Formal Reasoning - Institute for Computing and Information Sciences
... • I am someone. Someone painted the Mona Lisa. So, I painted the Mona Lisa. The first statement is correct, but the second is not. Even though they share the same form. And what about the sentence, This sentence is not true. Is it true, or not? To avoid these kinds of problems, we use formal languag ...
... • I am someone. Someone painted the Mona Lisa. So, I painted the Mona Lisa. The first statement is correct, but the second is not. Even though they share the same form. And what about the sentence, This sentence is not true. Is it true, or not? To avoid these kinds of problems, we use formal languag ...
Speaking Logic - SRI International
... Abstractions in computing include numbers, lists, channels, processes, protocols, and programming languages. These abstractions have algorithmic value in designing, representing, and reasoning about computational processes. Logic is the calculus of computing — it is used to delineate the precise mea ...
... Abstractions in computing include numbers, lists, channels, processes, protocols, and programming languages. These abstractions have algorithmic value in designing, representing, and reasoning about computational processes. Logic is the calculus of computing — it is used to delineate the precise mea ...
SOME AXIOMS FOR CONSTRUCTIVE ANALYSIS Introduction
... rich collection of recursively axiomatizable theories can be identified and explored. Our choice of M as a neutral base theory for reverse constructive analysis is motivated by practical and historical considerations. On the one hand, M (like the theory EL preferred by Troelstra and van Dalen) is st ...
... rich collection of recursively axiomatizable theories can be identified and explored. Our choice of M as a neutral base theory for reverse constructive analysis is motivated by practical and historical considerations. On the one hand, M (like the theory EL preferred by Troelstra and van Dalen) is st ...
Completeness and Decidability of a Fragment of Duration Calculus
... Duration Calculus (DC) was introduced by Zhou, Hoare and Ravn in 1991 as a logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a re ...
... Duration Calculus (DC) was introduced by Zhou, Hoare and Ravn in 1991 as a logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a re ...
Outline of Lecture 2 First Order Logic and Second Order Logic Basic
... • MSOL has no complete provability system: The Peano axioms are expressible in MSOL and characterize the structure h IN, +, ×, 0, 1i up to isomorphims. If there were a complete provability system, the set of MSOL(τarith )sentences true in h IN, +, ×, 0, 1i would be computable. But this contradicts G ...
... • MSOL has no complete provability system: The Peano axioms are expressible in MSOL and characterize the structure h IN, +, ×, 0, 1i up to isomorphims. If there were a complete provability system, the set of MSOL(τarith )sentences true in h IN, +, ×, 0, 1i would be computable. But this contradicts G ...
Knowledge Representation and Reasoning
... In other words, it tells us that if we accept as true a number of propositions — called premisses — which match certain patterns, we can deduce that some further proposition is true — this is called the conclusion. Thus we saw that from two propositions with the forms α → β and α we can deduce β. Th ...
... In other words, it tells us that if we accept as true a number of propositions — called premisses — which match certain patterns, we can deduce that some further proposition is true — this is called the conclusion. Thus we saw that from two propositions with the forms α → β and α we can deduce β. Th ...
ordinal logics and the characterization of informal concepts of proof
... Our proposal is to identify finitist proofs in arithmetic with the least class of systems 2^ containing primitive recursive arithmetic with a constructive existential quantifier, and if Px(k, m, n) is proved to be a proof predicate in 2^, then 2 also belongs to the class. Call this class JF ...
... Our proposal is to identify finitist proofs in arithmetic with the least class of systems 2^ containing primitive recursive arithmetic with a constructive existential quantifier, and if Px(k, m, n) is proved to be a proof predicate in 2^, then 2 also belongs to the class. Call this class JF ...
Logic and Computation Lecture notes Jeremy Avigad Assistant Professor, Philosophy
... 1998, while teaching a course called Logic and Computation in the Philosophy Department at Carnegie Mellon University. I distributed these notes to the class and then followed them almost word for word, in the hopes that doing so would enable students to pay more attention to the contents of the lec ...
... 1998, while teaching a course called Logic and Computation in the Philosophy Department at Carnegie Mellon University. I distributed these notes to the class and then followed them almost word for word, in the hopes that doing so would enable students to pay more attention to the contents of the lec ...
Suszko`s Thesis, Inferential Many-Valuedness, and the
... [L]ogicaltwo-valuedness ... is obviously related to the division of the universe of interpretation into two subsets of elements: distinguished ...
... [L]ogicaltwo-valuedness ... is obviously related to the division of the universe of interpretation into two subsets of elements: distinguished ...
Linear Contextual Modal Type Theory
... functional abstraction and hi for the proof term of >. This little example illustrates the complex nature of logic variables in linear logic and their role in higher-order linear unification. In the >-free case, every linear resource needs to be consumed by the same term on either side of the equati ...
... functional abstraction and hi for the proof term of >. This little example illustrates the complex nature of logic variables in linear logic and their role in higher-order linear unification. In the >-free case, every linear resource needs to be consumed by the same term on either side of the equati ...
A Hoare Logic for Linear Systems - School of Electronic Engineering
... be real-valued functions of time, i.e., members of the set V = T → R, where T is some set representing times; we obtain valuable algebraic structure by regarding V as a real vector space, scaling and adding functions pointwise: (af )(τ ) = a(f (τ )), (f + g)(τ ) = f (τ ) + g(τ ) for a ∈ R, τ ∈ T. Wh ...
... be real-valued functions of time, i.e., members of the set V = T → R, where T is some set representing times; we obtain valuable algebraic structure by regarding V as a real vector space, scaling and adding functions pointwise: (af )(τ ) = a(f (τ )), (f + g)(τ ) = f (τ ) + g(τ ) for a ∈ R, τ ∈ T. Wh ...
Document
... p ↔q denotes “I am at home if and only if it is raining.” If p denotes “You can take the flight.” and q denotes “You buy a ticket.” then p ↔q denotes “You can take the flight if and only ...
... p ↔q denotes “I am at home if and only if it is raining.” If p denotes “You can take the flight.” and q denotes “You buy a ticket.” then p ↔q denotes “You can take the flight if and only ...
The logic and mathematics of occasion sentences
... ABSTRACT. The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of ...
... ABSTRACT. The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of ...
The Emergence of First
... First-order logic—stripped of all infinitary operations—emerged only with Hilbert in (1917), where it remained a subsystem of logic, and with Skolem in (1923), who treated it as all of logic. During the nineteenth and early twentieth centuries, there was no generally accepted classification of the d ...
... First-order logic—stripped of all infinitary operations—emerged only with Hilbert in (1917), where it remained a subsystem of logic, and with Skolem in (1923), who treated it as all of logic. During the nineteenth and early twentieth centuries, there was no generally accepted classification of the d ...
Unification in Propositional Logic
... The substitutions θaA used in the Boolean case, contribute to the construction of minimal bases of unifiers in IP C too. θaA is indexed by a formula A ∈ F (x) and by a classical assignment a over x. How does the transformation (θaA)∗ act on a Kripke model u : P −→ P(x)? First, it does not change th ...
... The substitutions θaA used in the Boolean case, contribute to the construction of minimal bases of unifiers in IP C too. θaA is indexed by a formula A ∈ F (x) and by a classical assignment a over x. How does the transformation (θaA)∗ act on a Kripke model u : P −→ P(x)? First, it does not change th ...
A pragmatic dialogic interpretation of bi
... Closed Categories, in the interpretation of William Lawvere. Here model theory and proof theory meet at a new level, where also categorical proof theory plays an essential role. Indeed categorical proof theory is concerned not only with algorithm to establish the provability of formulas in given pro ...
... Closed Categories, in the interpretation of William Lawvere. Here model theory and proof theory meet at a new level, where also categorical proof theory plays an essential role. Indeed categorical proof theory is concerned not only with algorithm to establish the provability of formulas in given pro ...
Robot Morality and Review of classical logic.
... Analytic philosophy (like proving God’ Existence, free will, the problem of evil, etc) Many other… At this point I should ask all students to give another examples of similar problems that they want to solve ...
... Analytic philosophy (like proving God’ Existence, free will, the problem of evil, etc) Many other… At this point I should ask all students to give another examples of similar problems that they want to solve ...
Second-Order Logic of Paradox
... vagueness, and motion, and Buddhist philosophy and, in his recent book [14], metaphysical perplexities arising out the notion of parthood and in relation to the question of the unity of the proposition – all from a dialetheist perspective: one that considers it possible that there are true contradic ...
... vagueness, and motion, and Buddhist philosophy and, in his recent book [14], metaphysical perplexities arising out the notion of parthood and in relation to the question of the unity of the proposition – all from a dialetheist perspective: one that considers it possible that there are true contradic ...
Boolean Connectives and Formal Proofs - FB3
... rule allows you to introduce, for any name (or complex term) the proof, the assertion n = n. You are allowed to do this at an proof, and need not cite any earlier step as justification. We w our statement of this rule in the following way: Identity Introduction (= Intro): . n=n ...
... rule allows you to introduce, for any name (or complex term) the proof, the assertion n = n. You are allowed to do this at an proof, and need not cite any earlier step as justification. We w our statement of this rule in the following way: Identity Introduction (= Intro): . n=n ...
An Introduction to Prolog Programming
... A Prolog program corresponds to a set of formulas, all of which are assumed to be true. This restricts the range of possible interpretations of the predicate and function symbols appearing in these formulas. The formulas in the translated program may be thought of as the premises in a proof. If Prol ...
... A Prolog program corresponds to a set of formulas, all of which are assumed to be true. This restricts the range of possible interpretations of the predicate and function symbols appearing in these formulas. The formulas in the translated program may be thought of as the premises in a proof. If Prol ...
CHAPTER 1 The Foundations: Logic and Proof, Sets, and Functions
... 11. a) This is correct, using universal instantiation and modus ponens. b) This is invalid. After applying universal instantiation, it contains the fallacy of affirming the conclusion. c) This is invalid. After applying universal instantiation, it contains the fallacy of denying the hypothesis. d) T ...
... 11. a) This is correct, using universal instantiation and modus ponens. b) This is invalid. After applying universal instantiation, it contains the fallacy of affirming the conclusion. c) This is invalid. After applying universal instantiation, it contains the fallacy of denying the hypothesis. d) T ...
Quantifiers
... property, but we don’t know who or what this something is. • In order to perform some reasoning, we will give this something a name, and whatever we can infer from that point on, we can infer from the original ...
... property, but we don’t know who or what this something is. • In order to perform some reasoning, we will give this something a name, and whatever we can infer from that point on, we can infer from the original ...
How to tell the truth without knowing what you are talking about
... Leibniz’s works on logic were published only at the end of the 19th century, so the influence of his ideas on the emergence of modern logic has been very limited. In the 19th century, there was a main paradigm shift in mathematics: the perception that algebra could deal not only with numbers but, mo ...
... Leibniz’s works on logic were published only at the end of the 19th century, so the influence of his ideas on the emergence of modern logic has been very limited. In the 19th century, there was a main paradigm shift in mathematics: the perception that algebra could deal not only with numbers but, mo ...
Jesús Mosterín
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Jesús Mosterín (born 1941) is a leading Spanish philosopher and a thinker of broad spectrum, often at the frontier between science and philosophy.