term 1 - Teaching-WIKI
... • In propositional logic the smallest atoms represent whole propositions (propositions are atomic) – Propositional logic does not capture the internal structure of the propositions – It is not possible to work with units smaller than a proposition ...
... • In propositional logic the smallest atoms represent whole propositions (propositions are atomic) – Propositional logic does not capture the internal structure of the propositions – It is not possible to work with units smaller than a proposition ...
Propositional Calculus
... Logic helps to clarify the meanings of descriptions written, for example, in English. After all, one reason for our use of logic is to state precisely the requirements of computer systems. Descriptions in natural languages can be imprecise and ambiguous. An ambiguous sentence can have more than one ...
... Logic helps to clarify the meanings of descriptions written, for example, in English. After all, one reason for our use of logic is to state precisely the requirements of computer systems. Descriptions in natural languages can be imprecise and ambiguous. An ambiguous sentence can have more than one ...
completeness theorem for a first order linear
... that is a deductively closed set which does not contain all formulas, and as a consequence that is consistent. Suppose that . We can show that ...
... that is a deductively closed set which does not contain all formulas, and as a consequence that is consistent. Suppose that . We can show that ...
Propositional Logic .
... We will first see an example. Then we will define these notions formally. ...
... We will first see an example. Then we will define these notions formally. ...
Logic and Proof
... (1) It is not sunny this afternoon (¬ p) and it is colder than yesterday (q). (2) We will go swimming (r) only if it is sunny (p). (3) If we do not go swimming (¬ r), then we will take a canoe trip (s). (4) If we take a canoe trip (s), then we will be home by sunset (t). Therefore, we will be home b ...
... (1) It is not sunny this afternoon (¬ p) and it is colder than yesterday (q). (2) We will go swimming (r) only if it is sunny (p). (3) If we do not go swimming (¬ r), then we will take a canoe trip (s). (4) If we take a canoe trip (s), then we will be home by sunset (t). Therefore, we will be home b ...
Logic - Mathematical Institute SANU
... constants. In describing logical laws one keeps these constants and the rest is replaced by variables, which makes logical form appear. In algebraic laws, one has analogously constants for operations and variables for numbers. The usual logical constants are the connectives and, or, if, if and only ...
... constants. In describing logical laws one keeps these constants and the rest is replaced by variables, which makes logical form appear. In algebraic laws, one has analogously constants for operations and variables for numbers. The usual logical constants are the connectives and, or, if, if and only ...
Discrete Structure
... Applications of Predicate Logic • It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems (more on these in module 2) for any branch of mathematics. • Predicate logic with function symbols, the “=” operator, and a few proof-buil ...
... Applications of Predicate Logic • It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems (more on these in module 2) for any branch of mathematics. • Predicate logic with function symbols, the “=” operator, and a few proof-buil ...
Logic in Proofs (Valid arguments) A theorem is a hypothetical
... Logic in Proofs (Valid arguments) A theorem is a hypothetical statement of the form H 6 C, where H is a (compound) statement which is taken as being true, and C is a statement which follows from H by logical reasoning. Example: [(p 6 q) v (q 6 r) v (¬ r)] 6 (¬ p) An argument in logic is a way to rea ...
... Logic in Proofs (Valid arguments) A theorem is a hypothetical statement of the form H 6 C, where H is a (compound) statement which is taken as being true, and C is a statement which follows from H by logical reasoning. Example: [(p 6 q) v (q 6 r) v (¬ r)] 6 (¬ p) An argument in logic is a way to rea ...
PRESENTATION OF NATURAL DEDUCTION R. P. NEDERPELT
... of V xEA [P(x)], the natural first step is: "Let xEA". It will be clear that the latter sentence is not an assumption in the proper sense, as it also introduces the variable x. There is, however, a strong analogy with "normal" assumptions of the kind "Assume p", notably with respect to validity and ...
... of V xEA [P(x)], the natural first step is: "Let xEA". It will be clear that the latter sentence is not an assumption in the proper sense, as it also introduces the variable x. There is, however, a strong analogy with "normal" assumptions of the kind "Assume p", notably with respect to validity and ...
slides - National Taiwan University
... A set Σ of expressions is decidable iff there exists an effective procedure (algorithm) that, given an expression α, decides whether or not α ∈ Σ A set Σ of expressions is semidecidable iff there exists an effective procedure (semialgorithm) that, given an expression α, produces the answer “yes” iff α ∈ ...
... A set Σ of expressions is decidable iff there exists an effective procedure (algorithm) that, given an expression α, decides whether or not α ∈ Σ A set Σ of expressions is semidecidable iff there exists an effective procedure (semialgorithm) that, given an expression α, produces the answer “yes” iff α ∈ ...
Logic - Decision Procedures
... We will first see an example. Then we will define these notions formally. ...
... We will first see an example. Then we will define these notions formally. ...
PDF
... In this entry, we show that the deduction theorem below holds for intuitionistic propositional logic. We use the axiom system provided in this entry. Theorem 1. If ∆, A `i B, where ∆ is a set of wff ’s of the intuitionistic propositional logic, then ∆ `i A → B. The proof is very similar to that of t ...
... In this entry, we show that the deduction theorem below holds for intuitionistic propositional logic. We use the axiom system provided in this entry. Theorem 1. If ∆, A `i B, where ∆ is a set of wff ’s of the intuitionistic propositional logic, then ∆ `i A → B. The proof is very similar to that of t ...
Conditional and Indirect Proofs
... • A tautology will follow from any premises whatever. • This is because the negation of a tautology is a contradiction, so if we use IP by assuming the negation of a tautology, we can derive a contradiction independently of other premises. This is why this process is called a zeropremise deduction. ...
... • A tautology will follow from any premises whatever. • This is because the negation of a tautology is a contradiction, so if we use IP by assuming the negation of a tautology, we can derive a contradiction independently of other premises. This is why this process is called a zeropremise deduction. ...
slides
... Want a way to prove partial correctness statements valid... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional ...
... Want a way to prove partial correctness statements valid... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional ...
WhichQuantifiersLogical
... and quantifiers is to be established semantically in one way or another prior to their inferential role. Their meanings may be the primitives of our reasoning in general“and”, “or”, “not”, “if…then”, “all”, “some”or they may be understood informally like “most”, “has the same number as”, etc. in a ...
... and quantifiers is to be established semantically in one way or another prior to their inferential role. Their meanings may be the primitives of our reasoning in general“and”, “or”, “not”, “if…then”, “all”, “some”or they may be understood informally like “most”, “has the same number as”, etc. in a ...
Propositional Logic First Order Logic
... We can turn an inference rule into an axiom if we have ‘→’ in the logic. ...
... We can turn an inference rule into an axiom if we have ‘→’ in the logic. ...
First-Order Predicate Logic (2) - Department of Computer Science
... Undecidability of satisfiability (very informal!) We show that there does not exist any algorithm (no computer program) that decides whether a first-order predicate logic sentence is satisfiable; i.e., for input sentence G outputs “Yes” if G is satisfiable and “No” if G is not satisfiable. ...
... Undecidability of satisfiability (very informal!) We show that there does not exist any algorithm (no computer program) that decides whether a first-order predicate logic sentence is satisfiable; i.e., for input sentence G outputs “Yes” if G is satisfiable and “No” if G is not satisfiable. ...
Partial Correctness Specification
... The inference rules of Floyd-Hoare logic will be specified with a notation of the form ...
... The inference rules of Floyd-Hoare logic will be specified with a notation of the form ...
Comments on predicative logic
... This is a nice alternative, but we discuss another one. Namely, restrict the range of the ∀-elimination rule to atomic formulas. For lack of a better name, let us call this restricted calculus atomic PSOLi . Observe that Theorem 1 still goes through with atomic PSOLi (instead of predicative PSOLi ). ...
... This is a nice alternative, but we discuss another one. Namely, restrict the range of the ∀-elimination rule to atomic formulas. For lack of a better name, let us call this restricted calculus atomic PSOLi . Observe that Theorem 1 still goes through with atomic PSOLi (instead of predicative PSOLi ). ...
arguments (an upper and lower bound (integers) and a function
... Question 3. Modify the code above to numerically integrate between the lower bound a and the upper bound b. Hint, this is done by adding another parameter, width. The integral can then be calculated by summing up lots of rectangles of with area (width*f(x)). The narrower the width, the better the ap ...
... Question 3. Modify the code above to numerically integrate between the lower bound a and the upper bound b. Hint, this is done by adding another parameter, width. The integral can then be calculated by summing up lots of rectangles of with area (width*f(x)). The narrower the width, the better the ap ...
What is...Linear Logic? Introduction Jonathan Skowera
... the first connective will be called instead & (with), and the second connective will be called ⊗ (tensor ). The ⊗ rules apply in any context (they are contextfree), while the & rule R∧ requires the context of the two sequences to agree (it is context-sensitive). Another way to divide the rules? We m ...
... the first connective will be called instead & (with), and the second connective will be called ⊗ (tensor ). The ⊗ rules apply in any context (they are contextfree), while the & rule R∧ requires the context of the two sequences to agree (it is context-sensitive). Another way to divide the rules? We m ...
Propositional and predicate logic - Computing Science
... applying logic, for establishing and verifying facts, and changing or justifying practices, institutions, and beliefs based on new or existing ...
... applying logic, for establishing and verifying facts, and changing or justifying practices, institutions, and beliefs based on new or existing ...