The Origin of Proof Theory and its Evolution

... First-Order Number Theory - PA (Peano Arithmetic) First-order logic has sufficient expressive power for the formalization of virtually all of mathematics. A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them. Peano arithm ...

... First-Order Number Theory - PA (Peano Arithmetic) First-order logic has sufficient expressive power for the formalization of virtually all of mathematics. A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them. Peano arithm ...

Logic Logical Concepts Deduction Concepts Resolution

... Distinguished from propositional logic by its use of quantifiers Each interpretation of first-order logic includes a domain of discourse over which the quantifiers range ...

... Distinguished from propositional logic by its use of quantifiers Each interpretation of first-order logic includes a domain of discourse over which the quantifiers range ...

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... 1. Continue defining and exploring first-order theory of simple arithmetic, iQ. i Q is a first-order finite axiomatization of a “number-like” domain. Even though i Q is extremely weak as you see from Problem Set 3 from Boolos & Jeffrey, we can, nevertheless, show in constructive type theory, either ...

... 1. Continue defining and exploring first-order theory of simple arithmetic, iQ. i Q is a first-order finite axiomatization of a “number-like” domain. Even though i Q is extremely weak as you see from Problem Set 3 from Boolos & Jeffrey, we can, nevertheless, show in constructive type theory, either ...

1

... 1. (a) Identify the free and bound variable occurrences in the following logical formulas: • ∀x∃y(Rxz ∧ ∃zQyxz), • ∀x((∃yRxy→Ax)→Bxy), • ∀x(Ax→∃yBy) ∧ ∃z(Cxz→∃xDxyz). (b) Give the definition of a atomic formula of predicate logic and of a valuation of terms s based on a variable assignment s. (c) Pr ...

... 1. (a) Identify the free and bound variable occurrences in the following logical formulas: • ∀x∃y(Rxz ∧ ∃zQyxz), • ∀x((∃yRxy→Ax)→Bxy), • ∀x(Ax→∃yBy) ∧ ∃z(Cxz→∃xDxyz). (b) Give the definition of a atomic formula of predicate logic and of a valuation of terms s based on a variable assignment s. (c) Pr ...

Bound and Free Variables Theorems and Proofs

... Suppose you wanted to query a database. How do you do it? Modern database query language date back to SQL (structured query language), and are all based on first-order logic. • The idea goes back to Ted Codd, who invented the notion of relational databases. Suppose you’re a travel agent and want to ...

... Suppose you wanted to query a database. How do you do it? Modern database query language date back to SQL (structured query language), and are all based on first-order logic. • The idea goes back to Ted Codd, who invented the notion of relational databases. Suppose you’re a travel agent and want to ...

EECS 203-1 – Winter 2002 Definitions review sheet

... contradictory expression is false for all assignments of truth values to its variables. A satisfiable formula is an expression which is true for at least one assignment. • Logical equivalence and implication in propositional calculus: Two propositional expressions P and Q are logically equivalent if ...

... contradictory expression is false for all assignments of truth values to its variables. A satisfiable formula is an expression which is true for at least one assignment. • Logical equivalence and implication in propositional calculus: Two propositional expressions P and Q are logically equivalent if ...

Homework 5

... During the second half of this course you should work on a self-chosen project related to the topic of applied logic. This could, for instance, be a literature study about an interesting or the implementation (and documentation) of a proof environment. We will discuss a few possibilities in class. P ...

... During the second half of this course you should work on a self-chosen project related to the topic of applied logic. This could, for instance, be a literature study about an interesting or the implementation (and documentation) of a proof environment. We will discuss a few possibilities in class. P ...

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... substitution axiom that permits substitution on the level of atomic formulas. Since atomic formulas are built from predicate symbols, variables, parameters, and – in a theory with functions and equality – function applications and the equality predicate, we add a termsubstitution axiom for every arg ...

... substitution axiom that permits substitution on the level of atomic formulas. Since atomic formulas are built from predicate symbols, variables, parameters, and – in a theory with functions and equality – function applications and the equality predicate, we add a termsubstitution axiom for every arg ...

Propositional Logic Predicate Logic

... Informal Explanation: When it is True P (x1 , . . . , xn ) “P (x1 , . . . , xn )” A variable that represents a predicate with variables x1 , . . . , xn . ∀x.A “For any x, A” A is true for all individuals x. ∃x.A “There exists x s.t. A” B is true for some individual x. We also use individual constant ...

... Informal Explanation: When it is True P (x1 , . . . , xn ) “P (x1 , . . . , xn )” A variable that represents a predicate with variables x1 , . . . , xn . ∀x.A “For any x, A” A is true for all individuals x. ∃x.A “There exists x s.t. A” B is true for some individual x. We also use individual constant ...

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... It should be noted that there is a fine distinction between boolean valuations and first-order valuations. Boolean valuations can only analyze the propositional structure of formulas. They cannot evaluate quantified formulas and therefore have to treat them like propositional variables. In contrast ...

... It should be noted that there is a fine distinction between boolean valuations and first-order valuations. Boolean valuations can only analyze the propositional structure of formulas. They cannot evaluate quantified formulas and therefore have to treat them like propositional variables. In contrast ...

Practice Problem Set 1

... • These problems will not be graded. • Mutual discussion and discussion with the instructor/TA is strongly encouraged. 1. [From HW1, Autumn 2011] Use the proof system of first order logic studied in class to prove each of the following sequents. You must indicate which proof rule you are applying at ...

... • These problems will not be graded. • Mutual discussion and discussion with the instructor/TA is strongly encouraged. 1. [From HW1, Autumn 2011] Use the proof system of first order logic studied in class to prove each of the following sequents. You must indicate which proof rule you are applying at ...

INF3170 Logikk Spring 2011 Homework #8 Problems 2–6

... does not follow from the other deduction rules. e. Is this semantics complete? That is, is it the case that Γ I φ ⇒ Γ ` φ for Γ a finite set of formulas? Justify your answer. 8. Do problem 1 on page 60. ? 9. Do problem 4 on page 67. For each one, just indicate whether the term is “free” or “not fre ...

... does not follow from the other deduction rules. e. Is this semantics complete? That is, is it the case that Γ I φ ⇒ Γ ` φ for Γ a finite set of formulas? Justify your answer. 8. Do problem 1 on page 60. ? 9. Do problem 4 on page 67. For each one, just indicate whether the term is “free” or “not fre ...

Compactness Theorem for First-Order Logic

... G |-F j, ÿj • There is a proof in F from G for both j and ÿj ...

... G |-F j, ÿj • There is a proof in F from G for both j and ÿj ...

Lecture 16 Notes

... We have a function d : D → P (x) ∨ (P (x) →⊥) in any model. We need to know that there is evidence for (∃e.∀n(P (n) ⇔ ∃b.T (e, n, b)) ⇒⊥) ⇒⊥. ...

... We have a function d : D → P (x) ∨ (P (x) →⊥) in any model. We need to know that there is evidence for (∃e.∀n(P (n) ⇔ ∃b.T (e, n, b)) ⇒⊥) ⇒⊥. ...

Predicate Logic

... Q: What problem may occur if the same symbol is used to represent more than one variable in a formula? ambigous when proving/writing equivalences Q: Soln? do a variable substitution, i.e., above, replace the second y with an ”a” and the second ”z” with a ”b”. ...

... Q: What problem may occur if the same symbol is used to represent more than one variable in a formula? ambigous when proving/writing equivalences Q: Soln? do a variable substitution, i.e., above, replace the second y with an ”a” and the second ”z” with a ”b”. ...

x, y, x

... Q: What problem may occur if the same symbol is used to represent more than one variable in a formula? ambigous when proving/writing equivalences Q: Soln? do a variable substitution, i.e., above, replace the second y with an ”a” and the second ”z” with a ”b”. ...

... Q: What problem may occur if the same symbol is used to represent more than one variable in a formula? ambigous when proving/writing equivalences Q: Soln? do a variable substitution, i.e., above, replace the second y with an ”a” and the second ”z” with a ”b”. ...

T - RTU

... An inference rule is sound, if the conclusion is true in all cases where the premises are true. To prove the soundness, the truth table must be constructed with one line for each possible model of the proposition symbols in the premises. In all models where the premise is true, the conclusion must b ...

... An inference rule is sound, if the conclusion is true in all cases where the premises are true. To prove the soundness, the truth table must be constructed with one line for each possible model of the proposition symbols in the premises. In all models where the premise is true, the conclusion must b ...

Ch1 - COW :: Ceng

... Extend I to all formulas: 1. I(T) = 1 and I() = 0. 2. I(A1 ... An) = 1 if and only if I(Ai) = 1 for all i. 3. I(A1 ... An) = 1 if and only if I(Ai) = 1 for some i. 4. I(A) = 1 if and only if I(A) = 0. 5. I(A B) = 1 if and only if I(A) = 0 or I(B) = 1. 6. I(A B) = 1 if and only if I(A) ...

... Extend I to all formulas: 1. I(T) = 1 and I() = 0. 2. I(A1 ... An) = 1 if and only if I(Ai) = 1 for all i. 3. I(A1 ... An) = 1 if and only if I(Ai) = 1 for some i. 4. I(A) = 1 if and only if I(A) = 0. 5. I(A B) = 1 if and only if I(A) = 0 or I(B) = 1. 6. I(A B) = 1 if and only if I(A) ...

lec26-first-order

... But there exists first order theories defined by axioms which are not sufficient for proving all T-valid formulas. ...

... But there exists first order theories defined by axioms which are not sufficient for proving all T-valid formulas. ...

First order theories

... But there exists first order theories defined by axioms which are not sufficient for proving all T-valid formulas. ...

... But there exists first order theories defined by axioms which are not sufficient for proving all T-valid formulas. ...

First order theories - Decision Procedures

... But there exists first order theories defined by axioms which are not sufficient for proving all T-valid formulas. ...

... But there exists first order theories defined by axioms which are not sufficient for proving all T-valid formulas. ...

Howework 8

... P rov be a provability predicate for the theory Q and X and Y be formulas in the Q. Assume |=Q P rov(dXe) ⊃ Y and |=Q P rov(dY e) ⊃ X ...

... P rov be a provability predicate for the theory Q and X and Y be formulas in the Q. Assume |=Q P rov(dXe) ⊃ Y and |=Q P rov(dY e) ⊃ X ...

INTLOGS16 Test 2

... appearing within the equation as an S-expression, then (ii) give a yes or no answer as to whether the equation is true or not. In addition, (iii) for each of your affirmative verdicts, provide a clear, informal proof that confirms your verdict.1 (a) {∀x(Scared(x) ↔ Small(x)), ∃x¬Scared(x)} ` ∃x¬Smal ...

... appearing within the equation as an S-expression, then (ii) give a yes or no answer as to whether the equation is true or not. In addition, (iii) for each of your affirmative verdicts, provide a clear, informal proof that confirms your verdict.1 (a) {∀x(Scared(x) ↔ Small(x)), ∃x¬Scared(x)} ` ∃x¬Smal ...

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... Substitution is the key to describing the meaning of quantified formulas as well as to formal reasoning about them. A formula of the form (∀p)A means that A must be true no matter what we put in – or substitute – for the variable p. In order to explain substitution, we need to understand the role of ...

... Substitution is the key to describing the meaning of quantified formulas as well as to formal reasoning about them. A formula of the form (∀p)A means that A must be true no matter what we put in – or substitute – for the variable p. In order to explain substitution, we need to understand the role of ...

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 ...