Modus ponens
... ponens is sometimes called the rule of detachment. Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones", and Russell observes that "the process of the inference cannot be reduced to symbols. Its sole record is the occurrence of ⊦q [the consequent] . . . a ...
... ponens is sometimes called the rule of detachment. Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones", and Russell observes that "the process of the inference cannot be reduced to symbols. Its sole record is the occurrence of ⊦q [the consequent] . . . a ...
A HIGHER-ORDER FINE-GRAINED LOGIC FOR INTENSIONAL
... a preorder (a relation that is transitive, reflexive, but not antisymmetic) v, two nullary operations T and F , one unary operation 0 (written postfix), and four binary operations u, t, ⇒, ⇔, subject to the following conditions (here ≡ is the relation on B such that a ≡ b iff a v b and b v a). 1. T ...
... a preorder (a relation that is transitive, reflexive, but not antisymmetic) v, two nullary operations T and F , one unary operation 0 (written postfix), and four binary operations u, t, ⇒, ⇔, subject to the following conditions (here ≡ is the relation on B such that a ≡ b iff a v b and b v a). 1. T ...
deductive system
... A deductive system is a formal (mathematical) setup of reasoning. In order to describe a deductive system, a (formal) language system must first be in place, consisting of (well-formed) formulas, strings of symbols constructed according to some prescribed syntax. With the language in place, reasonin ...
... A deductive system is a formal (mathematical) setup of reasoning. In order to describe a deductive system, a (formal) language system must first be in place, consisting of (well-formed) formulas, strings of symbols constructed according to some prescribed syntax. With the language in place, reasonin ...
Propositional Logic
... A formula is in prenex form if it is of the form Q1 x1 .Q2 x2 . . . . Qn xn .ψ (possibly with n = 0) where each Qi is a quantifier (either ∀ or ∃) and ψ is a quantifier-free formula . Proposition For any formula of first-order logic, there exists an equivalent formula in prenex form. Proof. Such a p ...
... A formula is in prenex form if it is of the form Q1 x1 .Q2 x2 . . . . Qn xn .ψ (possibly with n = 0) where each Qi is a quantifier (either ∀ or ∃) and ψ is a quantifier-free formula . Proposition For any formula of first-order logic, there exists an equivalent formula in prenex form. Proof. Such a p ...
Combining Like
... Combining Like-Terms Objective: SWBAT simplify expressions by combining like-terms. ...
... Combining Like-Terms Objective: SWBAT simplify expressions by combining like-terms. ...
A short article for the Encyclopedia of Artificial Intelligence: Second
... then it is η-convertible to λx(M x), provided x is not free in M ). Many standard proof-theoretic results – such as cut-elimination (Girard, 1986), unification (Huet, 1975), resolution (Andrews, 1971), and Skolemization and Herbrand’s Theorem (Miller, 1987) – have been formulated for this fragment. ...
... then it is η-convertible to λx(M x), provided x is not free in M ). Many standard proof-theoretic results – such as cut-elimination (Girard, 1986), unification (Huet, 1975), resolution (Andrews, 1971), and Skolemization and Herbrand’s Theorem (Miller, 1987) – have been formulated for this fragment. ...
Jacques Herbrand (1908 - 1931) Principal writings in logic
... Herbrand domain D(F,p): an arbitrary item, and values for f and g (and any other function signs in R) iterated up to p times. Herbrand (validity) expansion: disjunction of instances of Herbrand functional form over D(F,p): ...
... Herbrand domain D(F,p): an arbitrary item, and values for f and g (and any other function signs in R) iterated up to p times. Herbrand (validity) expansion: disjunction of instances of Herbrand functional form over D(F,p): ...
MAT1193 – Notes on functions Functions are the fundamental
... We often write a function with a formula, e.g. F(x) = 3*x+5, where I have used * to represent multiplication. [e.g. is a latin abbreviation meaning “for example”]. This formula means that F i ...
... We often write a function with a formula, e.g. F(x) = 3*x+5, where I have used * to represent multiplication. [e.g. is a latin abbreviation meaning “for example”]. This formula means that F i ...
Philosophy as Logical Analysis of Science: Carnap, Schlick, Gödel
... the Circle, found inspiration in the Tractatus. As a former physicist, he took space-time points and objects or events occupying them to be fundamental physical entities. Following the Tractatus he also took the statements of physics to be truth functions of atomic statements about these entities. B ...
... the Circle, found inspiration in the Tractatus. As a former physicist, he took space-time points and objects or events occupying them to be fundamental physical entities. Following the Tractatus he also took the statements of physics to be truth functions of atomic statements about these entities. B ...
Discussion 2
... a sequence of characters a truth value (either true or false) a container object that holds a number of values of a data type ...
... a sequence of characters a truth value (either true or false) a container object that holds a number of values of a data type ...
(P Q). - Snistnote
... The theory associated with such rules is known as inference theory because it is concerned with the inferring of a conclusion from certain premises. When a conclusion is derived from a set of premises by using the accepted rules of reasoning, then such a process of ...
... The theory associated with such rules is known as inference theory because it is concerned with the inferring of a conclusion from certain premises. When a conclusion is derived from a set of premises by using the accepted rules of reasoning, then such a process of ...
LOGIC AND PSYCHOTHERAPY
... 1. In 1956 Bateson and his colleagues proposed the “double-bind theory”: “...an important factor in the development of schizophrenic thought disorder is the constant subjection of an individual to a so-called double-bind situation which includes the following elements. 1) The individual has an inten ...
... 1. In 1956 Bateson and his colleagues proposed the “double-bind theory”: “...an important factor in the development of schizophrenic thought disorder is the constant subjection of an individual to a so-called double-bind situation which includes the following elements. 1) The individual has an inten ...
Midterm Exam 1 Solutions, Comments, and Feedback
... ef (x) = e−x . With this choice, f (x) is unbounded, but g(x) is between 0 and 1 and hence bounded. 5. Let f : A → B be a function, and let C and D be subsets of A. (a) Give a careful, step-by-step proof of the relation f (C ∩ D) ⊆ f (C) ∩ f (D). Your write-up must include all necessary steps, with ...
... ef (x) = e−x . With this choice, f (x) is unbounded, but g(x) is between 0 and 1 and hence bounded. 5. Let f : A → B be a function, and let C and D be subsets of A. (a) Give a careful, step-by-step proof of the relation f (C ∩ D) ⊆ f (C) ∩ f (D). Your write-up must include all necessary steps, with ...
Propositional Logic Syntax of Propositional Logic
... Syntax of Predicate Logic • also based on sentences, but more complex – sentences can contain terms, which represent objects ...
... Syntax of Predicate Logic • also based on sentences, but more complex – sentences can contain terms, which represent objects ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.