Handout 14
... provable is a tautology. Thus, the formal system of propositional logic is not only sound (i.e. generates only valid formulas) but also generates all of them. Theorem 5.2 (completeness of propositional logic). Let T be a set of formulas and A a formula. Then T (A ...
... provable is a tautology. Thus, the formal system of propositional logic is not only sound (i.e. generates only valid formulas) but also generates all of them. Theorem 5.2 (completeness of propositional logic). Let T be a set of formulas and A a formula. Then T (A ...
Theories.Axioms,Rules of Inference
... Its amazing that one can decide if a statement is a theorem in this manner, instead of the standard way of checking all possibilities. This is how theorem provers work. Now we have a picture of where proofs of theorems come from: a theorem concerns a given theory in a given logic. That theory is a s ...
... Its amazing that one can decide if a statement is a theorem in this manner, instead of the standard way of checking all possibilities. This is how theorem provers work. Now we have a picture of where proofs of theorems come from: a theorem concerns a given theory in a given logic. That theory is a s ...
On a Symposium on the Foundations of Mathematics (1971) Paul
... names for infinitely many individuals, and indeed, for totalities of individuals of arbitrarily high transfinite cardinality. Infinitely long formulas are also considered. This sort of set-theoretical metamathematics shows, among other things, ...
... names for infinitely many individuals, and indeed, for totalities of individuals of arbitrarily high transfinite cardinality. Infinitely long formulas are also considered. This sort of set-theoretical metamathematics shows, among other things, ...
A Note on Assumptions about Skolem Functions
... We have presented an optimized Skolemization of existential quantifiers which moves information about the Skolem function from the local context of the occurrence of the existential quantifier to the top-level of the formula. Instances of this optimized Skolemization have been used implicitly or exp ...
... We have presented an optimized Skolemization of existential quantifiers which moves information about the Skolem function from the local context of the occurrence of the existential quantifier to the top-level of the formula. Instances of this optimized Skolemization have been used implicitly or exp ...
Biform Theories in Chiron
... clear demarcation between the algorithms that are primitive in the theory and those that are derived from the primitive algorithms. A biform theory T is a set Ω of formulas and rules in a language L. A rule in L consists of an algorithm called a transformer that transforms a tuple of input expressio ...
... clear demarcation between the algorithms that are primitive in the theory and those that are derived from the primitive algorithms. A biform theory T is a set Ω of formulas and rules in a language L. A rule in L consists of an algorithm called a transformer that transforms a tuple of input expressio ...
pdf
... (4) T is called axiomatizable, if there is a decidable subset of T whose logical consequences are exactly the theorems of T . T is finitely axiomatizable if it is axiomatizable with a finite set of axioms. (5) A set S ⊆N is called definable in T if there is unary predicate RS in the formal language ...
... (4) T is called axiomatizable, if there is a decidable subset of T whose logical consequences are exactly the theorems of T . T is finitely axiomatizable if it is axiomatizable with a finite set of axioms. (5) A set S ⊆N is called definable in T if there is unary predicate RS in the formal language ...
Sub-Birkhoff
... Two remarks on notation used in the definition: ≡[L] expresses that corresponding components are identical (as terms) except for one index where they are related by L. The notation s(~t) expresses that the sequence of terms ~t is substituted for the corresponding sequence of variables in s. Elements ...
... Two remarks on notation used in the definition: ≡[L] expresses that corresponding components are identical (as terms) except for one index where they are related by L. The notation s(~t) expresses that the sequence of terms ~t is substituted for the corresponding sequence of variables in s. Elements ...
31-3.pdf
... with how complicated it is to describe a set in terms of how many quantifiers you need and what symbols are needed in the language. There are many connections to complexity theory in that virtually all descriptive classes are equivalent to the more standard complexity classes. 2. Theory of Computing ...
... with how complicated it is to describe a set in terms of how many quantifiers you need and what symbols are needed in the language. There are many connections to complexity theory in that virtually all descriptive classes are equivalent to the more standard complexity classes. 2. Theory of Computing ...
Advanced Topics in Mathematics – Logic and Metamathematics Mr
... 2. Consider the following incorrect theorem: Suppose n is a natural number larger than 2 and n is not a prime number. Then 2n + 13 is not a prime number. (a) What are the hypotheses and conclusion of this theorem? (b) Show that the theorem is incorrect by finding a counterexample. 3. Complete the fo ...
... 2. Consider the following incorrect theorem: Suppose n is a natural number larger than 2 and n is not a prime number. Then 2n + 13 is not a prime number. (a) What are the hypotheses and conclusion of this theorem? (b) Show that the theorem is incorrect by finding a counterexample. 3. Complete the fo ...
the common rules of binary connectives are finitely based
... Theorem 2. If `1 , . . . , `n are independent and f.b. then `1 ∩ . . . ∩ `n is f.b. Example 2. As is well known, |=→ , |=← , |=↔ , |=↑ are f.b. Since these logics are independent according to the above, the common rules of →, ←, ↔, ↑ are f.b., by Theorem 2. This yields some special cases of Theorem ...
... Theorem 2. If `1 , . . . , `n are independent and f.b. then `1 ∩ . . . ∩ `n is f.b. Example 2. As is well known, |=→ , |=← , |=↔ , |=↑ are f.b. Since these logics are independent according to the above, the common rules of →, ←, ↔, ↑ are f.b., by Theorem 2. This yields some special cases of Theorem ...
powerpoint - IDA.LiU.se
... Rewrite (or p (or q r)) as (or p q r), with arbitrary number of arguments, and similarly for and The result is an expression on conjunctive normal form Consider the arguments of and as separate formulas, obtaining a set of or-expressions with literals as their arguments Consider these or-expressions ...
... Rewrite (or p (or q r)) as (or p q r), with arbitrary number of arguments, and similarly for and The result is an expression on conjunctive normal form Consider the arguments of and as separate formulas, obtaining a set of or-expressions with literals as their arguments Consider these or-expressions ...
The Closed World Assumption
... woman(bob) nor man(alice) hold, since if either of these were true then our program would not contain complete knowledge about true ground atomic formulas. Thus the CWA allows us to “complete” our knowledge base to the larger theory woman(alice), man(bob), ¬woman(bob), ¬man(alice) Note that we are h ...
... woman(bob) nor man(alice) hold, since if either of these were true then our program would not contain complete knowledge about true ground atomic formulas. Thus the CWA allows us to “complete” our knowledge base to the larger theory woman(alice), man(bob), ¬woman(bob), ¬man(alice) Note that we are h ...
Why the Sets of NF do not form a Cartesian-closed Category
... it f2 ) of the function sending {x} to ({∅} × {∅}) → x is a set. By the same token {∅} → ({∅} → x) is two types higher than x, and—by NF comprehension again—the graph (call it f3 ) of the function sending {∅} → ({∅} → x) to {{x}} is also a set. Then the composition f3 · f1 · f2 sends {x} to {{x}}. T ...
... it f2 ) of the function sending {x} to ({∅} × {∅}) → x is a set. By the same token {∅} → ({∅} → x) is two types higher than x, and—by NF comprehension again—the graph (call it f3 ) of the function sending {∅} → ({∅} → x) to {{x}} is also a set. Then the composition f3 · f1 · f2 sends {x} to {{x}}. T ...
Logic Logical Concepts Deduction Concepts Resolution
... "For every x there is a y s.t. R(x, y)" is converted into "There is a function f mapping every x into a y s.t. for every x R(x, f (x)) holds" ∀x1 . . . ∀xn ∃yR(x1 , . . . , xn , y) is satisfied by a model M iff For each possible value for x1 , . . . , xn there is a value for y that makes R(x1 , . . ...
... "For every x there is a y s.t. R(x, y)" is converted into "There is a function f mapping every x into a y s.t. for every x R(x, f (x)) holds" ∀x1 . . . ∀xn ∃yR(x1 , . . . , xn , y) is satisfied by a model M iff For each possible value for x1 , . . . , xn there is a value for y that makes R(x1 , . . ...
Decidable fragments of first-order logic Decidable fragments of first
... Fix any subset {b1 , b2 } of Bn of size r ∈ {1, 2} and recall that some r -table T{b1 ,b2 } from Tr is assiged to this subset. For any subset {b3 , . . . , bl+2 } of pairwise distinct elements of Bn that differ from b1 and b2 , consider the event that the table induced by b1 , . . . , bl+2 is equal ...
... Fix any subset {b1 , b2 } of Bn of size r ∈ {1, 2} and recall that some r -table T{b1 ,b2 } from Tr is assiged to this subset. For any subset {b3 , . . . , bl+2 } of pairwise distinct elements of Bn that differ from b1 and b2 , consider the event that the table induced by b1 , . . . , bl+2 is equal ...
study guide.
... ∧ the negation of its encoding to the CNF, and apply DeMorgan’s law. • A resolution proof system is used to find a contradiction in a formula (and, similarly, to prove that a formula is a tautology by finding a contradiction in its negation). Resolution starts with a formula in a CNF form, and appli ...
... ∧ the negation of its encoding to the CNF, and apply DeMorgan’s law. • A resolution proof system is used to find a contradiction in a formula (and, similarly, to prove that a formula is a tautology by finding a contradiction in its negation). Resolution starts with a formula in a CNF form, and appli ...
BEYOND FIRST ORDER LOGIC: FROM NUMBER OF
... distinguish it from first order model theory. We give more detailed examples accessible to model theorists of all sorts. We conclude with questions about countable models which require only a basic background in logic. For the past 50 years most research in model theory has focused on first order lo ...
... distinguish it from first order model theory. We give more detailed examples accessible to model theorists of all sorts. We conclude with questions about countable models which require only a basic background in logic. For the past 50 years most research in model theory has focused on first order lo ...
predicate
... • Let be a set of sentences of predicate calculus. If all finite subsets of are satisfiable, then so is . • Proof – uses soundness and completeness and finite length of proofs. ...
... • Let be a set of sentences of predicate calculus. If all finite subsets of are satisfiable, then so is . • Proof – uses soundness and completeness and finite length of proofs. ...
Homomorphism Preservation Theorem
... contains all atomic formulas R(x1; : : : ; x ) and x1 = x2, contains all negated atomic formulas :R(x1; : : : ; x ) and x1 6= x2, is closed under conjunction '1 ^ '2, is closed under disjunction '1 _ '2, is closed under existential quantification (9x)('(x)), is closed under universal qua ...
... contains all atomic formulas R(x1; : : : ; x ) and x1 = x2, contains all negated atomic formulas :R(x1; : : : ; x ) and x1 6= x2, is closed under conjunction '1 ^ '2, is closed under disjunction '1 _ '2, is closed under existential quantification (9x)('(x)), is closed under universal qua ...
hilbert systems - CSA
... S U {~X} is also Consistent If not, S U {~X} |- X S |- (~X > X) { Deduction Theorem } (~X > X) > X { Theorem } S |- X { Modus Ponens } S U {~X} is satisfiable { Model Existence Theorem } S U {X} is not valid. ...
... S U {~X} is also Consistent If not, S U {~X} |- X S |- (~X > X) { Deduction Theorem } (~X > X) > X { Theorem } S |- X { Modus Ponens } S U {~X} is satisfiable { Model Existence Theorem } S U {X} is not valid. ...
MATHEMATICAL LOGIC CLASS NOTE 1. Propositional logic A
... The proposition is proved using routine induction on formulas. By the proposition, without ambiguity we often write M |= φ[a1 , ..., an ] instead of M |= φ[s] where φ = φ(x1 , ..., xn ) and s(xi ) = ai . (Then this is more natural notation since M |= φ[a1 , ..., an ] means informally that φ holds in ...
... The proposition is proved using routine induction on formulas. By the proposition, without ambiguity we often write M |= φ[a1 , ..., an ] instead of M |= φ[s] where φ = φ(x1 , ..., xn ) and s(xi ) = ai . (Then this is more natural notation since M |= φ[a1 , ..., an ] means informally that φ holds in ...
Inference Tasks and Computational Semantics
... • Soundness is typically an easy property to prove. • Proofs typically have some kind of inductive structure. • One shows that if the first part of proof is true in a model then the rules only let us generate formulas that are also true in a model. • Proof follows by induction ...
... • Soundness is typically an easy property to prove. • Proofs typically have some kind of inductive structure. • One shows that if the first part of proof is true in a model then the rules only let us generate formulas that are also true in a model. • Proof follows by induction ...
Lecture 10 Notes
... philosophical side we hear phrases such as “mental constructions” and intuition used to account for human knowledge. On the technical side we see that computers are important factors in the technology of knowledge creation. For PC we have a clear computational semantics for understanding the logical ...
... philosophical side we hear phrases such as “mental constructions” and intuition used to account for human knowledge. On the technical side we see that computers are important factors in the technology of knowledge creation. For PC we have a clear computational semantics for understanding the logical ...
Propositional Logic
... A formula is in prenex form if it is of the form Q1 x1 .Q2 x2 . . . . Qn xn .ψ (possibly with n = 0) where each Qi is a quantifier (either ∀ or ∃) and ψ is a quantifier-free formula . Proposition For any formula of first-order logic, there exists an equivalent formula in prenex form. Proof. Such a p ...
... A formula is in prenex form if it is of the form Q1 x1 .Q2 x2 . . . . Qn xn .ψ (possibly with n = 0) where each Qi is a quantifier (either ∀ or ∃) and ψ is a quantifier-free formula . Proposition For any formula of first-order logic, there exists an equivalent formula in prenex form. Proof. Such a p ...