A short article for the Encyclopedia of Artificial Intelligence: Second
... model Peano’s axioms for the non-negative integers. As a corollary of Gödel’s incompleteness theorem, the set of true formulas in such a standard model are not recursively axiomatizable; that is, there is no theorem proving procedure that could (even theoretically) uncover all true formulas. This i ...
... model Peano’s axioms for the non-negative integers. As a corollary of Gödel’s incompleteness theorem, the set of true formulas in such a standard model are not recursively axiomatizable; that is, there is no theorem proving procedure that could (even theoretically) uncover all true formulas. This i ...
Propositional Logic
... The problem of finding at least one model of the set of formulas that is also a model of the formula , is known as the propositonal satisfiability (PSAT) problem. An exhaustive procedure for solving the PSAT problem is to try systematically all of the ways to assign True and False to the atoms in ...
... The problem of finding at least one model of the set of formulas that is also a model of the formula , is known as the propositonal satisfiability (PSAT) problem. An exhaustive procedure for solving the PSAT problem is to try systematically all of the ways to assign True and False to the atoms in ...
Monadic Predicate Logic is Decidable
... Sketch of a formal proof (by formula induction) • Base Cases: G is atomic. G is of the form Pi(t) or of the form t1=t2 (t, t1, and t2 are variables or constants) 1. Let G = Pi(t). We need to prove: Pi(t) is satisfied by d1 in M iff Pi(t) is satisfied by e1 in M’ But d1 and e1 are similar, hence the ...
... Sketch of a formal proof (by formula induction) • Base Cases: G is atomic. G is of the form Pi(t) or of the form t1=t2 (t, t1, and t2 are variables or constants) 1. Let G = Pi(t). We need to prove: Pi(t) is satisfied by d1 in M iff Pi(t) is satisfied by e1 in M’ But d1 and e1 are similar, hence the ...
Definition - Rogelio Davila
... An exhaustive procedure for solving the PSAT problem is to try systematically all of the ways to assign True and False to the atoms in the formula, checking the assignment to see if all formulas have value True under that assignment. If there are n atoms in the formula, there are 2n different assign ...
... An exhaustive procedure for solving the PSAT problem is to try systematically all of the ways to assign True and False to the atoms in the formula, checking the assignment to see if all formulas have value True under that assignment. If there are n atoms in the formula, there are 2n different assign ...
Predicate logic, motivation
... (Dictionary: P_: _ is a philosopher; F_: _ is female; s: Simone) Even if we don’t yet have a way of proving this argument is valid, we can see the reasoning. Use &I and generalize (if Simone is a female philosopher, then there has to exist at least one female philosopher). ...
... (Dictionary: P_: _ is a philosopher; F_: _ is female; s: Simone) Even if we don’t yet have a way of proving this argument is valid, we can see the reasoning. Use &I and generalize (if Simone is a female philosopher, then there has to exist at least one female philosopher). ...
valid - Informatik Uni Leipzig
... Let us consider the following axiom schemata: T: 2ϕ → ϕ (knowledge axiom) 4: 2ϕ → 22ϕ (positive introspection) 5: 3ϕ → 23ϕ (negative introspection: equivalently ¬2ϕ → 2¬2ϕ) B: ϕ → 23ϕ D: 2ϕ → 3ϕ (disbelief in the negation, equivalently 2ϕ → ¬2¬ϕ) . . . and the following classes of frames, for which ...
... Let us consider the following axiom schemata: T: 2ϕ → ϕ (knowledge axiom) 4: 2ϕ → 22ϕ (positive introspection) 5: 3ϕ → 23ϕ (negative introspection: equivalently ¬2ϕ → 2¬2ϕ) B: ϕ → 23ϕ D: 2ϕ → 3ϕ (disbelief in the negation, equivalently 2ϕ → ¬2¬ϕ) . . . and the following classes of frames, for which ...
What is Logic?
... Propositional Calculus is the language we use to reason about propositional logic. A sentence in propositional logic is called a well-formed formula (wff). ...
... Propositional Calculus is the language we use to reason about propositional logic. A sentence in propositional logic is called a well-formed formula (wff). ...
Creativity and Artificial Intelligence
... Does this imply that logic is irrelevant for planning and for artificial intelligence for that matter? While intelligence implies the ability for planning, the converse has not necessarily to be true. It very much depends on what kind of planning is meant. In a fixed and relatively restricted domain ...
... Does this imply that logic is irrelevant for planning and for artificial intelligence for that matter? While intelligence implies the ability for planning, the converse has not necessarily to be true. It very much depends on what kind of planning is meant. In a fixed and relatively restricted domain ...
Elements of Finite Model Theory
... The most common are based on inductive definitions of relations that always reach a fixed-point, either in a monotone or inflationary semantics. Monotone inductive definitions always give rise to a relational operator which determines a least fixed-point (LFP), whereas inflationary inductive definit ...
... The most common are based on inductive definitions of relations that always reach a fixed-point, either in a monotone or inflationary semantics. Monotone inductive definitions always give rise to a relational operator which determines a least fixed-point (LFP), whereas inflationary inductive definit ...
Modal Logic
... The canonical frame for System K is the pair Fk = (Wk,Rk) where (1) Wk = {X | X is an MCS } (2) If X and Y are MCSs, then X Rk Y iff {❏X} Y. The canonical model for System K is given by Mk = (Fk,Vk) where for each X Wk, Vk(X) = X P. Lemma For each MCS X Wk and for each formula ,Mk ...
... The canonical frame for System K is the pair Fk = (Wk,Rk) where (1) Wk = {X | X is an MCS } (2) If X and Y are MCSs, then X Rk Y iff {❏X} Y. The canonical model for System K is given by Mk = (Fk,Vk) where for each X Wk, Vk(X) = X P. Lemma For each MCS X Wk and for each formula ,Mk ...
Chapter 9
... Typically: we first do the proof for Case 1 where = and function symbols are absent, then Case 2 where identity is present, and finally Case 3 where both identity and function symbols are present. In this last case, there is often a subsidiary induction on complexity to prove that all terms have som ...
... Typically: we first do the proof for Case 1 where = and function symbols are absent, then Case 2 where identity is present, and finally Case 3 where both identity and function symbols are present. In this last case, there is often a subsidiary induction on complexity to prove that all terms have som ...
first order logic
... First order logic is much more expressive than propositional logic. The topics on first order logic are: Quantifiers Negation Multiple quantifiers Arguments of quantified statements ...
... First order logic is much more expressive than propositional logic. The topics on first order logic are: Quantifiers Negation Multiple quantifiers Arguments of quantified statements ...
PDF
... such that each Bj (where j ≤ m) is either an axiom, or a formula in ∆0 . Then certainly this is also a deduction with assumptions in ∆ and conclusion A → B. Therefore, ∆ ` A → B. The deduction theorem holds in most of the widely studied logical systems, such as classical propositional logic and pre ...
... such that each Bj (where j ≤ m) is either an axiom, or a formula in ∆0 . Then certainly this is also a deduction with assumptions in ∆ and conclusion A → B. Therefore, ∆ ` A → B. The deduction theorem holds in most of the widely studied logical systems, such as classical propositional logic and pre ...
Jacques Herbrand (1908 - 1931) Principal writings in logic
... ES(A,p) that makes the expansion true and assigns the numerical value q to the constant c. œxœy∑zı(x,y,z) expresses the existence, for any p and q, of interpretations that make ES(A,p) true, and give the constant c the value q. If the theory is a true theory of arithmetic, then œxœy∑zı(x,y,z) is tru ...
... ES(A,p) that makes the expansion true and assigns the numerical value q to the constant c. œxœy∑zı(x,y,z) expresses the existence, for any p and q, of interpretations that make ES(A,p) true, and give the constant c the value q. If the theory is a true theory of arithmetic, then œxœy∑zı(x,y,z) is tru ...
Chapter 7 Propositional and Predicate Logic
... Relationship between the quantifiers: ∃x P(x) ≡ ¬(∀x)¬P(x) “If There exists an x for which P holds, then it is not true that for all x P does not hold”. ∃x Like(x, War) ≡ ¬(∀x) ¬Like(x, War) ...
... Relationship between the quantifiers: ∃x P(x) ≡ ¬(∀x)¬P(x) “If There exists an x for which P holds, then it is not true that for all x P does not hold”. ∃x Like(x, War) ≡ ¬(∀x) ¬Like(x, War) ...
Syntax of first order logic.
... a model. By compactness, Φ ∪ Ψ has a model. But this must be infinite. q.e.d. ...
... a model. By compactness, Φ ∪ Ψ has a model. But this must be infinite. q.e.d. ...
byd.1 Second-Order logic
... In addition, one can define the class of well-orderings, by adding the following to the definition of a linear ordering: ∀P (∃x P (x) → ∃x (P (x) ∧ ∀y (y < x → ¬P (y)))). This asserts that every non-empty set has a least element, modulo the identification of “set” with “one-place relation”. For ano ...
... In addition, one can define the class of well-orderings, by adding the following to the definition of a linear ordering: ∀P (∃x P (x) → ∃x (P (x) ∧ ∀y (y < x → ¬P (y)))). This asserts that every non-empty set has a least element, modulo the identification of “set” with “one-place relation”. For ano ...
deductive system
... ingredient, and there are only one or two rules of inference (modus ponens is usually one of them). Theorems in a Hilbert system are those formulas which are conclusions of deductions from axioms. • natural deduction: as opposed to a Hilbert system, a natural deduction system consists of all rules a ...
... ingredient, and there are only one or two rules of inference (modus ponens is usually one of them). Theorems in a Hilbert system are those formulas which are conclusions of deductions from axioms. • natural deduction: as opposed to a Hilbert system, a natural deduction system consists of all rules a ...
Propositional Logic
... Γ is valid in a model M (or M satisfies Γ), iff M |= A for every formula A ∈ Γ. We denote this by M |= Γ. Γ is satisfiable iff there exists a model M such that M |= Γ, and it is refutable iff there exists a model M such that M �|= Γ. Γ is valid, denoted by |= Γ, iff M |= Γ for every model M, and it is u ...
... Γ is valid in a model M (or M satisfies Γ), iff M |= A for every formula A ∈ Γ. We denote this by M |= Γ. Γ is satisfiable iff there exists a model M such that M |= Γ, and it is refutable iff there exists a model M such that M �|= Γ. Γ is valid, denoted by |= Γ, iff M |= Γ for every model M, and it is u ...
Problem Set 3
... Write your solutions to the following problems and submit them electronically by Monday, October 13 th at the start of class. These problems will be graded on a 0/1/2 basis based on the effort you have demon strated in solving all of the problems. We will try to get these problems returned to you w ...
... Write your solutions to the following problems and submit them electronically by Monday, October 13 th at the start of class. These problems will be graded on a 0/1/2 basis based on the effort you have demon strated in solving all of the problems. We will try to get these problems returned to you w ...