Logic: Introduction - Department of information engineering and
... • Design Validation and verification: to verify the correctness of a design with a certainty beyond that of conventional testing. It uses temporal logic . • AI: mechanized reasoning and expert systems. • Security: With increasing use of network, security has become a big issue. Hence, the concept o ...
... • Design Validation and verification: to verify the correctness of a design with a certainty beyond that of conventional testing. It uses temporal logic . • AI: mechanized reasoning and expert systems. • Security: With increasing use of network, security has become a big issue. Hence, the concept o ...
Comparing Constructive Arithmetical Theories Based - Math
... Kripke model does not force IS2 . The reason is that S2 is ∀Σb1 -conservative over IS21 (see e.g., [A, Th. 3.17]) and so if the Kripke model forces IS21 , using forcing definition, its root would be a model of ∀x, y∃z ≤ x(x−̇y = z). In the theory IP V which is the natural conservative extension of ...
... Kripke model does not force IS2 . The reason is that S2 is ∀Σb1 -conservative over IS21 (see e.g., [A, Th. 3.17]) and so if the Kripke model forces IS21 , using forcing definition, its root would be a model of ∀x, y∃z ≤ x(x−̇y = z). In the theory IP V which is the natural conservative extension of ...
Logic and Resolution - Institute for Computing and Information
... Logic and Resolution One of the earliest formalisms for the representation of knowledge is logic. The formalism is characterized by a well-defined syntax and semantics, and provides a number of inference rules to manipulate logical formulas on the basis of their form in order to derive new knowledge ...
... Logic and Resolution One of the earliest formalisms for the representation of knowledge is logic. The formalism is characterized by a well-defined syntax and semantics, and provides a number of inference rules to manipulate logical formulas on the basis of their form in order to derive new knowledge ...
Modal Logic
... In (classical) propositional and predicate logic, every formula is either true or false in any model. But there are situations were we need to distinguish between different modes of truth, such as necessarily true, known to be true, believed to be true and always true in the future (with respect to ...
... In (classical) propositional and predicate logic, every formula is either true or false in any model. But there are situations were we need to distinguish between different modes of truth, such as necessarily true, known to be true, believed to be true and always true in the future (with respect to ...
CLASSICAL LOGIC and FUZZY LOGIC
... contained within a universe of elements, X, that can be identified as being a collection of elements in X that are strictly true or strictly false. The veracity (truth) of an element in the proposition P can be assigned a binary truth value, called T (P), For binary (Boolean) classical logic, T (P) ...
... contained within a universe of elements, X, that can be identified as being a collection of elements in X that are strictly true or strictly false. The veracity (truth) of an element in the proposition P can be assigned a binary truth value, called T (P), For binary (Boolean) classical logic, T (P) ...
PDF
... formal theory of arithmetic is incomplete”, where a formal theory is viewed as one whose theorems are derivable from an axiom system. For such theories there will always be formulas that are true (, for instance, in the standard interpretation of arithmetic) but not theorems of the theories. When it ...
... formal theory of arithmetic is incomplete”, where a formal theory is viewed as one whose theorems are derivable from an axiom system. For such theories there will always be formulas that are true (, for instance, in the standard interpretation of arithmetic) but not theorems of the theories. When it ...
The initial question: “What is the meaning of a first
... The following can be perceived as shortcomings. There is neither (1) mature semantics nor (2) the proof theory for FOL under the principle of the alphabetic innocence (what about the rule of UG?). The concept of the meaning of a formula is highly intuitive and it stands in need of detailed investiga ...
... The following can be perceived as shortcomings. There is neither (1) mature semantics nor (2) the proof theory for FOL under the principle of the alphabetic innocence (what about the rule of UG?). The concept of the meaning of a formula is highly intuitive and it stands in need of detailed investiga ...
On Perfect Introspection with Quantifying-in
... and T~pR~M,respectively. A formula is called s u b j e c t i v e if all predicate and function symbols appear within the scope of a B, and o b j e c t i v e if it does not contain any B's. L i t e r a l s and c l a u s e s have their usual meaning. Sequences of terms or variables are sometimes writt ...
... and T~pR~M,respectively. A formula is called s u b j e c t i v e if all predicate and function symbols appear within the scope of a B, and o b j e c t i v e if it does not contain any B's. L i t e r a l s and c l a u s e s have their usual meaning. Sequences of terms or variables are sometimes writt ...
A HIGHER-ORDER FINE-GRAINED LOGIC FOR INTENSIONAL
... only alphabetically in their bound variables are assigned the same intensions. To summarize, an intensional model assigns intensions to terms in such a way that logical constants are interpreted as designated operations, term application and abstraction are interpreted in the standard way, and lambd ...
... only alphabetically in their bound variables are assigned the same intensions. To summarize, an intensional model assigns intensions to terms in such a way that logical constants are interpreted as designated operations, term application and abstraction are interpreted in the standard way, and lambd ...
Homomorphism Preservation Theorem
... Preservation under Extensions? Theorem (A., Dawar and Grohe 2005) The extension preservation property holds on the following classes: ...
... Preservation under Extensions? Theorem (A., Dawar and Grohe 2005) The extension preservation property holds on the following classes: ...
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 ...
Propositional Dynamic Logic of Regular Programs*+
... if and only if a executed in state s can terminate in state t. The truth of an assertion is determined relative to a program state, so we say “p is true in state s.” The formula (ai p is true in state s if there is a state t such that (s, t) E p(a) and p is true in state 2. The formula p v 4 is true ...
... if and only if a executed in state s can terminate in state t. The truth of an assertion is determined relative to a program state, so we say “p is true in state s.” The formula (ai p is true in state s if there is a state t such that (s, t) E p(a) and p is true in state 2. The formula p v 4 is true ...
First-Order Logic with Dependent Types
... In this signature, S is a type, the type of sorts declared in a DFOL signature. Univ is a dependent type family that returns a new type for each sort S, namely the type of terms of sort S; models will interpret the type Univ S as the universe for the sort S. o is the type of formulas. The remainder ...
... In this signature, S is a type, the type of sorts declared in a DFOL signature. Univ is a dependent type family that returns a new type for each sort S, namely the type of terms of sort S; models will interpret the type Univ S as the universe for the sort S. o is the type of formulas. The remainder ...
A Brief Introduction to Propositional Logic
... A proof is valid only if every assumption is eventually discharged. This must happen below the point where an assumption has been made, in the proof tree. If an assumption is used more than once, it must be discharged in all those paths in the proof tree. Rule 8: implies-elimination (modus ponens) ...
... A proof is valid only if every assumption is eventually discharged. This must happen below the point where an assumption has been made, in the proof tree. If an assumption is used more than once, it must be discharged in all those paths in the proof tree. Rule 8: implies-elimination (modus ponens) ...
Guarded negation
... as a syntactic fragment of first-order logic, it is also natural to ask for syntactic explanations: what syntactic features of modal formulas (viewed as first-order formulas) are responsible for their good behavior? And can we generalize modal logic, preserving these features, while at the same tim ...
... as a syntactic fragment of first-order logic, it is also natural to ask for syntactic explanations: what syntactic features of modal formulas (viewed as first-order formulas) are responsible for their good behavior? And can we generalize modal logic, preserving these features, while at the same tim ...
pdf
... theory of arithmetic is incomplete”, where a formal theory is viewed as one whose theorems are derivable from an axiom system. For such theories there will always be formulas that are true (for instance, in the standard interpretation of arithmetic) but not theorems of the theories. When it comes to ...
... theory of arithmetic is incomplete”, where a formal theory is viewed as one whose theorems are derivable from an axiom system. For such theories there will always be formulas that are true (for instance, in the standard interpretation of arithmetic) but not theorems of the theories. When it comes to ...
Logical Prior Probability - Institute for Creative Technologies
... The purpose of this paper is to present a prior over theories in first-order logic, similar in nature to the priors of algorithmic probability. There are several possible motivations for such a prior. First, it is hoped that the study of priors over logics will be useful to the study of realistic re ...
... The purpose of this paper is to present a prior over theories in first-order logic, similar in nature to the priors of algorithmic probability. There are several possible motivations for such a prior. First, it is hoped that the study of priors over logics will be useful to the study of realistic re ...
Partial Correctness Specification
... These specifications are ‘partial’ because for {P } C {Q} to be true it is not necessary for the execution of C to terminate when started in a state satisfying P It is only required that if the execution terminates, then Q holds {X = 1} WHILE T DO X := X {Y = 2} – this specification is true! ...
... These specifications are ‘partial’ because for {P } C {Q} to be true it is not necessary for the execution of C to terminate when started in a state satisfying P It is only required that if the execution terminates, then Q holds {X = 1} WHILE T DO X := X {Y = 2} – this specification is true! ...
PREPOSITIONAL LOGIS
... • Logic is a great knowledge representation language for many AI problems • Propositional logic is the simple foundation and fine for some AI problems • First order logic (FOL) is much more expressive as a KR language and more commonly used in AI • There are many variations: horn logic, higher order ...
... • Logic is a great knowledge representation language for many AI problems • Propositional logic is the simple foundation and fine for some AI problems • First order logic (FOL) is much more expressive as a KR language and more commonly used in AI • There are many variations: horn logic, higher order ...
paper by David Pierce
... (2) to prove that all elements of those sets have certain properties; (3) to define functions on those sets. These three techniques are often confused, but they should not be. Clarity here can prevent mathematical mistakes; it can also highlight important concepts and results such as Fermat’s (Little ...
... (2) to prove that all elements of those sets have certain properties; (3) to define functions on those sets. These three techniques are often confused, but they should not be. Clarity here can prevent mathematical mistakes; it can also highlight important concepts and results such as Fermat’s (Little ...
A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC–Part
... , . . .–and the primary logical symbols. The latter are the Boolean variables p, q, p0 , p00 , q13 , . . ., and the connectives: ¬, ∨, >, ⊥, 2, (, ), =, ∀, and the comma. We note two slight deviations from the standard definitions: One is that we add an induction clause “if A is formula, then so is ...
... , . . .–and the primary logical symbols. The latter are the Boolean variables p, q, p0 , p00 , q13 , . . ., and the connectives: ¬, ∨, >, ⊥, 2, (, ), =, ∀, and the comma. We note two slight deviations from the standard definitions: One is that we add an induction clause “if A is formula, then so is ...
Modal Logics Definable by Universal Three
... that P is possible) – in the class of symmetric frames, and the axiom ♦P → ♦P (if P is possible, then it is necessary that P is possible) – in the class of Euclidean frames. Thus we may think that every modal formula ϕ defines a class of frames, namely the class of those frames in which ϕ is valid. ...
... that P is possible) – in the class of symmetric frames, and the axiom ♦P → ♦P (if P is possible, then it is necessary that P is possible) – in the class of Euclidean frames. Thus we may think that every modal formula ϕ defines a class of frames, namely the class of those frames in which ϕ is valid. ...
FIRST-ORDER QUERY EVALUATION ON STRUCTURES OF
... the substructure of A induced by Nr (ā) and expanded with one constant for each element of ā. Given two tuples of elements ā and b̄ we say that they have the same r-neighborhood type, written Nr (ā) ' Nr (b̄), if there is an isomorphism between Nr (ā) and Nr (b̄). We consider first-order logic ...
... the substructure of A induced by Nr (ā) and expanded with one constant for each element of ā. Given two tuples of elements ā and b̄ we say that they have the same r-neighborhood type, written Nr (ā) ' Nr (b̄), if there is an isomorphism between Nr (ā) and Nr (b̄). We consider first-order logic ...
PARADOX AND INTUITION
... only with respect to truth-functional connectives, the part of meaning of quantifiers which is independent of the specification of domain, and the juxtaposition of symbols cannot force the interpretation of any of its predicate-letters as a relation with a nondenumerable field. Some connections bet ...
... only with respect to truth-functional connectives, the part of meaning of quantifiers which is independent of the specification of domain, and the juxtaposition of symbols cannot force the interpretation of any of its predicate-letters as a relation with a nondenumerable field. Some connections bet ...