Propositional logic - Computing Science
... Argument 1: If the program syntax is faulty or if program execution results in division by zero, then the computer will generate an error message. Therefore, if the computer does not generate, then the program syntax is correct and program execution does not result in division by zero. Argument 2: I ...
... Argument 1: If the program syntax is faulty or if program execution results in division by zero, then the computer will generate an error message. Therefore, if the computer does not generate, then the program syntax is correct and program execution does not result in division by zero. Argument 2: I ...
IS IT EASY TO LEARN THE LOGIC
... operations. The problem is still worse in Natural Deduction proofs because the effort of reasoning is richer in possible alternatives to obtain the desired result by applying logical rules. However, the logical operations are necessary both in the decision as in the proof, because they put in motion ...
... operations. The problem is still worse in Natural Deduction proofs because the effort of reasoning is richer in possible alternatives to obtain the desired result by applying logical rules. However, the logical operations are necessary both in the decision as in the proof, because they put in motion ...
Proof Theory in Type Theory
... Intuitively, we use arbitrary proof of ` B(σ) instead of using ordinals. For instance, we have clearly ∀∆ ⊆ Pos[`S0 ∆, B(σ) → `S1 ∆, B(σ)] because S0 is a subsystem of S1 and hence, by the Ω-rule `S1 B(σ), B0 (σ). It follos by the usual argument that we have `S1 A, ¬A for any formula A. It is then d ...
... Intuitively, we use arbitrary proof of ` B(σ) instead of using ordinals. For instance, we have clearly ∀∆ ⊆ Pos[`S0 ∆, B(σ) → `S1 ∆, B(σ)] because S0 is a subsystem of S1 and hence, by the Ω-rule `S1 B(σ), B0 (σ). It follos by the usual argument that we have `S1 A, ¬A for any formula A. It is then d ...
unit 8 - WordPress.com
... forms and functions. Arguments to functions are always passed by sharing and are evaluated before they are passed (i.e., in applicative order). Arguments to special forms are passed unevaluated¡ªin other words, by name. Each special form is free to choose internally when (and if) to evaluate its par ...
... forms and functions. Arguments to functions are always passed by sharing and are evaluated before they are passed (i.e., in applicative order). Arguments to special forms are passed unevaluated¡ªin other words, by name. Each special form is free to choose internally when (and if) to evaluate its par ...
full text (.pdf)
... interpreted as universal Horn sentences over relational models. 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 sho ...
... interpreted as universal Horn sentences over relational models. 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 sho ...
4 slides/page
... • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the following: Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence thi ...
... • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the following: Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence thi ...
Chapter 15 Functional Programming
... If the free variable of N have no bound occurrences in M, then the term M[N/x] is formed by replacing all free occurrences of x in M by N. Otherwise, assume that the variable y is free in N and bound in M. Then consistently replace the binding and corresponding bound occurrences of y in M by a new v ...
... If the free variable of N have no bound occurrences in M, then the term M[N/x] is formed by replacing all free occurrences of x in M by N. Otherwise, assume that the variable y is free in N and bound in M. Then consistently replace the binding and corresponding bound occurrences of y in M by a new v ...
Tautologies Arguments Logical Implication
... axioms and inference rules from which it is possible to derive all and only the tautologies? • Soundness says that only tautologies are derivable ...
... axioms and inference rules from which it is possible to derive all and only the tautologies? • Soundness says that only tautologies are derivable ...
The Diagonal Lemma Fails in Aristotelian Logic
... exist. However, the formulae in Table 2 are implausible translations of the natural language sentences. (Strawson, 1952, p. 173) So he proposed to take the term (∃x)Fx as a presupposition. It means that ~(Ex)Fx does not imply that A is false, but rather (Ex)Fx “is a necessary precondition not merely ...
... exist. However, the formulae in Table 2 are implausible translations of the natural language sentences. (Strawson, 1952, p. 173) So he proposed to take the term (∃x)Fx as a presupposition. It means that ~(Ex)Fx does not imply that A is false, but rather (Ex)Fx “is a necessary precondition not merely ...
Syntax and Semantics of Propositional and Predicate Logic
... When doing formal logic, we use ordinary mathematical concepts and tools—functions, variables, deductions, etc. This creates a potential confusion: when we mention a mathematical notion, do we mean that we are using the notion, or that we are talking about it? For example, consider the variable ’x’. ...
... When doing formal logic, we use ordinary mathematical concepts and tools—functions, variables, deductions, etc. This creates a potential confusion: when we mention a mathematical notion, do we mean that we are using the notion, or that we are talking about it? For example, consider the variable ’x’. ...
Functional programming
... Functional languages are statically (Haskell) or dynamically (Lisp) typed ...
... Functional languages are statically (Haskell) or dynamically (Lisp) typed ...
PPT
... • Program – S-expression and consists of Sexpressions, e.g. (A (B C …) (S (G H …) K)) – The basic elements of s-expressions are lists and atoms. – S-expression may be interpreted as list (data structure) – Or function with arguments, e.g. (ADD (SUB 4 3) 6) return 7. Sexpression may be evaluated and ...
... • Program – S-expression and consists of Sexpressions, e.g. (A (B C …) (S (G H …) K)) – The basic elements of s-expressions are lists and atoms. – S-expression may be interpreted as list (data structure) – Or function with arguments, e.g. (ADD (SUB 4 3) 6) return 7. Sexpression may be evaluated and ...
PDF
... 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 predicate logic, intuitionistic logic, normal modal logics, to name a few. On the ...
... 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 predicate logic, intuitionistic logic, normal modal logics, to name a few. On the ...
The Decision Problem for Standard Classes
... Below 0, 1n, 1 and 21 denote the place sequences (0, 0, * ), (n, 0..0 .), (W, 0 0. ... ), and (0, 1, 0, 0,. ..), respectively. We say that a class K of formulas is decidable if both satisfiability and finite satisfiability (that is, satisfiability in a finite model) are decidable for formulas in K. ...
... Below 0, 1n, 1 and 21 denote the place sequences (0, 0, * ), (n, 0..0 .), (W, 0 0. ... ), and (0, 1, 0, 0,. ..), respectively. We say that a class K of formulas is decidable if both satisfiability and finite satisfiability (that is, satisfiability in a finite model) are decidable for formulas in K. ...
Jean Van Heijenoort`s View of Modern Logic
... the proposition into subject and predicate had been replaced by its analysis into function and argument(s). A preliminary accomplishment was the propositional calculus, with a truth-functional definition of the connectives, including the conditional. Of cardinal importance was the realization that, ...
... the proposition into subject and predicate had been replaced by its analysis into function and argument(s). A preliminary accomplishment was the propositional calculus, with a truth-functional definition of the connectives, including the conditional. Of cardinal importance was the realization that, ...
Ch1516rev
... that is closer to Pascal than to LISP Uses type declarations, but also does type inferencing to determine the types of undeclared variables (will see in Chapter 5) It is strongly typed (whereas Scheme is essentially typeless) and has no type coercions Includes exception handling and a module f ...
... that is closer to Pascal than to LISP Uses type declarations, but also does type inferencing to determine the types of undeclared variables (will see in Chapter 5) It is strongly typed (whereas Scheme is essentially typeless) and has no type coercions Includes exception handling and a module f ...
Chapter 1
... that is closer to Pascal than to LISP Uses type declarations, but also does type inferencing to determine the types of undeclared variables (will see in Chapter 5) It is strongly typed (whereas Scheme is essentially typeless) and has no type coercions Includes exception handling and a module f ...
... that is closer to Pascal than to LISP Uses type declarations, but also does type inferencing to determine the types of undeclared variables (will see in Chapter 5) It is strongly typed (whereas Scheme is essentially typeless) and has no type coercions Includes exception handling and a module f ...
Default Rules for Curry
... Abstract. In functional logic programs, rules are applicable independently of textual order, i.e., any rule can potentially be used to evaluate an expression. This is similar to logic languages and opposite to functional languages, e.g., Haskell enforces a strict sequential interpretation of rules. ...
... Abstract. In functional logic programs, rules are applicable independently of textual order, i.e., any rule can potentially be used to evaluate an expression. This is similar to logic languages and opposite to functional languages, e.g., Haskell enforces a strict sequential interpretation of rules. ...
Modal_Logics_Eyal_Ariel_151107
... holds” or in other terms “after every terminating execution of , holds”. ...
... holds” or in other terms “after every terminating execution of , holds”. ...
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 ...
MUltseq: a Generic Prover for Sequents and Equations*
... logics. This means that it takes as input the rules of a many-valued sequent calculus as well as a many-sided sequent and searches – automatically or interactively – for a proof of the latter. For the sake of readability, the output of MUltseq is typeset as a LATEX document. Though the sequent rules ...
... logics. This means that it takes as input the rules of a many-valued sequent calculus as well as a many-sided sequent and searches – automatically or interactively – for a proof of the latter. For the sake of readability, the output of MUltseq is typeset as a LATEX document. Though the sequent rules ...
Lecture 3
... than Q and Q is weaker than P. This because Q is true in more (or at least the same) states than P. P imposes more restrictions on a state. The strongest formula is false (true in fewer case) and the weakest is true (true in more case). • The following theorems are called weakening or strengthen dep ...
... than Q and Q is weaker than P. This because Q is true in more (or at least the same) states than P. P imposes more restrictions on a state. The strongest formula is false (true in fewer case) and the weakest is true (true in more case). • The following theorems are called weakening or strengthen dep ...
Definition - Rogelio Davila
... “Ideography, a Formula language, Modeled upon that of Arithmetic, for Pure Thought” (1879), introduced the Quantification Logic. Alfred Tarsky (1902-1983), mathematician and logician, remarked the importance of distinguishing between the object language and the metalanguage. ...
... “Ideography, a Formula language, Modeled upon that of Arithmetic, for Pure Thought” (1879), introduced the Quantification Logic. Alfred Tarsky (1902-1983), mathematician and logician, remarked the importance of distinguishing between the object language and the metalanguage. ...
Lecture 21 - FSU Computer Science
... • Pure functional programming languages only allow pure functions in a program. • A pure function can be counted on to return the same output each time we invoke it with the same input parameter values. • Can be emulated in traditional languages: expressions behave like pure functions; many routines ...
... • Pure functional programming languages only allow pure functions in a program. • A pure function can be counted on to return the same output each time we invoke it with the same input parameter values. • Can be emulated in traditional languages: expressions behave like pure functions; many routines ...