Chapter 5 Predicate Logic
... Consider predicates first. An expression like G(a) is true just in case f (a) is in the subset of D that f assigns G to. For example, if a is paired with Ernie and Ernie is in the set of D that G is paired with, F (a) = Ernie and Ernie ∈ f (G), then G(a) is true. Likewise, H(a, b) is true just in ca ...
... Consider predicates first. An expression like G(a) is true just in case f (a) is in the subset of D that f assigns G to. For example, if a is paired with Ernie and Ernie is in the set of D that G is paired with, F (a) = Ernie and Ernie ∈ f (G), then G(a) is true. Likewise, H(a, b) is true just in ca ...
Quantified Equilibrium Logic and the First Order Logic of Here
... This report continues the work of [26] on first-order, or, as we shall say, quantified equilibrium logic (or QEL for short) and its relation to non-ground answer set programming. The report has three main contributions. First, we present a slightly different version of QEL where the so-called unique n ...
... This report continues the work of [26] on first-order, or, as we shall say, quantified equilibrium logic (or QEL for short) and its relation to non-ground answer set programming. The report has three main contributions. First, we present a slightly different version of QEL where the so-called unique n ...
x - Stanford University
... arguments, but each function has a fixed arity. Functions evaluate to objects, not propositions. There is no syntactic way to distinguish functions and predicates; you'll have to look at how they're used. ...
... arguments, but each function has a fixed arity. Functions evaluate to objects, not propositions. There is no syntactic way to distinguish functions and predicates; you'll have to look at how they're used. ...
Logic and Existential Commitment
... constant in Bill is a human and Bill married Hillary determines that it is only true on possible uses for Bill is a human and Bill married Hillary according to which they both are true. A possible use for a sentence will be any coordinated use of its non-logical elements consistent with its logical ...
... constant in Bill is a human and Bill married Hillary determines that it is only true on possible uses for Bill is a human and Bill married Hillary according to which they both are true. A possible use for a sentence will be any coordinated use of its non-logical elements consistent with its logical ...
Propositional Logic What is logic? Propositions Negation
... • This is a contentious question! We will play it safe, and stick to: – “The systematic use of symbolic techniques and mathematical methods to determine the forms of valid deductive argument.” [Shorter Oxford Dictionary]. ...
... • This is a contentious question! We will play it safe, and stick to: – “The systematic use of symbolic techniques and mathematical methods to determine the forms of valid deductive argument.” [Shorter Oxford Dictionary]. ...
Non-classical metatheory for non-classical logics
... language - the propositional calculus - for a large class of weak non-classical logics. The class includes fuzzy logics such Lukasiewicz, Gödel and product logic [7], BCK [6], intuitionistic logics, quantum logic [8], among others. The list does not include logics which do not allow you to infer φ ...
... language - the propositional calculus - for a large class of weak non-classical logics. The class includes fuzzy logics such Lukasiewicz, Gödel and product logic [7], BCK [6], intuitionistic logics, quantum logic [8], among others. The list does not include logics which do not allow you to infer φ ...
Lecture01 - Mathematics
... has few prerequisites. One can study quite a bit of interesting number theory, enumeration, and graph theory without knowing any more mathematics than the arithmetic of whole numbers. One can even find and write valid proofs with such a background. A little algebra is helpful for studying logic, set ...
... has few prerequisites. One can study quite a bit of interesting number theory, enumeration, and graph theory without knowing any more mathematics than the arithmetic of whole numbers. One can even find and write valid proofs with such a background. A little algebra is helpful for studying logic, set ...
Structural Multi-type Sequent Calculus for Inquisitive Logic
... under uniform substitution, which is a hurdle for a smooth proof-theoretic treatment for inquisitive logic. In [22], a labelled calculus was introduced for an earlier version of inquisitive logic, defined on the basis of the so called pair semantics [13, 19]. The calculus in [22] makes use of extra ...
... under uniform substitution, which is a hurdle for a smooth proof-theoretic treatment for inquisitive logic. In [22], a labelled calculus was introduced for an earlier version of inquisitive logic, defined on the basis of the so called pair semantics [13, 19]. The calculus in [22] makes use of extra ...
Discrete Mathematics and Logic II. Formal Logic
... non-formal standards, criteria, procedures for the analysis, interpretation, evaluation, criticism and construction of argumentation in everyday discourse." is used to reason about events in the human and social sciences Most reasoning from known facts to unknown facts that uses natural language can ...
... non-formal standards, criteria, procedures for the analysis, interpretation, evaluation, criticism and construction of argumentation in everyday discourse." is used to reason about events in the human and social sciences Most reasoning from known facts to unknown facts that uses natural language can ...
Chapter 2 - Princeton University Press
... If some sets “really” do exist in some sense, perhaps they are not described accurately by our axioms. We can’t be sure about that. But at least we can investigate the properties of any objects that do satisfy our axioms. We find it convenient to call those objects “sets” because we believe that tho ...
... If some sets “really” do exist in some sense, perhaps they are not described accurately by our axioms. We can’t be sure about that. But at least we can investigate the properties of any objects that do satisfy our axioms. We find it convenient to call those objects “sets” because we believe that tho ...
Least and greatest fixed points in linear logic
... Focusing is not restricted to linear logic. It has been extended to intuitionistic and classical logics. There are two approaches for doing so: either start from scratch, or use an encoding. ⊢ [F] o ...
... Focusing is not restricted to linear logic. It has been extended to intuitionistic and classical logics. There are two approaches for doing so: either start from scratch, or use an encoding. ⊢ [F] o ...
Is the Liar Sentence Both True and False? - NYU Philosophy
... to be real numbers in the interval [0, 1].) To the same degree of approximation, rejecting A is having a low degree of belief in it: one lower than the co-threshold 1 − T . This has the desired result that rejection precludes acceptance. (And it allows, as of course we should, for sentences that we ...
... to be real numbers in the interval [0, 1].) To the same degree of approximation, rejecting A is having a low degree of belief in it: one lower than the co-threshold 1 − T . This has the desired result that rejection precludes acceptance. (And it allows, as of course we should, for sentences that we ...
Identity in modal logic theorem proving
... In the realm of modal logics, almost all presentations of the logic of these systems are given in terms of axioms. But no one who is interested in providing automated proofs within modal logic uses an axiomatic system, and so it would therefore seem that all these methods of implementing t h e m mus ...
... In the realm of modal logics, almost all presentations of the logic of these systems are given in terms of axioms. But no one who is interested in providing automated proofs within modal logic uses an axiomatic system, and so it would therefore seem that all these methods of implementing t h e m mus ...
pdf
... if ϕ ∈ L is PSPACE-hard. (Of course, if we think of a modal logic as being characterized by an axiom system, then ϕ ∈ L iff ϕ is provable from the axioms characterizing L.) We say that ϕ is consistent with L if ¬ϕ ∈ / L. Since consistency is just the dual of provability, it follows from Ladner’s res ...
... if ϕ ∈ L is PSPACE-hard. (Of course, if we think of a modal logic as being characterized by an axiom system, then ϕ ∈ L iff ϕ is provable from the axioms characterizing L.) We say that ϕ is consistent with L if ¬ϕ ∈ / L. Since consistency is just the dual of provability, it follows from Ladner’s res ...
sentential logic
... In considering arguments formally, we care about what would be true if the premises were true. Generally, we are not concerned with actual truth values of any particular sentences- whether they are actually true or false. Yet there are some sentences that must be true, just as a matter of logic. Con ...
... In considering arguments formally, we care about what would be true if the premises were true. Generally, we are not concerned with actual truth values of any particular sentences- whether they are actually true or false. Yet there are some sentences that must be true, just as a matter of logic. Con ...
pdf file
... but that is not strictly formal in the sense of a purely syntactic derivation using a very precise and circumscribed formal set of rules of inference. In other words, I have in mind the type of proof found in a typical textbook on algebra, analysis, number theory, etc. Grantham goes on to say that T ...
... but that is not strictly formal in the sense of a purely syntactic derivation using a very precise and circumscribed formal set of rules of inference. In other words, I have in mind the type of proof found in a typical textbook on algebra, analysis, number theory, etc. Grantham goes on to say that T ...
A Partially Truth Functional Approach to
... paradigm cases have the form “If A then B” where A and B are statements that can “stand alone” (unlike “I were rich” in “If I were rich then I would own a yacht”). Such constructions will here be denoted “A > B.” Interpreting A > B as the material conditional implies that it is either true or false ...
... paradigm cases have the form “If A then B” where A and B are statements that can “stand alone” (unlike “I were rich” in “If I were rich then I would own a yacht”). Such constructions will here be denoted “A > B.” Interpreting A > B as the material conditional implies that it is either true or false ...
Discrete Mathematics - Lecture 4: Propositional Logic and Predicate
... less than 20 faculty and at least one noble laureate. B. All universities in USA where every department has at least 20 faculty and at least one noble laureate. C. For all universities in USA there is a department has less than 20 faculty or at most one noble laureate. D. For all universities in USA ...
... less than 20 faculty and at least one noble laureate. B. All universities in USA where every department has at least 20 faculty and at least one noble laureate. C. For all universities in USA there is a department has less than 20 faculty or at most one noble laureate. D. For all universities in USA ...
1 The calculus of “predicates”
... universe of discourse; variables standing for these names (ranging over the domain), predicate symbols, and quantifiers. In first-order logic there are also function symbols, but we concentrate for present just on the predicate part of the calculus. A typical formula of the predicate calculus is jus ...
... universe of discourse; variables standing for these names (ranging over the domain), predicate symbols, and quantifiers. In first-order logic there are also function symbols, but we concentrate for present just on the predicate part of the calculus. A typical formula of the predicate calculus is jus ...
A brief introduction to Logic and its applications
... Another reason why one could not prove P ∨ ¬P ? When you prove a statement such as A ∨ B you can extract a proof that answers whether A or B holds. If we were able to prove the excluded middle, we could extract an algorithm that, given some proposition tells us whether it is valid or not (Curry-Howa ...
... Another reason why one could not prove P ∨ ¬P ? When you prove a statement such as A ∨ B you can extract a proof that answers whether A or B holds. If we were able to prove the excluded middle, we could extract an algorithm that, given some proposition tells us whether it is valid or not (Curry-Howa ...
Lectures on Laws of Supply and Demand, Simple and Compound
... Definition A compound proposition is two or more propositions combined by a logical connective. Example 2 “ If Brian and Angela are not both happy then either Brian is not happy or Angela is not happy”. This is an example of a compound proposition. Logic is not concerned with determining the truth ...
... Definition A compound proposition is two or more propositions combined by a logical connective. Example 2 “ If Brian and Angela are not both happy then either Brian is not happy or Angela is not happy”. This is an example of a compound proposition. Logic is not concerned with determining the truth ...
CHAPTER 5 SOME EXTENSIONAL SEMANTICS
... If T is the only designated value, the third value ⊥ corresponds to some notion of incomplete information, like undefined or unknown and is often denoted by the symbol U or I. If, on the other hand, ⊥ corresponds to inconsistent information, i.e. its meaning is something like known to be both true a ...
... If T is the only designated value, the third value ⊥ corresponds to some notion of incomplete information, like undefined or unknown and is often denoted by the symbol U or I. If, on the other hand, ⊥ corresponds to inconsistent information, i.e. its meaning is something like known to be both true a ...
Completeness through Flatness in Two
... An advantage of the presence of this constant is that it makes it easy to express properties of the ordering relation in the language. Compared to other twodimensional temporal logics, TAL is a quite expressive formalism, being able to express the operators of most of the systems that are known fro ...
... An advantage of the presence of this constant is that it makes it easy to express properties of the ordering relation in the language. Compared to other twodimensional temporal logics, TAL is a quite expressive formalism, being able to express the operators of most of the systems that are known fro ...
Lesson 12
... There is a subtle difference between entailment and inference. Version 2 CSE IIT, Kharagpur ...
... There is a subtle difference between entailment and inference. Version 2 CSE IIT, Kharagpur ...
The Future of Post-Human Mathematical Logic
... (and Other Mental States) ............................................................67 Table 1.11. The Theoretical Levels of Consciousness (and Other Mental States) ............................................................68 Table 1.12. The Thematic Issues of Consciousness (and Other Mental Sta ...
... (and Other Mental States) ............................................................67 Table 1.11. The Theoretical Levels of Consciousness (and Other Mental States) ............................................................68 Table 1.12. The Thematic Issues of Consciousness (and Other Mental Sta ...
Willard Van Orman Quine
Willard Van Orman Quine (/kwaɪn/; June 25, 1908 – December 25, 2000) (known to intimates as ""Van"") was an American philosopher and logician in the analytic tradition, recognized as ""one of the most influential philosophers of the twentieth century."" From 1930 until his death 70 years later, Quine was continually affiliated with Harvard University in one way or another, first as a student, then as a professor of philosophy and a teacher of logic and set theory, and finally as a professor emeritus who published or revised several books in retirement. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978. A recent poll conducted among analytic philosophers named Quine as the fifth most important philosopher of the past two centuries. He won the first Schock Prize in Logic and Philosophy in 1993 for ""his systematical and penetrating discussions of how learning of language and communication are based on socially available evidence and of the consequences of this for theories on knowledge and linguistic meaning."" In 1996 he was awarded the Kyoto Prize in Arts and Philosophy for his ""outstanding contributions to the progress of philosophy in the 20th century by proposing numerous theories based on keen insights in logic, epistemology, philosophy of science and philosophy of language.""Quine falls squarely into the analytic philosophy tradition while also being the main proponent of the view that philosophy is not conceptual analysis but the abstract branch of the empirical sciences. His major writings include ""Two Dogmas of Empiricism"" (1951), which attacked the distinction between analytic and synthetic propositions and advocated a form of semantic holism, and Word and Object (1960), which further developed these positions and introduced Quine's famous indeterminacy of translation thesis, advocating a behaviorist theory of meaning. He also developed an influential naturalized epistemology that tried to provide ""an improved scientific explanation of how we have developed elaborate scientific theories on the basis of meager sensory input."" He is also important in philosophy of science for his ""systematic attempt to understand science from within the resources of science itself"" and for his conception of philosophy as continuous with science. This led to his famous quip that ""philosophy of science is philosophy enough."" In philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the ""Quine–Putnam indispensability thesis,"" an argument for the reality of mathematical entities.