HOARE`S LOGIC AND PEANO`S ARITHMETIC
... Induction scheme : for each assertion p E L, containing free variable X,the foilowingisanaxiom[p(O)AVx l(p(x)+p(x+l))]+Vx *p(x), Thus, we may observe that equations (3)-(6) alone define N under initial algebra semantics and so we may consider (1) and (2) as additions, making a first refinement of th ...
... Induction scheme : for each assertion p E L, containing free variable X,the foilowingisanaxiom[p(O)AVx l(p(x)+p(x+l))]+Vx *p(x), Thus, we may observe that equations (3)-(6) alone define N under initial algebra semantics and so we may consider (1) and (2) as additions, making a first refinement of th ...
Symbolic Logic II
... As Sider says, the idea behind Kleene’s “strong” truth table is that if there is enough classical information to determine the truth value of the conditional then we can put the classical truth value in the table, if not, then not — it must remain #. Notice, however, that there are two important di ...
... As Sider says, the idea behind Kleene’s “strong” truth table is that if there is enough classical information to determine the truth value of the conditional then we can put the classical truth value in the table, if not, then not — it must remain #. Notice, however, that there are two important di ...
Higher-order Logic: Foundations
... • If M is a general model and σ a substitution, then V M(σ, t) is uniquely determined, for every term t. V M(σ, t) is value of t in M w.r.t. σ. • Gives rise to the standard notion of satisfiability/validity: ◦ We write V M, σ |= φ for V M(σ, φ) = T . ◦ φ is satisfiable in M if V M, σ |= φ, for some ...
... • If M is a general model and σ a substitution, then V M(σ, t) is uniquely determined, for every term t. V M(σ, t) is value of t in M w.r.t. σ. • Gives rise to the standard notion of satisfiability/validity: ◦ We write V M, σ |= φ for V M(σ, φ) = T . ◦ φ is satisfiable in M if V M, σ |= φ, for some ...
02157 Functional Programming - A brief introduction to Lambda
... scope of an abstraction λx.M in t; otherwise it is free. If x has at least one free occurrence in t, then it is called a free variable of t. ...
... scope of an abstraction λx.M in t; otherwise it is free. If x has at least one free occurrence in t, then it is called a free variable of t. ...
Aristotle, Boole, and Categories
... It follows from all this that a syllogism contains six occurrences of terms, two in each of the three sentences. A further requirement is that there be three terms each having two occurrences in distinct sentences. The following naming convention uniquely identifies the syllogistic form. The conclus ...
... It follows from all this that a syllogism contains six occurrences of terms, two in each of the three sentences. A further requirement is that there be three terms each having two occurrences in distinct sentences. The following naming convention uniquely identifies the syllogistic form. The conclus ...
Least and greatest fixed points in linear logic
... Exponentials As shown above, µMALL= can be encoded using exponentials and second-order quantifiers. But at first-order, exponentials and fixed points are incomparable. We could add exponentials in further work, but conjecture that the essential observations done in this work would stay the same. Non ...
... Exponentials As shown above, µMALL= can be encoded using exponentials and second-order quantifiers. But at first-order, exponentials and fixed points are incomparable. We could add exponentials in further work, but conjecture that the essential observations done in this work would stay the same. Non ...
CHAPTER 1 The main subject of Mathematical Logic is
... For the human reader such representations are less convenient, so we shall stick to the use of bound variables. In the definition of “substitution of expression E 0 for variable x in expression E”, either one requires that no variable free in E 0 becomes bound by a variable-binding operator in E, wh ...
... For the human reader such representations are less convenient, so we shall stick to the use of bound variables. In the definition of “substitution of expression E 0 for variable x in expression E”, either one requires that no variable free in E 0 becomes bound by a variable-binding operator in E, wh ...
When is Metric Temporal Logic Expressively Complete?
... Given an N -bounded FOK formula with one free variable x, we show that it is equivalent to a N 0 -bounded formula (over a possibly larger set of monadic predicates, suitably interpreted) in which the unary functions are only applied to x. We can remove occurrences of unary functions within the scope ...
... Given an N -bounded FOK formula with one free variable x, we show that it is equivalent to a N 0 -bounded formula (over a possibly larger set of monadic predicates, suitably interpreted) in which the unary functions are only applied to x. We can remove occurrences of unary functions within the scope ...
Reasoning without Contradiction
... Adding or subtracting a tautology to its premises will have no effect on the validity of an argument, so it is reasonable to believe that tautologies are not required for reasoning. But contradictions, it seems, feature in tried and trusted proof procedures, so one might suppose that, were contradic ...
... Adding or subtracting a tautology to its premises will have no effect on the validity of an argument, so it is reasonable to believe that tautologies are not required for reasoning. But contradictions, it seems, feature in tried and trusted proof procedures, so one might suppose that, were contradic ...
Logical nihilism - University of Notre Dame
... φ ⊃ ψ, “means” that `IPC ψ in the event that `IPC φ, then one should expect `IPC φ ⊃ ψ in every situation in which the set of theorems of IPC is closed under the rule “from φ, infer ψ.” However, the disjunction property implies that these expectations will not be met. To see this, consider the Kreis ...
... φ ⊃ ψ, “means” that `IPC ψ in the event that `IPC φ, then one should expect `IPC φ ⊃ ψ in every situation in which the set of theorems of IPC is closed under the rule “from φ, infer ψ.” However, the disjunction property implies that these expectations will not be met. To see this, consider the Kreis ...
Chapter 2
... For instance in the following code the output of read is used by square function as argument (read function converts string into numeric form) square (read “4”) Notice the usage of parenthesis which are used to associate the argument with read method. If you do not use these, the interpreter will no ...
... For instance in the following code the output of read is used by square function as argument (read function converts string into numeric form) square (read “4”) Notice the usage of parenthesis which are used to associate the argument with read method. If you do not use these, the interpreter will no ...
Admissible rules in the implication-- negation fragment of intuitionistic logic
... 1. Γ ⊢LW ∆ iff Π ⊢LW ∆, m m 2. Γ |∼L ∆ iff Π |∼L ∆. 2 This follows from the fact that any such fragment has the classical deduction theorem and an extension of the implication–negation fragment of IPC has the classical deduction theorem iff it is an axiomatic extension (a folklore result; for an exp ...
... 1. Γ ⊢LW ∆ iff Π ⊢LW ∆, m m 2. Γ |∼L ∆ iff Π |∼L ∆. 2 This follows from the fact that any such fragment has the classical deduction theorem and an extension of the implication–negation fragment of IPC has the classical deduction theorem iff it is an axiomatic extension (a folklore result; for an exp ...
02/06
... Neumann architecture Efficiency is the primary concern, rather than the suitability of the language for software development ...
... Neumann architecture Efficiency is the primary concern, rather than the suitability of the language for software development ...
connections to higher type Recursion Theory, Proof-Theory
... possible, one may have the wings burned: the first system invented by Church led to contradictions. Inconsistencies, though, frequently occur in early versions of interesting formal systems: Frege's set theory, Church's "set of postulates", Martin-Löf's type theory were all inconsistent. This was du ...
... possible, one may have the wings burned: the first system invented by Church led to contradictions. Inconsistencies, though, frequently occur in early versions of interesting formal systems: Frege's set theory, Church's "set of postulates", Martin-Löf's type theory were all inconsistent. This was du ...
Lecture 4 - Michael De
... Weak 3-valued Kleene/Bochvar logic Another three-valued non-bivalent logic is weak 3-valued Kleene logic. Unlike K3 , we have that a sentence takes the value i whenever any part of it takes i. That means e.g. that A ∧ B takes the value i even when A or B takes i. One interpretation of this logic is ...
... Weak 3-valued Kleene/Bochvar logic Another three-valued non-bivalent logic is weak 3-valued Kleene logic. Unlike K3 , we have that a sentence takes the value i whenever any part of it takes i. That means e.g. that A ∧ B takes the value i even when A or B takes i. One interpretation of this logic is ...
BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION
... • When two or more sum terms are multiplied by Boolean multiplication, the result is a • Product-of-Sum or POS expression. • • (A + B)(A + B + C) • • (A + B + C)(C + D + E)(B + C + D) • • (A + B)(A + B + C)(A + C) • The Domain of a POS expression is the set of variables contained in the expression, ...
... • When two or more sum terms are multiplied by Boolean multiplication, the result is a • Product-of-Sum or POS expression. • • (A + B)(A + B + C) • • (A + B + C)(C + D + E)(B + C + D) • • (A + B)(A + B + C)(A + C) • The Domain of a POS expression is the set of variables contained in the expression, ...
Multi-Agent Only
... (Other variants such as Halpern & Moses, Ben-David & Gafni, Waaler not discussed here.) ...
... (Other variants such as Halpern & Moses, Ben-David & Gafni, Waaler not discussed here.) ...
Logical Argument
... Less subjective criteria for validity of arguments are often clearly desirable, and in some cases we should even expect an argument to be rigorous, that is, to adhere to precise rules of validity. This is the case for arguments used in mathematical proofs. Note that a rigorous proof does not have to ...
... Less subjective criteria for validity of arguments are often clearly desirable, and in some cases we should even expect an argument to be rigorous, that is, to adhere to precise rules of validity. This is the case for arguments used in mathematical proofs. Note that a rigorous proof does not have to ...
T - STI Innsbruck
... • A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True • A valid sentence or tautology is a sentence that is True under all interpretations, no matter what the world is actually like or how the semantics are defined (ex ...
... • A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True • A valid sentence or tautology is a sentence that is True under all interpretations, no matter what the world is actually like or how the semantics are defined (ex ...
02_Artificial_Intelligence-PropositionalLogic
... • A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True • A valid sentence or tautology is a sentence that is True under all interpretations, no matter what the world is actually like or how the semantics are defined (ex ...
... • A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True • A valid sentence or tautology is a sentence that is True under all interpretations, no matter what the world is actually like or how the semantics are defined (ex ...
F - Teaching-WIKI
... • A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True • A valid sentence or tautology is a sentence that is True under all interpretations, no matter what the world is actually like or how the semantics are defined (ex ...
... • A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True • A valid sentence or tautology is a sentence that is True under all interpretations, no matter what the world is actually like or how the semantics are defined (ex ...
