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PHILOSOPHY 326 / MATHEMATICS 307 SYMBOLIC LOGIC This
... decision procedures for determining truth functional validity and logical equivalence, and (most importantly) proficiency in constructing proofs in a system of natural deduction for the logic of propositions. This course does not have a fixed and predetermined syllabus. There is core content which w ...
... decision procedures for determining truth functional validity and logical equivalence, and (most importantly) proficiency in constructing proofs in a system of natural deduction for the logic of propositions. This course does not have a fixed and predetermined syllabus. There is core content which w ...
Scoring Rubric for Assignment 1
... unclear. Theory is not relevant or only relevant for some aspects; theory is not clearly articulated and/or has incorrect or incomplete components. Relationship between theory and research is unclear or inaccurate, major errors in the logic are present. 0 – 4 pts Conclusion may not be clear and the ...
... unclear. Theory is not relevant or only relevant for some aspects; theory is not clearly articulated and/or has incorrect or incomplete components. Relationship between theory and research is unclear or inaccurate, major errors in the logic are present. 0 – 4 pts Conclusion may not be clear and the ...
Predicate Calculus pt. 2
... Exercise 2 A (symmetric, irreflexive) graph G = (V, E) consists of a set of vertices V and a binary, symmetric, irreflexive relation E on V , the edge relation of the graph. If xEy, we say that the vertices x and y are connected (by an edge). An N -coloring of G assigns to each vertex one of the col ...
... Exercise 2 A (symmetric, irreflexive) graph G = (V, E) consists of a set of vertices V and a binary, symmetric, irreflexive relation E on V , the edge relation of the graph. If xEy, we say that the vertices x and y are connected (by an edge). An N -coloring of G assigns to each vertex one of the col ...
What is Logic?
... Not logically valid, BUT can still be useful. In fact, it models the way humans reason all the time: Every non-flying bird I’ve seen before has been a penguin; hence that non-flying bird must be a penguin. ...
... Not logically valid, BUT can still be useful. In fact, it models the way humans reason all the time: Every non-flying bird I’ve seen before has been a penguin; hence that non-flying bird must be a penguin. ...
byd.1 Second-Order logic
... the notion of connectedness for graphs, by saying that there is no nontrivial separation of the vertices into disconnected parts: ¬∃A (∃x A(x) ∧ ∃y ¬A(y) ∧ ∀w ∀z ((A(w) ∧ ¬A(z)) → ¬R(w, z))). For yet another example, you might try as an exercise to define the class of finite structures whose domain ...
... the notion of connectedness for graphs, by saying that there is no nontrivial separation of the vertices into disconnected parts: ¬∃A (∃x A(x) ∧ ∃y ¬A(y) ∧ ∀w ∀z ((A(w) ∧ ¬A(z)) → ¬R(w, z))). For yet another example, you might try as an exercise to define the class of finite structures whose domain ...
The Closed World Assumption
... Prolog goes into a loop on this, but under the declarative reading, p is a logical consequence of the theory (this was covered in Question 3 of Tutorial 3). The problem here is the inclusion of negated formulas in the program. ...
... Prolog goes into a loop on this, but under the declarative reading, p is a logical consequence of the theory (this was covered in Question 3 of Tutorial 3). The problem here is the inclusion of negated formulas in the program. ...
Difficulties of the set of natural numbers
... entity i.e. a set is a critical issue in mathematics and philosophy. Around it two opposite concepts of infinity have been developed, which are potential infinity and actual infinity. The former regards the infinite series 0, 1, 2, ... is potentially endless and the process of adding more and more n ...
... entity i.e. a set is a critical issue in mathematics and philosophy. Around it two opposite concepts of infinity have been developed, which are potential infinity and actual infinity. The former regards the infinite series 0, 1, 2, ... is potentially endless and the process of adding more and more n ...
From proof theory to theories theory
... any particular theory, and the simplicity of this formalism, compared to any particular theory such as geometry, arithmetic, or set theory, has lead to the development of a branch of proof theory that focuses on predicate logic. A central result in this branch of proof theory is the cut elimination ...
... any particular theory, and the simplicity of this formalism, compared to any particular theory such as geometry, arithmetic, or set theory, has lead to the development of a branch of proof theory that focuses on predicate logic. A central result in this branch of proof theory is the cut elimination ...
notes
... Let P be a propositions containing the (distinct) atomic formulas A 1 , . . . , An and v1 , . . . v2n its interpretations. We denote with v P the boolean function associated with P , i.e. vP : {0, 1}n → {0, 1} is defined as follows: for each (a 1 , . . . , an ), ai ∈ {0, 1}, there exists i ∈ {1, . ...
... Let P be a propositions containing the (distinct) atomic formulas A 1 , . . . , An and v1 , . . . v2n its interpretations. We denote with v P the boolean function associated with P , i.e. vP : {0, 1}n → {0, 1} is defined as follows: for each (a 1 , . . . , an ), ai ∈ {0, 1}, there exists i ∈ {1, . ...
Sets with dependent elements: Elaborating on Castoriadis` notion of
... “We must think of a multiplicity which is not one in the traditional sense of the word, but which we point out as one, and which is not a multiplicity in the sense that we could enumerate, actually or potentially, its ‘elements’, yet we can point out within it terms not altogether confused. (...) A ...
... “We must think of a multiplicity which is not one in the traditional sense of the word, but which we point out as one, and which is not a multiplicity in the sense that we could enumerate, actually or potentially, its ‘elements’, yet we can point out within it terms not altogether confused. (...) A ...
A simple proof of Parsons` theorem
... the theorem simplifies if the universal theory U admits definition by cases,7 as it is the case with PRA. In this case, we may take k = 1. Note, however, that no such simplification is forthcoming for part (2) of the theorem! The above theorem (in general, Herbrand’s theorem for prenex formulas) can ...
... the theorem simplifies if the universal theory U admits definition by cases,7 as it is the case with PRA. In this case, we may take k = 1. Note, however, that no such simplification is forthcoming for part (2) of the theorem! The above theorem (in general, Herbrand’s theorem for prenex formulas) can ...
Early_Term_Test Comments
... • Be able to translate logic statements into English statements without variables and symbols. Conversely, be able to translate English statements into logic statements. • Be able to verify correctness of reasoning involving logic statements. • Also, be able to deduce conclusions from logic statemen ...
... • Be able to translate logic statements into English statements without variables and symbols. Conversely, be able to translate English statements into logic statements. • Be able to verify correctness of reasoning involving logic statements. • Also, be able to deduce conclusions from logic statemen ...
Partial Correctness Specification
... A proof in Floyd-Hoare logic is a sequence of lines, each of which is either an axiom of the logic or follows from earlier lines by a rule of inference of the logic u ...
... A proof in Floyd-Hoare logic is a sequence of lines, each of which is either an axiom of the logic or follows from earlier lines by a rule of inference of the logic u ...
P - Department of Computer Science
... inference rules R such that, given any set of axioms A and a sentence c, there is a proof of c, starting with A and applying the rules in R, iff c is entailed by A. • Incompleteness Theorem: any theory that is derived from a decidable set of axioms and that characterizes the standard behavior of the ...
... inference rules R such that, given any set of axioms A and a sentence c, there is a proof of c, starting with A and applying the rules in R, iff c is entailed by A. • Incompleteness Theorem: any theory that is derived from a decidable set of axioms and that characterizes the standard behavior of the ...
Probability Captures the Logic of Scientific
... Now we will explicate the concept that is expressed by the word ‘confirms’ in statements like the following. (These are from news reports on the web.) ‘New evidence confirms last year’s indication that one type of neutrino emerging from the Sun’s core does switch to another type en route to the eart ...
... Now we will explicate the concept that is expressed by the word ‘confirms’ in statements like the following. (These are from news reports on the web.) ‘New evidence confirms last year’s indication that one type of neutrino emerging from the Sun’s core does switch to another type en route to the eart ...
Coordinate-free logic - Utrecht University Repository
... (ii) if ϕ, ψ are formulas, then (ϕ ∧ ψ), ¬ϕ are formulas, (iii) if ϕ is a formula and x is a simple term, then ∀x ϕ is a formula. We will assume that ∨, →, ↔, ∃ are defined in an obvious way. For example, ∃x ϕ denotes ¬∀x ¬ϕ. As the definitions show, we have no terms with more than one argument-pla ...
... (ii) if ϕ, ψ are formulas, then (ϕ ∧ ψ), ¬ϕ are formulas, (iii) if ϕ is a formula and x is a simple term, then ∀x ϕ is a formula. We will assume that ∨, →, ↔, ∃ are defined in an obvious way. For example, ∃x ϕ denotes ¬∀x ¬ϕ. As the definitions show, we have no terms with more than one argument-pla ...
1 Chapter III Set Theory as a Theory of First Order Predicate Logic
... are generated by different sets of Urelements there is one that is special. This is the hierachy which results when we start with nothing, so to speak, i.e. when we begin with the empty set. It may not be immediately obvious that this will get us anything at all, but only a little reflection shows t ...
... are generated by different sets of Urelements there is one that is special. This is the hierachy which results when we start with nothing, so to speak, i.e. when we begin with the empty set. It may not be immediately obvious that this will get us anything at all, but only a little reflection shows t ...
PRESERVATION THEOREMS IN LUKASIEWICZ MODEL THEORY
... 4. Basic Facts in Lukasiewicz Model Theory Our first task in this section is to prove the compactness theorem which is the cornerstone of model theory. Our proof is based on the ultraproduct construction and we do not use Henkin construction or completeness theorem. In fact, Lukasiewicz predicate lo ...
... 4. Basic Facts in Lukasiewicz Model Theory Our first task in this section is to prove the compactness theorem which is the cornerstone of model theory. Our proof is based on the ultraproduct construction and we do not use Henkin construction or completeness theorem. In fact, Lukasiewicz predicate lo ...
Frege`s Other Program
... that, as far as is currently known, it is too weak as it does not entail Peano Arithmetic, but only weaker systems, in particular Robinson’s Q. Alternatively, unlike the approach just described, we can break the bond of logicism and extensionalism, rejecting one while maintaining the other. The most ...
... that, as far as is currently known, it is too weak as it does not entail Peano Arithmetic, but only weaker systems, in particular Robinson’s Q. Alternatively, unlike the approach just described, we can break the bond of logicism and extensionalism, rejecting one while maintaining the other. The most ...
THE HISTORY OF LOGIC
... philosophical programmes concerning mathematics and language. He held that arithmetic and analysis are parts of logic, and made great strides in casting number theory within the system of the Begriffsschrift. To capture mathematical induction, minimal closures, and a host of other mathematc Peter Ki ...
... philosophical programmes concerning mathematics and language. He held that arithmetic and analysis are parts of logic, and made great strides in casting number theory within the system of the Begriffsschrift. To capture mathematical induction, minimal closures, and a host of other mathematc Peter Ki ...
Biform Theories in Chiron
... in the metalanguage of L, not in L itself. This is because deduction and computation rules cannot directly manipulate values such as numbers, functions, and sets; they can only manipulate the expressions that denote these values. Traditional logics do not usually provide a facility for formalizing t ...
... in the metalanguage of L, not in L itself. This is because deduction and computation rules cannot directly manipulate values such as numbers, functions, and sets; they can only manipulate the expressions that denote these values. Traditional logics do not usually provide a facility for formalizing t ...
A Short Glossary of Metaphysics
... The view that common-or-garden objects like tables, animals and planets persist through time by enduring. Contrast perdurantism. entity (from Latin ens, object) Object of any kind that exists. Often used more widely than ‘thing’ (Latin res). Entities comprise any object taken to exist, not just indi ...
... The view that common-or-garden objects like tables, animals and planets persist through time by enduring. Contrast perdurantism. entity (from Latin ens, object) Object of any kind that exists. Often used more widely than ‘thing’ (Latin res). Entities comprise any object taken to exist, not just indi ...