formula is evaluated at a local state, and the temporal modalities relate causal precedences among local states. Examples of such logics include TRPTL[13] and TLC [I]. In global partial order logics ...
Modal Logic
... logic. What we add are two unary connectives and ♦. We have a set Atoms of propositional letters p, q, r, . . ., also called atomic formulas or atoms. Definition 1. Formulas of basic modal logic are given by the following rule ϕ ::= ⊥ | > | p | ¬ϕ | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | (ϕ → ϕ) | (ϕ ⇔ ϕ) | ϕ | ♦ϕ. ...
... logic. What we add are two unary connectives and ♦. We have a set Atoms of propositional letters p, q, r, . . ., also called atomic formulas or atoms. Definition 1. Formulas of basic modal logic are given by the following rule ϕ ::= ⊥ | > | p | ¬ϕ | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | (ϕ → ϕ) | (ϕ ⇔ ϕ) | ϕ | ♦ϕ. ...
On Gabbay`s temporal fixed point operator
... there is nothing to prove. Assume the statement for n. Clearly, gn =q hn . By remark 2.5 and the inductive hypothesis, gn and hn are A-similar up to n. As A is local, gn (A) and hn (A) agree up to n: i.e., gn+1 (q) and hn+1 (q) agree before n + 1. This completes the induction. 2. By (1) we can repla ...
... there is nothing to prove. Assume the statement for n. Clearly, gn =q hn . By remark 2.5 and the inductive hypothesis, gn and hn are A-similar up to n. As A is local, gn (A) and hn (A) agree up to n: i.e., gn+1 (q) and hn+1 (q) agree before n + 1. This completes the induction. 2. By (1) we can repla ...
Propositional Logic: Why? soning Starts with George Boole around 1850
... It represents a class of common valid arguments ...
... It represents a class of common valid arguments ...
Predicate Calculus - National Taiwan University
... How can we check if a formula is a tautology? If the domain is finite, then we can try all the possible interpretations (all the possible functions and predicates). But if the domain is infinite? Intuitively, this is why a computer cannot be programmed to determine if an arbitrary formula in predica ...
... How can we check if a formula is a tautology? If the domain is finite, then we can try all the possible interpretations (all the possible functions and predicates). But if the domain is infinite? Intuitively, this is why a computer cannot be programmed to determine if an arbitrary formula in predica ...
Modal Logics Definable by Universal Three
... formulas. For a given first-order sentence Φ over the signature consisting of a single binary symbol R we define KΦ to be the set of those frames which satisfy Φ. In this paper we are interested in the satisfiability problem for modal logic over classes of frames definable by universal first-order f ...
... formulas. For a given first-order sentence Φ over the signature consisting of a single binary symbol R we define KΦ to be the set of those frames which satisfy Φ. In this paper we are interested in the satisfiability problem for modal logic over classes of frames definable by universal first-order f ...
Propositional Dynamic Logic of Regular Programs*+
... The new Q-variables are given meaning in $9 when they are introduced into the closure. The definition of n@(Q) depends only on @(Y) which has been already well defined previously. The new structure 9 is over @i and & , where A,, = CD;- CD,,is the set of Q-variables introduced in taking the closure o ...
... The new Q-variables are given meaning in $9 when they are introduced into the closure. The definition of n@(Q) depends only on @(Y) which has been already well defined previously. The new structure 9 is over @i and & , where A,, = CD;- CD,,is the set of Q-variables introduced in taking the closure o ...
Lesson 2
... Arguments • 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 premi ...
... Arguments • 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 premi ...
Lecture 14 Notes
... In the other case, we have formulas of the form F (∀x)A and, by duality, T (∃x)A, which we call formulas of type δ of existential type. δ-formulas are decomposed into F B[a/x] (and T B[a/x], respectively), where a is a new parameter. These formulas are often denoted by δ(a) and the requirement that ...
... In the other case, we have formulas of the form F (∀x)A and, by duality, T (∃x)A, which we call formulas of type δ of existential type. δ-formulas are decomposed into F B[a/x] (and T B[a/x], respectively), where a is a new parameter. These formulas are often denoted by δ(a) and the requirement that ...
The semantics of predicate logic
... each atom. In predicate logic, the smallest unit to which we can assign a truth value is a predicate P (t1 , t2 , . . . , tn ) applied to terms. But we cannot arbitrarily assign a truth value, as we did for propositional atoms. There needs to be some consistency. We need to assign values to variable ...
... each atom. In predicate logic, the smallest unit to which we can assign a truth value is a predicate P (t1 , t2 , . . . , tn ) applied to terms. But we cannot arbitrarily assign a truth value, as we did for propositional atoms. There needs to be some consistency. We need to assign values to variable ...
Rules of inference
... “It is below freezing now (p). Therefore, it is either below freezing or raining now (q).” “It is below freezing (p). It is raining now (q). Therefore, it is below freezing and it is raining now. “if it rains today (p), then we will not have a barbecue today (q). if we do not have a barbecue t ...
... “It is below freezing now (p). Therefore, it is either below freezing or raining now (q).” “It is below freezing (p). It is raining now (q). Therefore, it is below freezing and it is raining now. “if it rains today (p), then we will not have a barbecue today (q). if we do not have a barbecue t ...
Lesson 2
... modus ponens • A B, B |= A, modus ponens + transposition • A B, B |= A modus ponens + transposition • A B, B |= A elimination of disjunction (disjunctive syllogism) Introduction to Logic ...
... modus ponens • A B, B |= A, modus ponens + transposition • A B, B |= A modus ponens + transposition • A B, B |= A elimination of disjunction (disjunctive syllogism) Introduction to Logic ...
How to Prove Properties by Induction on Formulas
... in S. Thus, any implication “If ¬β uses only atoms in S, then . . . ” is definitely true. β ∨ γ: Either (i) both β and γ use only atoms in S, or (ii) it is not the case that both β and γ use only atoms in S. In case (i), the induction hypothesis implies M |= β if and only if M0 |= β, and likewise fo ...
... in S. Thus, any implication “If ¬β uses only atoms in S, then . . . ” is definitely true. β ∨ γ: Either (i) both β and γ use only atoms in S, or (ii) it is not the case that both β and γ use only atoms in S. In case (i), the induction hypothesis implies M |= β if and only if M0 |= β, and likewise fo ...
On Linear Inference
... may never pick up a given block, even if the rule pickup would permit us to do so. This is more important in this new setting because inferences may be irreversible, so making an inference may constitute a real commitment. If all truths are persistent (and hence inference is monotonic) we can always ...
... may never pick up a given block, even if the rule pickup would permit us to do so. This is more important in this new setting because inferences may be irreversible, so making an inference may constitute a real commitment. If all truths are persistent (and hence inference is monotonic) we can always ...
An Instantiation-Based Theorem Prover for First
... These limitations in MILP can be addressed by switching to a more expressive language like first-order logic (FOL). FOL lets us specify a problem in terms of classes and relations, and reason about these classes and relations directly—we can do lifted reasoning. Lifted reasoning lets us work with st ...
... These limitations in MILP can be addressed by switching to a more expressive language like first-order logic (FOL). FOL lets us specify a problem in terms of classes and relations, and reason about these classes and relations directly—we can do lifted reasoning. Lifted reasoning lets us work with st ...
When is Metric Temporal Logic Expressively Complete?
... Given an N -bounded FOK formula with one free variable x, we show that it is equivalent to a N 0 -bounded formula (over a possibly larger set of monadic predicates, suitably interpreted) in which the unary functions are only applied to x. We can remove occurrences of unary functions within the scope ...
... Given an N -bounded FOK formula with one free variable x, we show that it is equivalent to a N 0 -bounded formula (over a possibly larger set of monadic predicates, suitably interpreted) in which the unary functions are only applied to x. We can remove occurrences of unary functions within the scope ...
AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic
... A PROPOSITION LETTER is any symbol from following list: A, ...Z, A0...Z0, A1...Z1... The PROPOSITIONAL CONNECTIVES are ¬, ∨, ∧, →, ↔ An EXPRESSION of propositional logic is any sequence of sentence letters, propositional connectives, or left and right parentheses. METAVARIABLES such as Φ and Ψ are n ...
... A PROPOSITION LETTER is any symbol from following list: A, ...Z, A0...Z0, A1...Z1... The PROPOSITIONAL CONNECTIVES are ¬, ∨, ∧, →, ↔ An EXPRESSION of propositional logic is any sequence of sentence letters, propositional connectives, or left and right parentheses. METAVARIABLES such as Φ and Ψ are n ...
Notes on Propositional and Predicate Logic
... Each premise is converted to conjunctive normal form in this way. Then the surroundig and- expression is removed so that each premise becomes a set of or- expressions, and one takes the union of those sets. In this way, one obtains the entire set of premises as a set of or- expressions. The same ope ...
... Each premise is converted to conjunctive normal form in this way. Then the surroundig and- expression is removed so that each premise becomes a set of or- expressions, and one takes the union of those sets. In this way, one obtains the entire set of premises as a set of or- expressions. The same ope ...
Modus ponens
... of definition" and the "rule of substitution". Modus ponens allows one to eliminate a conditional statement from a logical proof or argument (the antecedents) and thereby not carry these antecedents forward in an everlengthening string of symbols; for this reason modus ponens is sometimes called the ...
... of definition" and the "rule of substitution". Modus ponens allows one to eliminate a conditional statement from a logical proof or argument (the antecedents) and thereby not carry these antecedents forward in an everlengthening string of symbols; for this reason modus ponens is sometimes called the ...
Knowledge Representation
... • There is a precise meaning to expressions in predicate logic. • Like in propositional logic, it is all about determining whether something is true or false. • X P(X) means that P(X) must be true for every object X in the domain of interest. • X P(X) means that P(X) must be true for at least on ...
... • There is a precise meaning to expressions in predicate logic. • Like in propositional logic, it is all about determining whether something is true or false. • X P(X) means that P(X) must be true for every object X in the domain of interest. • X P(X) means that P(X) must be true for at least on ...
Autoepistemic Logic and Introspective Circumscription
... formula F is entailed by Tiff, for every I E $2, (I, $2) satisfies F. For instance, T entails P2, -~BP1 and BP2. It is clear that a nonmodM formula is entailed by T iff it is entMled by Pz in classical logic. E x a m p l e 2. Let T be {P1 V P2,P1 D BP1,P2 =- BP2}. As will be seen later, this theory ...
... formula F is entailed by Tiff, for every I E $2, (I, $2) satisfies F. For instance, T entails P2, -~BP1 and BP2. It is clear that a nonmodM formula is entailed by T iff it is entMled by Pz in classical logic. E x a m p l e 2. Let T be {P1 V P2,P1 D BP1,P2 =- BP2}. As will be seen later, this theory ...