Finite-variable fragments of first
... Finite-variable fragments of first-order logic We explained in Chapter 1 that the counting quantifiers are of interest primarily in terms of their computational effect on certain decidable fragments of first-order logic. The purpose of this chapter is to lay the groundwork for our investigation of c ...
... Finite-variable fragments of first-order logic We explained in Chapter 1 that the counting quantifiers are of interest primarily in terms of their computational effect on certain decidable fragments of first-order logic. The purpose of this chapter is to lay the groundwork for our investigation of c ...
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... We immediately obtain this sequent: hyp : (A(0)&∀x.(∀y.(y ≤ x ⇒ A(y)) ⇒ A(x + 1))) ` ∀x.∀y.(y ≤ x ⇒ A(y)) We can decompose the hypothesis into a0 : A(0), all : ∀x.(∀y.(y ≤ x ⇒ A(y)) ⇒ A(x + 1))), x : N ` ∀y.(y ≤ x ⇒ A(y)) by λ(x.ind(x; ...; u, v. ...)) ` ∀y.(y ≤ 0 ⇒ A(y)) by λ(y.λ(le. ...)) y : N, l ...
... We immediately obtain this sequent: hyp : (A(0)&∀x.(∀y.(y ≤ x ⇒ A(y)) ⇒ A(x + 1))) ` ∀x.∀y.(y ≤ x ⇒ A(y)) We can decompose the hypothesis into a0 : A(0), all : ∀x.(∀y.(y ≤ x ⇒ A(y)) ⇒ A(x + 1))), x : N ` ∀y.(y ≤ x ⇒ A(y)) by λ(x.ind(x; ...; u, v. ...)) ` ∀y.(y ≤ 0 ⇒ A(y)) by λ(y.λ(le. ...)) y : N, l ...
Lecture notes from 5860
... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
Linearizing some recursive logic programs
... Q(X1 , . . . , Xn ) ←− Q1 (Y1,1 , . . . , Y1,n1 ), . . . , Qp (Yp,1 , . . . , Yp,np ) where X1 , . . . , Xn are variables, the Yi,j ’s are either variables or constants, Q is an intensional predicate, the Qi ’s are either intensional database predicates (i.e. relations defined by logical rules) or e ...
... Q(X1 , . . . , Xn ) ←− Q1 (Y1,1 , . . . , Y1,n1 ), . . . , Qp (Yp,1 , . . . , Yp,np ) where X1 , . . . , Xn are variables, the Yi,j ’s are either variables or constants, Q is an intensional predicate, the Qi ’s are either intensional database predicates (i.e. relations defined by logical rules) or e ...
Section.8.3
... A logic is higher-order if it allows predicate names or function names to be quantified or to be arguments of a predicate. Example. The sentence, “There is a function with a fixed point.” can be formalized as f x p(ƒ(x), x), where p denotes equality. Example. Given the sentence, “ Every binary rel ...
... A logic is higher-order if it allows predicate names or function names to be quantified or to be arguments of a predicate. Example. The sentence, “There is a function with a fixed point.” can be formalized as f x p(ƒ(x), x), where p denotes equality. Example. Given the sentence, “ Every binary rel ...
How to Prove Properties by Induction on Formulas
... A few comments may be helpful. First, the propositional logic meaning of implies is crucial for making this proof work in case (ii) for each of the connectives. As soon as the hypothesis is false, the truth of the implication “comes for free.” Second, in the induction, I’ve tried to make it clear wh ...
... A few comments may be helpful. First, the propositional logic meaning of implies is crucial for making this proof work in case (ii) for each of the connectives. As soon as the hypothesis is false, the truth of the implication “comes for free.” Second, in the induction, I’ve tried to make it clear wh ...
Probability Captures the Logic of Scientific
... Turning now to the choice of λ, Carnap (1980, pp. 111–119) considered the rate of learning from experience that different values of λ induce and came to the conclusion that λ should be about 1 or 2. Another consideration is this: So far as we know a priori, the statistical probability (roughly, long ...
... Turning now to the choice of λ, Carnap (1980, pp. 111–119) considered the rate of learning from experience that different values of λ induce and came to the conclusion that λ should be about 1 or 2. Another consideration is this: So far as we know a priori, the statistical probability (roughly, long ...
8 predicate logic
... a straightforward way. Thus, the proposition “If Socrates is altruistic, then Plato is altruistic” can be represented as As ⊃ Ap; the proposition “Socrates is altruistic but Plato is not” can be represented as As · ~Ap, and so on. Representing quantified propositions in predicate logic requires a li ...
... a straightforward way. Thus, the proposition “If Socrates is altruistic, then Plato is altruistic” can be represented as As ⊃ Ap; the proposition “Socrates is altruistic but Plato is not” can be represented as As · ~Ap, and so on. Representing quantified propositions in predicate logic requires a li ...
A Crevice on the Crane Beach: Finite-Degree
... approaches, especially the topological ones of [8], [9], have the former has the Crane Beach Property. Notably, all these yet to yield strong lower bounds. logics are quite far from full FO[ARB], and in that sense, fail A different take originated from a conjecture of Lautemann to identify the part ...
... approaches, especially the topological ones of [8], [9], have the former has the Crane Beach Property. Notably, all these yet to yield strong lower bounds. logics are quite far from full FO[ARB], and in that sense, fail A different take originated from a conjecture of Lautemann to identify the part ...
Monadic Second-Order Logic with Arbitrary Monadic Predicates⋆
... 1. It has an equivalent automaton model: automata with advice. 2. It has an equivalent algebraic model: one-scan programs. 3. It has a machine-independent characterization, based on generalizations of Myhill-Nerode equivalence relations. This extends the equivalence between automata with advice and ...
... 1. It has an equivalent automaton model: automata with advice. 2. It has an equivalent algebraic model: one-scan programs. 3. It has a machine-independent characterization, based on generalizations of Myhill-Nerode equivalence relations. This extends the equivalence between automata with advice and ...
Predicate Logic
... Often, one does not want to associate the arguments of an atomic formula with a particular individual. To avoid this, variables are used. ...
... Often, one does not want to associate the arguments of an atomic formula with a particular individual. To avoid this, variables are used. ...
1 The calculus of “predicates”
... sets. The predicates are number-theoretic or set-theoretic predicates – for example, “... is even” or “... is less than ...”. We can introduce such predicates into a finite language. For example, if the base set is given by A 1,2,3,4 where 1,2,3,4 now really are numbers and no longer mere partit ...
... sets. The predicates are number-theoretic or set-theoretic predicates – for example, “... is even” or “... is less than ...”. We can introduce such predicates into a finite language. For example, if the base set is given by A 1,2,3,4 where 1,2,3,4 now really are numbers and no longer mere partit ...
PROOFS BY INDUCTION AND CONTRADICTION, AND WELL
... Proposition (Principle of strong induction). If S ⊂ N is a subset of the natural numbers such that (i) 0 ∈ S, and (ii) whenever {0, . . . , k} ⊂ S, then k + 1 ∈ S, then S = N. Remark. Note the difference from the principle of induction above. In the second property we require the stronger assumption ...
... Proposition (Principle of strong induction). If S ⊂ N is a subset of the natural numbers such that (i) 0 ∈ S, and (ii) whenever {0, . . . , k} ⊂ S, then k + 1 ∈ S, then S = N. Remark. Note the difference from the principle of induction above. In the second property we require the stronger assumption ...
Relational Predicate Logic
... “Sometime,” and So On The word “someone” can be misleading, for it sometimes functions as an existential quantifier and sometimes as a universal quantifier. The words “somewhere,” “something,” “sometime,” and so on can also be misleading in this way. ...
... “Sometime,” and So On The word “someone” can be misleading, for it sometimes functions as an existential quantifier and sometimes as a universal quantifier. The words “somewhere,” “something,” “sometime,” and so on can also be misleading in this way. ...
Review - Gerry O nolan
... inference is in relation to its premisses (43). After all, in its strongest form, the sceptical thesis is just the denial of the proposition that a conclusion's probability is ever altered as a result of observational evidence (40, 42, 44). It is in this sense that the premisses of an inductive infe ...
... inference is in relation to its premisses (43). After all, in its strongest form, the sceptical thesis is just the denial of the proposition that a conclusion's probability is ever altered as a result of observational evidence (40, 42, 44). It is in this sense that the premisses of an inductive infe ...
Predicate logic, motivation
... --if there is a tilde, the quantifier applies to the tilde and to whatever the tilde applies to --if there is a parenthesis (or bracket), the quantifier applies to everything in that pair of parentheses (or brackets) A variable is bound if and only if it is within the scope of a quantifier that con ...
... --if there is a tilde, the quantifier applies to the tilde and to whatever the tilde applies to --if there is a parenthesis (or bracket), the quantifier applies to everything in that pair of parentheses (or brackets) A variable is bound if and only if it is within the scope of a quantifier that con ...
Slides from 10/20/14
... --if there is a tilde, the quantifier applies to the tilde and to whatever the tilde applies to --if there is a parenthesis (or bracket), the quantifier applies to everything in that pair of parentheses (or brackets) A variable is bound if and only if it is within the scope of a quantifier that con ...
... --if there is a tilde, the quantifier applies to the tilde and to whatever the tilde applies to --if there is a parenthesis (or bracket), the quantifier applies to everything in that pair of parentheses (or brackets) A variable is bound if and only if it is within the scope of a quantifier that con ...
Computer Science 202a Homework #2, due in class
... integer d such that b = d · a. Express each of the following false statements about the domain of integers as a closed predicate logic formula, and give a counterexample to show that it is false. Use the binary predicates divides(a, b) for “a divides b” and (a = b) for “a is equal to b”, and the bin ...
... integer d such that b = d · a. Express each of the following false statements about the domain of integers as a closed predicate logic formula, and give a counterexample to show that it is false. Use the binary predicates divides(a, b) for “a divides b” and (a = b) for “a is equal to b”, and the bin ...
The Problem of Induction
... argument in the form of a dilemma (sometimes referred to as “Hume’s fork”) to show that there can be no such reasoning. Such reasoning would, he argues, have to be either a priori demonstrative reasoning concerning relations of ideas or “experimental” (i.e. empirical) reasoning concerning matters of ...
... argument in the form of a dilemma (sometimes referred to as “Hume’s fork”) to show that there can be no such reasoning. Such reasoning would, he argues, have to be either a priori demonstrative reasoning concerning relations of ideas or “experimental” (i.e. empirical) reasoning concerning matters of ...
Propositional Logic Predicate Logic
... Name of Symbols ∀ (universal quantifier), and ∃ (existential quantifier). Definition. A formula A is valid if A is true no matter how we replace the individual constants in A with concrete individuals and the predicate variables in A with concrete predicates. Note. The set of individuals must be in ...
... Name of Symbols ∀ (universal quantifier), and ∃ (existential quantifier). Definition. A formula A is valid if A is true no matter how we replace the individual constants in A with concrete individuals and the predicate variables in A with concrete predicates. Note. The set of individuals must be in ...