A Proof of Cut-Elimination Theorem for U Logic.
... Basic Propositional Logic, BPL, was invented by Albert Visser in 1981 [5]. He wanted to interpret implication as formal provability. To protect his system against the liar paradox, modus ponens is weakened. His axiomatization of BPL uses natural deduction[3, p. 8]. The first sequent calculus for BPL ...
... Basic Propositional Logic, BPL, was invented by Albert Visser in 1981 [5]. He wanted to interpret implication as formal provability. To protect his system against the liar paradox, modus ponens is weakened. His axiomatization of BPL uses natural deduction[3, p. 8]. The first sequent calculus for BPL ...
T - UTH e
... raining.” then p →q denotes “If I am at home then it is raining.” In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). ...
... raining.” then p →q denotes “If I am at home then it is raining.” In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). ...
Lecture 23 Notes
... that we can find evidence for either P or for ∼ P , and that evidence indicates the disjunct it validates. The constructive meaning of this logical law is that we have the computational ability to decide which case holds. On the other hand, the standard classical interpretation does not refer to hum ...
... that we can find evidence for either P or for ∼ P , and that evidence indicates the disjunct it validates. The constructive meaning of this logical law is that we have the computational ability to decide which case holds. On the other hand, the standard classical interpretation does not refer to hum ...
Propositional Logic: Why? soning Starts with George Boole around 1850
... The connections between the elements of the argument is lost in propositional logic Here we are talking about general properties (also called predicates) and individuals of a domain of discourse who may or may not have those properties Instead of introducing names for complete propositions -like in ...
... The connections between the elements of the argument is lost in propositional logic Here we are talking about general properties (also called predicates) and individuals of a domain of discourse who may or may not have those properties Instead of introducing names for complete propositions -like in ...
A puzzle about de rebus beliefs
... and he found that their logic cannot be adequately formalized within the first-order predicate calculus. If we try to formalize the sentence by a paraphrase using individual variables that range over critics, or over sets or collections or fusions of critics, we misrepresent its logical structure. T ...
... and he found that their logic cannot be adequately formalized within the first-order predicate calculus. If we try to formalize the sentence by a paraphrase using individual variables that range over critics, or over sets or collections or fusions of critics, we misrepresent its logical structure. T ...
(pdf)
... One important distinction to make is that fuzzy logic is NOT probability. Although both employ values between 0 and 1 that represent something about the symbol or event, it is the meaning of this number that differs. In probability, the number represents the likelihood of an event’s occurrence. In f ...
... One important distinction to make is that fuzzy logic is NOT probability. Although both employ values between 0 and 1 that represent something about the symbol or event, it is the meaning of this number that differs. In probability, the number represents the likelihood of an event’s occurrence. In f ...
The Foundations: Logic and Proofs - UTH e
... If P(x) denotes “x > 0” and U is the integers, then x P(x) is true. It is also true if U is the positive integers. If P(x) denotes “x < 0” and U is the positive integers, then x P(x) is false. If P(x) denotes “x is even” and U is the integers, then x P(x) is true. ...
... If P(x) denotes “x > 0” and U is the integers, then x P(x) is true. It is also true if U is the positive integers. If P(x) denotes “x < 0” and U is the positive integers, then x P(x) is false. If P(x) denotes “x is even” and U is the integers, then x P(x) is true. ...
What is...Linear Logic? Introduction Jonathan Skowera
... by formulas. The edge labels must be compatible with the vertex labels, e.g., the edges of a ` should be labelled A, B and A ` B. Exactly three edges must be attached to ` and ⊗ vertices, exactly two edges to a and c vertices, and exactly one edge to i and o vertices. The only exception is the i whi ...
... by formulas. The edge labels must be compatible with the vertex labels, e.g., the edges of a ` should be labelled A, B and A ` B. Exactly three edges must be attached to ` and ⊗ vertices, exactly two edges to a and c vertices, and exactly one edge to i and o vertices. The only exception is the i whi ...
January 12
... A. express all (and only) propositions, i.e., all and only things that are either true or false; and B. state all logical relations that propositions have to each other. Thus if some proposition p logically implies some proposition q, then both these propositions and the inference rule by which p lo ...
... A. express all (and only) propositions, i.e., all and only things that are either true or false; and B. state all logical relations that propositions have to each other. Thus if some proposition p logically implies some proposition q, then both these propositions and the inference rule by which p lo ...
Tactical and Strategic Challenges to Logic (KAIST
... The theoretical core of the Hewitt-Woods book is an Inconsistency Robust Direct Logic,7 or IRDL for short. IRDL embodies a formidable heavy-equipment mathematical machinery, and is still very much a work in progress. There is no need here to absorb its many technicalities. It is perfectly possible ...
... The theoretical core of the Hewitt-Woods book is an Inconsistency Robust Direct Logic,7 or IRDL for short. IRDL embodies a formidable heavy-equipment mathematical machinery, and is still very much a work in progress. There is no need here to absorb its many technicalities. It is perfectly possible ...
A Logic of Explicit Knowledge - Lehman College
... family of things we might call reasons are mathematical proofs. Sergei Artemov has introduced a kind of propositional modal logic of explicit proofs, LP, [1, 2], and in this there are various operations introduced for what he calls proof polynomials. These operations make intuitive sense in a broade ...
... family of things we might call reasons are mathematical proofs. Sergei Artemov has introduced a kind of propositional modal logic of explicit proofs, LP, [1, 2], and in this there are various operations introduced for what he calls proof polynomials. These operations make intuitive sense in a broade ...
Propositional Calculus
... Logic helps to clarify the meanings of descriptions written, for example, in English. After all, one reason for our use of logic is to state precisely the requirements of computer systems. Descriptions in natural languages can be imprecise and ambiguous. An ambiguous sentence can have more than one ...
... Logic helps to clarify the meanings of descriptions written, for example, in English. After all, one reason for our use of logic is to state precisely the requirements of computer systems. Descriptions in natural languages can be imprecise and ambiguous. An ambiguous sentence can have more than one ...
Chapter 1, Part I: Propositional Logic
... raining.” then p →q denotes “If I am at home then it is raining.” In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). ...
... raining.” then p →q denotes “If I am at home then it is raining.” In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). ...
Chapter1_Parts2
... forced to assume otherwise. These atoms are called assumables.! The assumables (ok_cb1, ok_s1, ok_s2, ok_s3, ok_l1, ok_l2) represent the assumption that we assume that the switches, lights, and circuit breakers are ok.! If the system is working correctly (all assumables are true), the observations a ...
... forced to assume otherwise. These atoms are called assumables.! The assumables (ok_cb1, ok_s1, ok_s2, ok_s3, ok_l1, ok_l2) represent the assumption that we assume that the switches, lights, and circuit breakers are ok.! If the system is working correctly (all assumables are true), the observations a ...
Bilattices In Logic Programming
... also that no requirement is imposed that these pairs, or ‘truth values,’ be consistent. Anything is allowed; after the fact we can try to identify those truth values that are consistent. Consistent truth values are those in which belief and doubt complement each other in some reasonable sense. This ...
... also that no requirement is imposed that these pairs, or ‘truth values,’ be consistent. Anything is allowed; after the fact we can try to identify those truth values that are consistent. Consistent truth values are those in which belief and doubt complement each other in some reasonable sense. This ...
Symbolic Logic II
... that P → Q, Q → R 2LP P → R, we need to show that there is some interpretation in which KVI (P → Q) 6= 0, KVI (Q → R) 6= 0 and KVI (P → R) = 0. If KVI (P → R) = 0, then KVI (P) = 1 and KVI (R) = 0. But if KVI (R) = 0 and KVI (Q → R) 6= 0, then KVI (Q) 6= 1. If KVI (Q) 6= 1 and KVI (P) = 1 and KVI (P ...
... that P → Q, Q → R 2LP P → R, we need to show that there is some interpretation in which KVI (P → Q) 6= 0, KVI (Q → R) 6= 0 and KVI (P → R) = 0. If KVI (P → R) = 0, then KVI (P) = 1 and KVI (R) = 0. But if KVI (R) = 0 and KVI (Q → R) 6= 0, then KVI (Q) 6= 1. If KVI (Q) 6= 1 and KVI (P) = 1 and KVI (P ...
Analysis of the paraconsistency in some logics
... satisfying, on this paper, Con1, Con2 and Con3 and a set of formulas. We will say that Γ is a theory of L if Γ ⊆ L. We will also say that Γ is closed if it contains all of its consequences (the converse of Con1.) For our purposes, F or is a numerable set of symbols from the language that contains ¬ ...
... satisfying, on this paper, Con1, Con2 and Con3 and a set of formulas. We will say that Γ is a theory of L if Γ ⊆ L. We will also say that Γ is closed if it contains all of its consequences (the converse of Con1.) For our purposes, F or is a numerable set of symbols from the language that contains ¬ ...
A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC–Part
... It is immediate by the conservation requirement that, conversely, a connected modal chain of type (3) can be replaced by a classical, possibly disconnected, chain of type (2). Modal extensions of propositional ([2], [8], [9], [10], [12], [17]) and predicate ([1], [3], [5], [9], [10], [11]) classical ...
... It is immediate by the conservation requirement that, conversely, a connected modal chain of type (3) can be replaced by a classical, possibly disconnected, chain of type (2). Modal extensions of propositional ([2], [8], [9], [10], [12], [17]) and predicate ([1], [3], [5], [9], [10], [11]) classical ...
First-Order Logic, Second-Order Logic, and Completeness
... a sentence false, again by soundness I know that I won’t be able to prove it. Completeness allows us to make these transitions in the reverse direction. A model-theoretic argument can establish that a sentence is a consequence of some other sentences. If completeness holds one knows that there is al ...
... a sentence false, again by soundness I know that I won’t be able to prove it. Completeness allows us to make these transitions in the reverse direction. A model-theoretic argument can establish that a sentence is a consequence of some other sentences. If completeness holds one knows that there is al ...
mathematical logic: constructive and non
... ranging over a collection M of subsets of D. A standard model for a set of sentences AQ, Av A2,... is one with M = {the set 2D of all subsets of D}. The above results do not extend when only standard models are used, in view of the categoricity of Peano's axioms for the natural numbers (using a vari ...
... ranging over a collection M of subsets of D. A standard model for a set of sentences AQ, Av A2,... is one with M = {the set 2D of all subsets of D}. The above results do not extend when only standard models are used, in view of the categoricity of Peano's axioms for the natural numbers (using a vari ...
slides
... In later lectures we shall see that it extends to predicate (i.e., first-order) logic (and beyond!) (Indeed Prolog implements predicate logic, not only propositional logic.) ...
... In later lectures we shall see that it extends to predicate (i.e., first-order) logic (and beyond!) (Indeed Prolog implements predicate logic, not only propositional logic.) ...
Reducing Propositional Theories in Equilibrium Logic to
... programs of a special kind. Answer set semantics was already generalised and extended to arbitrary propositional theories with two negations in the system of equilibrium logic, defined in [17] and further studied in [18,19,20]. Equilibrium logic is based on a simple, minimal model construction in th ...
... programs of a special kind. Answer set semantics was already generalised and extended to arbitrary propositional theories with two negations in the system of equilibrium logic, defined in [17] and further studied in [18,19,20]. Equilibrium logic is based on a simple, minimal model construction in th ...
Dialetheic truth theory: inconsistency, non-triviality, soundness, incompleteness
... with the assumption, and note that, because PA* has a recursive proof relation (PA*derivations form a recursive set of sequences of strings on the alphabet of L), and because all recursive relations can be represented in PA (and therefore, by assumption, also in PA*), it is possible to formulate a p ...
... with the assumption, and note that, because PA* has a recursive proof relation (PA*derivations form a recursive set of sequences of strings on the alphabet of L), and because all recursive relations can be represented in PA (and therefore, by assumption, also in PA*), it is possible to formulate a p ...
Predicate Logic - Teaching-WIKI
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
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.