A(x)
... Formula B logically follows from A1, …, An, denoted A1,…,An |= B, iff B is true in every model of {A1,…,An}. Thus for every interpretation I in which the formulas A1, …, An are true it holds that the formula B is true as well: A1,…,An |= B: If |=I A1,…, |=I An then |=I B, for all I. Note that the “c ...
... Formula B logically follows from A1, …, An, denoted A1,…,An |= B, iff B is true in every model of {A1,…,An}. Thus for every interpretation I in which the formulas A1, …, An are true it holds that the formula B is true as well: A1,…,An |= B: If |=I A1,…, |=I An then |=I B, for all I. Note that the “c ...
Introduction to Proofs, Rules of Equivalence, Rules of
... Valid Argument Forms and Rules • Valid argument forms allow us to establish legitimate patterns of inference, legitimate rules. • If a pattern of inference is valid, we can rest assured that any instance of that pattern is valid without having to the instance. • Never forget: Validity is a matter o ...
... Valid Argument Forms and Rules • Valid argument forms allow us to establish legitimate patterns of inference, legitimate rules. • If a pattern of inference is valid, we can rest assured that any instance of that pattern is valid without having to the instance. • Never forget: Validity is a matter o ...
Problem Set 3
... In each of the following, you will be given a list of first-order predicates and functions along with an English sentence. In each case, write a statement in first-order logic that expresses the indicated sentence. Your statement may use any first-order construct (equality, connectives, quantifiers, ...
... In each of the following, you will be given a list of first-order predicates and functions along with an English sentence. In each case, write a statement in first-order logic that expresses the indicated sentence. Your statement may use any first-order construct (equality, connectives, quantifiers, ...
Propositional Logic: Part I - Semantics
... “If pigs could fly then I’d enjoy brussel sprouts!” p : Pigs fly; b : Enjoy sprouts This (p |= b) is an invalid argument. Why use it? The real argument is: p, ¬p |= b which is a valid argument. Why is it valid? There is no counter example where p ∧ ¬p is true and b is false. Ex falso quod libet! i. ...
... “If pigs could fly then I’d enjoy brussel sprouts!” p : Pigs fly; b : Enjoy sprouts This (p |= b) is an invalid argument. Why use it? The real argument is: p, ¬p |= b which is a valid argument. Why is it valid? There is no counter example where p ∧ ¬p is true and b is false. Ex falso quod libet! i. ...
A short proof of the strong normalization of the simply typed
... types of Girard’s system F was done by Parigot in [6] in two different ways : by using reducibility candidates and by a CPS transformation to the λ-calculus. The technique we present here can also be used to prove the strong normalization of the cut elimination procedure for the classical natural de ...
... types of Girard’s system F was done by Parigot in [6] in two different ways : by using reducibility candidates and by a CPS transformation to the λ-calculus. The technique we present here can also be used to prove the strong normalization of the cut elimination procedure for the classical natural de ...
Introduction to Theoretical Computer Science, lesson 3
... home work a) 1. case – by means of a table: home work b) Indirect: premises + negated conclusion (p r) (p r) and we assume that every f. is true: ...
... home work a) 1. case – by means of a table: home work b) Indirect: premises + negated conclusion (p r) (p r) and we assume that every f. is true: ...
completeness theorem for a first order linear
... system for PLTL was given in [8], while its rst order extension, FOLTL, was presented in [13]. There are many complete axiomatizations of di erent rst order temporal logics. For example, some kinds of such logics with F and P operators over various classes of time ows were axiomatized in [9], whi ...
... system for PLTL was given in [8], while its rst order extension, FOLTL, was presented in [13]. There are many complete axiomatizations of di erent rst order temporal logics. For example, some kinds of such logics with F and P operators over various classes of time ows were axiomatized in [9], whi ...
PDF
... remains is the case when A has the form D. We do induction on the number n of ’s in A. The case when n = 0 means that A is a wff of PLc , and has already been proved. Now suppose A has n + 1 ’s. Then D has n ’s, and so by induction, ` D[B/p] ↔ D[C/p], and therefore ` D[B/p] ↔ D[C/p] by 2. This ...
... remains is the case when A has the form D. We do induction on the number n of ’s in A. The case when n = 0 means that A is a wff of PLc , and has already been proved. Now suppose A has n + 1 ’s. Then D has n ’s, and so by induction, ` D[B/p] ↔ D[C/p], and therefore ` D[B/p] ↔ D[C/p] by 2. This ...
pdf - Consequently.org
... On these grounds, if the background account of deducibility included the commitment that some p did not entail some q, then, relative to that background, tonk fails the demand of consistency. This is one of the tests Belnap considers in the paper. In the case of a natural deduction proof theory or a ...
... On these grounds, if the background account of deducibility included the commitment that some p did not entail some q, then, relative to that background, tonk fails the demand of consistency. This is one of the tests Belnap considers in the paper. In the case of a natural deduction proof theory or a ...
Exercise
... P(x) it is not enough to show that P(a) is true for one or some a’s. 2. To show that a statement of the form x P(x) is FALSE, it is enough to show that P(a) is false for one a ...
... P(x) it is not enough to show that P(a) is true for one or some a’s. 2. To show that a statement of the form x P(x) is FALSE, it is enough to show that P(a) is false for one a ...
handout
... exactly to the proof rules of propositional intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Two of its main pr ...
... exactly to the proof rules of propositional intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Two of its main pr ...
Stephen Cook and Phuong Nguyen. Logical foundations of proof
... book. Let PH be the class of languages in the polynomial-time hierarchy. In logic terms, these are the sets of binary strings that are described by a bounded first-order formula in a two-sorted language, with one sort for natural numbers and one sort for binary strings indexed by these numbers, equi ...
... book. Let PH be the class of languages in the polynomial-time hierarchy. In logic terms, these are the sets of binary strings that are described by a bounded first-order formula in a two-sorted language, with one sort for natural numbers and one sort for binary strings indexed by these numbers, equi ...
Notes and exercises on First Order Logic
... Variables are placeholders so we must have some means of replacing them with more concrete information. We often need to replace a free variable v by an entire term t, i.e. substitute t for v. In substituting t for v we have to leave untouched the bound occurrences of v since they are in the scope o ...
... Variables are placeholders so we must have some means of replacing them with more concrete information. We often need to replace a free variable v by an entire term t, i.e. substitute t for v. In substituting t for v we have to leave untouched the bound occurrences of v since they are in the scope o ...
Lecture Notes 2
... Example Informal Proof Prove: If a is smaller than b and c is identical to b then c is larger than a. Since we are given that a is smaller than b, it follows that b must be larger than a. Moreover, since c is identical to b, it follows that c must be larger than a. ...
... Example Informal Proof Prove: If a is smaller than b and c is identical to b then c is larger than a. Since we are given that a is smaller than b, it follows that b must be larger than a. Moreover, since c is identical to b, it follows that c must be larger than a. ...
CS 2742 (Logic in Computer Science) Lecture 6
... The truth table showed us a situation when both premises (p → q) and q are true, but the conclusion p is false. Therefore, ((p → q) ∧ q) → p is not a tautology and thus the argument based on it is not a valid argument. However, note that if any of the premises are false, a valid argument can produce ...
... The truth table showed us a situation when both premises (p → q) and q are true, but the conclusion p is false. Therefore, ((p → q) ∧ q) → p is not a tautology and thus the argument based on it is not a valid argument. However, note that if any of the premises are false, a valid argument can produce ...
full text (.pdf)
... We consider two related decision problems: given a rule of the form (1), (i) is it relationally valid? That is, is it true in all relational models? (ii) is it derivable in PHL? The paper Kozen 2000] considered problem (i) only. We show that both of these problems are PSPACE -hard by a single reduc ...
... We consider two related decision problems: given a rule of the form (1), (i) is it relationally valid? That is, is it true in all relational models? (ii) is it derivable in PHL? The paper Kozen 2000] considered problem (i) only. We show that both of these problems are PSPACE -hard by a single reduc ...
PDF
... where V is the set of variables and V (Σ) is the set of variables and constants, with modus ponens as its rule of inference: from A and A → B we may infer B. The first three axiom schemas and the modus ponens tell us that predicate logic is an extension of the propositional logic. On the other hand, ...
... where V is the set of variables and V (Σ) is the set of variables and constants, with modus ponens as its rule of inference: from A and A → B we may infer B. The first three axiom schemas and the modus ponens tell us that predicate logic is an extension of the propositional logic. On the other hand, ...
Lecture 3
... using mathematics and logic. The models of these theories often match the actual world is some way. But our scientific theories can be wrong. ...
... using mathematics and logic. The models of these theories often match the actual world is some way. But our scientific theories can be wrong. ...
Mathematical Logic
... axioms and inference rules from which it was possible to derive all the tautologies. Unnatural Proofs and deductions in Hilbert axiomatization are awkward and unnatural. Other proof styles, such as Natural Deductions, are more intuitive. As a matter of facts, nobody is practically using Hilbert calc ...
... axioms and inference rules from which it was possible to derive all the tautologies. Unnatural Proofs and deductions in Hilbert axiomatization are awkward and unnatural. Other proof styles, such as Natural Deductions, are more intuitive. As a matter of facts, nobody is practically using Hilbert calc ...
Propositional and First Order Reasoning
... DPLL Proof is Resolution Proof • Why is each reasoning step resolution • When DPLL terminates, it can emit a proof • Claim: – it can always emit a resolution proof – emitting proofs is only polynomial overhead, a natural extension of the algorithm ...
... DPLL Proof is Resolution Proof • Why is each reasoning step resolution • When DPLL terminates, it can emit a proof • Claim: – it can always emit a resolution proof – emitting proofs is only polynomial overhead, a natural extension of the algorithm ...
Quiz Game Midterm
... consequence of some premises. Give an informal or formal proof of the argument (your choice), and then construct a truth table to show this result. Make sure to write down your explanation of why it’s a logical but not tautological truth. ...
... consequence of some premises. Give an informal or formal proof of the argument (your choice), and then construct a truth table to show this result. Make sure to write down your explanation of why it’s a logical but not tautological truth. ...
HW 12
... 4. The difference between two sets A and B is the set of all objects that belong to set A but not to B. This is written as A \ B a. Provide a definitional axiom for A \ B (use a 2-place function symbol diff(x,y)) b. Construct a formal proof that shows that for any sets A, B, and C: A (B \ C) = (A ...
... 4. The difference between two sets A and B is the set of all objects that belong to set A but not to B. This is written as A \ B a. Provide a definitional axiom for A \ B (use a 2-place function symbol diff(x,y)) b. Construct a formal proof that shows that for any sets A, B, and C: A (B \ C) = (A ...
INTRODUCTION TO LOGIC Lecture 6 Natural Deduction Proofs in
... Proofs in Natural Deduction Proofs in Natural Deduction are trees of L2 -sentences ...
... Proofs in Natural Deduction Proofs in Natural Deduction are trees of L2 -sentences ...
Predicate logic
... • Intuitionistic first-order logic uses intuitionistic rather than classical propositional calculus; for example, ¬¬φ need not be equivalent to φ • Infinitary logic allows infinitely long sentences; for example, one may allow a conjunction or disjunction of infinitely many formulas, or quantificatio ...
... • Intuitionistic first-order logic uses intuitionistic rather than classical propositional calculus; for example, ¬¬φ need not be equivalent to φ • Infinitary logic allows infinitely long sentences; for example, one may allow a conjunction or disjunction of infinitely many formulas, or quantificatio ...
Document
... • P : it-is-raining-here-now • since this is either a true or false statement about the world, the value of P is either true or false ...
... • P : it-is-raining-here-now • since this is either a true or false statement about the world, the value of P is either true or false ...