A novel approach to giving an interpretation of logic inside category theory. This work has been developed as part of my sabbatical Marie Curie fellowship in Leeds.
Presented at the Logic Seminar, School of Mathematics, University of Leeds (2012).
The document discusses the trace-method for visualizing quadric surfaces. A trace is the intersection of a surface with a plane. For a given surface S, traces parallel to the coordinate planes reveal the shape of S. Specifically, the intersection of S with planes where x=c, y=c, or z=c produce traces perpendicular to the respective axes. The traces of quadric surfaces are conic sections, such as ellipses or circles, which can be used to sketch the overall surface shape.
The document discusses tangent planes to surfaces. A surface is differentiable at a point if it is smooth and has a well-defined tangent plane at that point. The tangent plane approximates the surface near the point of tangency. To find the equation of the tangent plane, we calculate the partial derivatives of the surface function at the point to determine the slopes in the x- and y-directions. These slopes and the point define vectors in the tangent plane, and their cross product gives the normal vector. The equation of the tangent plane is then (z - c) = M(x - a) + L(y - b), where M and L are the partial derivatives and (a, b, c) is the
1. This document provides step-by-step instructions for solving various calculus problems including finding zeros, derivatives, integrals, limits, continuity, asymptotes, extrema, and solving differential equations.
2. For each type of problem, it lists the key steps to take in a "You think..." prompt followed by more detailed explanations and formulas.
3. The document is intended as a reference for a student to know the general approach and techniques for multiple calculus problems at a glance.
The document discusses polar equations and their use in representing conic sections. It defines key terms like focus, directrix, and eccentricity used to describe ellipses, parabolas, and hyperbolas. Ellipses and hyperbolas are defined geometrically as all points where the distance to one focus (PF) divided by the distance to the corresponding directrix (PD) is a constant (the eccentricity). Examples are given of the polar forms of different conic sections for varying eccentricities.
The document provides an introduction to tensors and their properties. It discusses:
1) Tensors map vectors to other vectors and can be represented by matrices. Common examples are the stress tensor and deformation gradient tensor.
2) Operations performed on matrices, such as addition, multiplication, and transposition, can also be performed on tensors when expressed in terms of their components.
3) Tensors differ from ordinary matrices in that their components must transform in a certain way under a change of basis to maintain the tensor's physical meaning.
The document discusses 3D coordinate systems. It explains that a 3D coordinate system adds a z-axis perpendicular to the x and y axes. There are two ways to add the z-axis, resulting in either a right-hand or left-hand system. The right-hand system is used in math and science, while the left-hand system is used in computer graphics. Points in 3D space are identified by ordered triples (x,y,z). Basic graphs in 3D include planes defined by constant equations like x=k, as well as linear equations that define planes.
The document discusses 3D coordinate systems. It explains that a z-axis is added perpendicular to the x- and y-axes to form a 3D coordinate system. There are two ways to orient the z-axis, known as the right-hand system and left-hand system. Every point in 3D space can be located using an ordered triple (x, y, z). The document also discusses the three coordinate planes and provides an example of sketching the graph of an equation in 3D space.
The document discusses multi-variable functions and how to visualize them. It defines Rn as the set of n-tuples of real numbers. A n-variable real valued function maps from Rn to the real numbers. Level curves are used to visualize functions of two variables by setting z equal to constants and plotting the resulting equations in the x-y plane. This produces families of curves that depict the shape of the surface. Examples show how different functions produce different level curve shapes like ellipses, hyperbolas or parabolas.
The document discusses the trace-method for visualizing quadric surfaces. A trace is the intersection of a surface with a plane. For a given surface S, traces parallel to the coordinate planes reveal the shape of S. Specifically, the intersection of S with planes where x=c, y=c, or z=c produce traces perpendicular to the respective axes. The traces of quadric surfaces are conic sections, such as ellipses or circles, which can be used to sketch the overall surface shape.
The document discusses tangent planes to surfaces. A surface is differentiable at a point if it is smooth and has a well-defined tangent plane at that point. The tangent plane approximates the surface near the point of tangency. To find the equation of the tangent plane, we calculate the partial derivatives of the surface function at the point to determine the slopes in the x- and y-directions. These slopes and the point define vectors in the tangent plane, and their cross product gives the normal vector. The equation of the tangent plane is then (z - c) = M(x - a) + L(y - b), where M and L are the partial derivatives and (a, b, c) is the
1. This document provides step-by-step instructions for solving various calculus problems including finding zeros, derivatives, integrals, limits, continuity, asymptotes, extrema, and solving differential equations.
2. For each type of problem, it lists the key steps to take in a "You think..." prompt followed by more detailed explanations and formulas.
3. The document is intended as a reference for a student to know the general approach and techniques for multiple calculus problems at a glance.
The document discusses polar equations and their use in representing conic sections. It defines key terms like focus, directrix, and eccentricity used to describe ellipses, parabolas, and hyperbolas. Ellipses and hyperbolas are defined geometrically as all points where the distance to one focus (PF) divided by the distance to the corresponding directrix (PD) is a constant (the eccentricity). Examples are given of the polar forms of different conic sections for varying eccentricities.
The document provides an introduction to tensors and their properties. It discusses:
1) Tensors map vectors to other vectors and can be represented by matrices. Common examples are the stress tensor and deformation gradient tensor.
2) Operations performed on matrices, such as addition, multiplication, and transposition, can also be performed on tensors when expressed in terms of their components.
3) Tensors differ from ordinary matrices in that their components must transform in a certain way under a change of basis to maintain the tensor's physical meaning.
The document discusses 3D coordinate systems. It explains that a 3D coordinate system adds a z-axis perpendicular to the x and y axes. There are two ways to add the z-axis, resulting in either a right-hand or left-hand system. The right-hand system is used in math and science, while the left-hand system is used in computer graphics. Points in 3D space are identified by ordered triples (x,y,z). Basic graphs in 3D include planes defined by constant equations like x=k, as well as linear equations that define planes.
The document discusses 3D coordinate systems. It explains that a z-axis is added perpendicular to the x- and y-axes to form a 3D coordinate system. There are two ways to orient the z-axis, known as the right-hand system and left-hand system. Every point in 3D space can be located using an ordered triple (x, y, z). The document also discusses the three coordinate planes and provides an example of sketching the graph of an equation in 3D space.
The document discusses multi-variable functions and how to visualize them. It defines Rn as the set of n-tuples of real numbers. A n-variable real valued function maps from Rn to the real numbers. Level curves are used to visualize functions of two variables by setting z equal to constants and plotting the resulting equations in the x-y plane. This produces families of curves that depict the shape of the surface. Examples show how different functions produce different level curve shapes like ellipses, hyperbolas or parabolas.
The document discusses the properties of parabolas that open up or down and parabolas that open left or right. It provides the general form and standard form of each, noting that a positive value of a in the standard form indicates the direction the parabola opens (up if discussing vertical parabolas, right if discussing horizontal parabolas) while a negative a indicates the opposite direction. It also identifies the axis of symmetry formula for each. Examples are given to demonstrate finding the direction a parabola opens, its vertex, and axis of symmetry.
This document contains a summary of a survey given to cadets at the AFA (Air Force Academy) regarding their participation in various sports. The survey found that:
- 66 cadets play volleyball, with 25 not playing another sport
- 68 cadets play swimming, with 29 not playing another sport
- 70 cadets play athletics, with 26 not playing another sport
- 6 cadets play all three sports
The number of cadets that play at least two of the sports is 59.
The document discusses parabolas and their key properties:
- A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- The vertex is the point where the axis of symmetry intersects the parabola. The focus and directrix are a fixed distance (p) from the vertex.
- The latus rectum is the line segment from the focus to the parabola, perpendicular to the axis of symmetry. Its length is determined by the equation of the parabola.
The document discusses partial derivatives. It defines a partial derivative as the slope of a curve intersecting a surface at a point, where the curve is obtained by fixing one of the variables in the surface equation. The partial derivative with respect to x is the slope of the curve intersecting when y is fixed, and vice versa for the partial derivative with respect to y. Examples are provided to demonstrate calculating partial derivatives algebraically and finding equations of tangent lines using partial derivatives.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
27 triple integrals in spherical and cylindrical coordinatesmath267
The document discusses cylindrical and spherical coordinate systems. It defines cylindrical coordinates as using polar coordinates in the xy-plane with z as the third coordinate. It provides an example of converting between rectangular and cylindrical coordinates. Spherical coordinates represent a point as (ρ, θ, φ) where ρ is the distance from the origin and θ and φ specify the direction. Conversion rules between the different systems are given.
The document discusses calculating the area swept out by a polar function r=f(θ) between the angles θ=a and θ=b. The polar area formula is given as A = (1/2) ∫f(θ)2 dθ. Formulas are derived to integrate the squares of sine and cosine in terms of the cosine double angle. These integrals are summarized. An example problem finds the area swept by r=2sin(θ) between 0 and 2π, which is 2π, the area of a circle with radius 1 swept out twice.
The document discusses graphs in polar coordinates. It begins by reviewing how polar coordinates (r, θ) represent a point P, where r is the distance from the origin and θ is the angle relative to the x-axis. It then discusses how to convert between rectangular (x, y) and polar (r, θ) coordinates. The document explains that polar equations relate r and θ, giving the example of r = θ which graphs as an Archimedean spiral. It also discusses the graphs of constant equations like r = c, which is a circle, and θ = c, which is a line. The document concludes by explaining how to graph other polar equations by plotting points.
The document describes polar coordinates, which specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. Conversion formulas between polar (r, θ) and rectangular (x, y) coordinates are provided. An example problem illustrates plotting points from their polar coordinates and finding the corresponding rectangular coordinates.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
The document discusses first degree (linear) functions. It states that most real-world mathematical functions can be composed of formulas from three families: algebraic, trigonometric, and exponential-logarithmic. It focuses on linear functions of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Examples are given of equations and how to determine the slope and y-intercept to write the equation in slope-intercept form as a linear function.
The document discusses polar equations and their use in describing curves. Polar equations are defined as equations involving the variables r and θ. Common polar equations like r = c, θ = c, r = ±c*cos(θ), and r = ±c*sin(θ) are presented along with examples of how they describe geometric shapes like circles and lines. In particular, the equations r = ±c*cos(θ) and r = ±c*sin(θ) always describe circles.
1. The document discusses parabolas and their key characteristics including focus, directrix, and standard equation forms.
2. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
3. The standard equation forms for parabolas are provided depending on the orientation and location of the vertex.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses quadratic functions and parabolas. It begins by defining quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It then provides an example of graphing the quadratic function y = x2 - 4x - 12. To do this, it finds the vertex by setting x = -b/2a, and uses the vertex and other points like the y-intercept to sketch the parabolic shape. It also discusses general properties of parabolas, such as being symmetric around a center line and having a highest/lowest point called the vertex that sits on this line.
This document defines categories, functors, and natural transformations in category theory. It begins by discussing the "size problem" in naively defining categories and introduces the concept of a universe to address this. Categories are then defined as classes of objects and sets of arrows between objects, satisfying composition and identity laws. Functors map categories to categories by mapping objects and arrows, preserving structure. Natural transformations relate functors by assigning morphisms between their actions on objects. The Yoneda lemma and Godement products of natural transformations are also introduced.
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
The document presents solutions to 6 problems:
1) Finding the maximum area of a quadrilateral inscribed in a circle.
2) Proving that 30 divides the difference of two prime number sets with differences of 8.
3) Showing polynomials with integer coefficients satisfy a recursive relation.
4) Determining the number of points dividing a triangle into equal area triangles.
5) Proving an acute triangle is equilateral given conditions on angle bisectors and altitudes.
6) Characterizing a function on integers satisfying two properties.
The document provides three questions from a past exam on Engineering Mathematics IV. Question 1a asks to find the third order Taylor approximation of the differential equation dy/dx = y + 1 with the initial condition y(0) = 0. Question 1b asks to solve a differential equation using the modified Euler's method at two points. Question 1c asks to find the value of y(0.4) using Milne's predictor-corrector method for a given differential equation.
- The document discusses planes and their representations in 3D geometry.
- A plane can be defined by an equation of the form P=P0 + sX + tY, where X and Y are non-zero, non-parallel vectors.
- There is exactly one plane passing through any three non-collinear points A, B, C. This plane is given by the vector equation P = A + sAB + tAC.
Jarrar: First Order Logic- Inference MethodsMustafa Jarrar
Lecture slides by Mustafa Jarrar at Birzeit University, Palestine.
See the course webpage at: http://jarrar-courses.blogspot.com/2011/11/artificial-intelligence-fall-2011.html and http://www.jarrar.info
and on Youtube:
http://www.youtube.com/watch?v=v92oPUYxCQQ&list=PLCC05105BA39E9BC0
Last time we talked about propositional logic, a logic on simple statements.
This time we will talk about first order logic, a logic on quantified statements.
First order logic is much more expressive than propositional logic.
The topics on first order logic are:
1-Quantifiers
2-Negation
3-Multiple quantifiers
4-Arguments of quantified statements
The document discusses the properties of parabolas that open up or down and parabolas that open left or right. It provides the general form and standard form of each, noting that a positive value of a in the standard form indicates the direction the parabola opens (up if discussing vertical parabolas, right if discussing horizontal parabolas) while a negative a indicates the opposite direction. It also identifies the axis of symmetry formula for each. Examples are given to demonstrate finding the direction a parabola opens, its vertex, and axis of symmetry.
This document contains a summary of a survey given to cadets at the AFA (Air Force Academy) regarding their participation in various sports. The survey found that:
- 66 cadets play volleyball, with 25 not playing another sport
- 68 cadets play swimming, with 29 not playing another sport
- 70 cadets play athletics, with 26 not playing another sport
- 6 cadets play all three sports
The number of cadets that play at least two of the sports is 59.
The document discusses parabolas and their key properties:
- A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- The vertex is the point where the axis of symmetry intersects the parabola. The focus and directrix are a fixed distance (p) from the vertex.
- The latus rectum is the line segment from the focus to the parabola, perpendicular to the axis of symmetry. Its length is determined by the equation of the parabola.
The document discusses partial derivatives. It defines a partial derivative as the slope of a curve intersecting a surface at a point, where the curve is obtained by fixing one of the variables in the surface equation. The partial derivative with respect to x is the slope of the curve intersecting when y is fixed, and vice versa for the partial derivative with respect to y. Examples are provided to demonstrate calculating partial derivatives algebraically and finding equations of tangent lines using partial derivatives.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
27 triple integrals in spherical and cylindrical coordinatesmath267
The document discusses cylindrical and spherical coordinate systems. It defines cylindrical coordinates as using polar coordinates in the xy-plane with z as the third coordinate. It provides an example of converting between rectangular and cylindrical coordinates. Spherical coordinates represent a point as (ρ, θ, φ) where ρ is the distance from the origin and θ and φ specify the direction. Conversion rules between the different systems are given.
The document discusses calculating the area swept out by a polar function r=f(θ) between the angles θ=a and θ=b. The polar area formula is given as A = (1/2) ∫f(θ)2 dθ. Formulas are derived to integrate the squares of sine and cosine in terms of the cosine double angle. These integrals are summarized. An example problem finds the area swept by r=2sin(θ) between 0 and 2π, which is 2π, the area of a circle with radius 1 swept out twice.
The document discusses graphs in polar coordinates. It begins by reviewing how polar coordinates (r, θ) represent a point P, where r is the distance from the origin and θ is the angle relative to the x-axis. It then discusses how to convert between rectangular (x, y) and polar (r, θ) coordinates. The document explains that polar equations relate r and θ, giving the example of r = θ which graphs as an Archimedean spiral. It also discusses the graphs of constant equations like r = c, which is a circle, and θ = c, which is a line. The document concludes by explaining how to graph other polar equations by plotting points.
The document describes polar coordinates, which specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. Conversion formulas between polar (r, θ) and rectangular (x, y) coordinates are provided. An example problem illustrates plotting points from their polar coordinates and finding the corresponding rectangular coordinates.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
The document discusses first degree (linear) functions. It states that most real-world mathematical functions can be composed of formulas from three families: algebraic, trigonometric, and exponential-logarithmic. It focuses on linear functions of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Examples are given of equations and how to determine the slope and y-intercept to write the equation in slope-intercept form as a linear function.
The document discusses polar equations and their use in describing curves. Polar equations are defined as equations involving the variables r and θ. Common polar equations like r = c, θ = c, r = ±c*cos(θ), and r = ±c*sin(θ) are presented along with examples of how they describe geometric shapes like circles and lines. In particular, the equations r = ±c*cos(θ) and r = ±c*sin(θ) always describe circles.
1. The document discusses parabolas and their key characteristics including focus, directrix, and standard equation forms.
2. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
3. The standard equation forms for parabolas are provided depending on the orientation and location of the vertex.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses quadratic functions and parabolas. It begins by defining quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It then provides an example of graphing the quadratic function y = x2 - 4x - 12. To do this, it finds the vertex by setting x = -b/2a, and uses the vertex and other points like the y-intercept to sketch the parabolic shape. It also discusses general properties of parabolas, such as being symmetric around a center line and having a highest/lowest point called the vertex that sits on this line.
This document defines categories, functors, and natural transformations in category theory. It begins by discussing the "size problem" in naively defining categories and introduces the concept of a universe to address this. Categories are then defined as classes of objects and sets of arrows between objects, satisfying composition and identity laws. Functors map categories to categories by mapping objects and arrows, preserving structure. Natural transformations relate functors by assigning morphisms between their actions on objects. The Yoneda lemma and Godement products of natural transformations are also introduced.
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
The document presents solutions to 6 problems:
1) Finding the maximum area of a quadrilateral inscribed in a circle.
2) Proving that 30 divides the difference of two prime number sets with differences of 8.
3) Showing polynomials with integer coefficients satisfy a recursive relation.
4) Determining the number of points dividing a triangle into equal area triangles.
5) Proving an acute triangle is equilateral given conditions on angle bisectors and altitudes.
6) Characterizing a function on integers satisfying two properties.
The document provides three questions from a past exam on Engineering Mathematics IV. Question 1a asks to find the third order Taylor approximation of the differential equation dy/dx = y + 1 with the initial condition y(0) = 0. Question 1b asks to solve a differential equation using the modified Euler's method at two points. Question 1c asks to find the value of y(0.4) using Milne's predictor-corrector method for a given differential equation.
- The document discusses planes and their representations in 3D geometry.
- A plane can be defined by an equation of the form P=P0 + sX + tY, where X and Y are non-zero, non-parallel vectors.
- There is exactly one plane passing through any three non-collinear points A, B, C. This plane is given by the vector equation P = A + sAB + tAC.
Jarrar: First Order Logic- Inference MethodsMustafa Jarrar
Lecture slides by Mustafa Jarrar at Birzeit University, Palestine.
See the course webpage at: http://jarrar-courses.blogspot.com/2011/11/artificial-intelligence-fall-2011.html and http://www.jarrar.info
and on Youtube:
http://www.youtube.com/watch?v=v92oPUYxCQQ&list=PLCC05105BA39E9BC0
Last time we talked about propositional logic, a logic on simple statements.
This time we will talk about first order logic, a logic on quantified statements.
First order logic is much more expressive than propositional logic.
The topics on first order logic are:
1-Quantifiers
2-Negation
3-Multiple quantifiers
4-Arguments of quantified statements
Lecture slides by Mustafa Jarrar at Birzeit University, Palestine.
See the course webpage at: http://jarrar-courses.blogspot.com/2011/11/artificial-intelligence-fall-2011.html and http://www.jarrar.info
and on Youtube:
http://www.youtube.com/watch?v=v92oPUYxCQQ&list=PLCC05105BA39E9BC0
The document provides an overview of first-order logic (FOL) as presented by Professor Padhraic Smyth in CS 271. It discusses the syntax and semantics of FOL, including constants, predicates, functions, variables, quantifiers, and how FOL allows for more expressive representations than propositional logic by incorporating objects, relations, and functions. Examples are given throughout to illustrate FOL concepts such as terms, atomic sentences, complex sentences, universal and existential quantification, and how FOL can be used to represent domains like the Wumpus world.
First-order logic allows for more expressive power than propositional logic by representing objects, relations, and functions in the world. It includes constants like names, predicates that relate objects, functions, variables, logical connectives, equality, and quantifiers. Relations can represent properties of single objects or facts about multiple objects. Models represent interpretations of first-order logic statements graphically. Terms refer to objects as constants or functions. Atomic sentences make statements about objects using predicates. Complex sentences combine atomic sentences with connectives. Universal quantification asserts something is true for all objects, while existential quantification asserts something is true for at least one object.
The document discusses first-order logic (FOL) and its advantages over propositional logic for representing knowledge. It introduces the basic elements of FOL syntax, such as constants, predicates, functions, variables, and connectives. It provides examples of FOL expressions and discusses how objects and relations between objects can be represented. It also covers quantification in FOL using universal and existential quantifiers.
Using Controlled Natural Language and First Order Logic to improve e-consulta...jodischneider
A reading group talk about 3 papers from the IMPACT project.
Taken together, they demonstrate how online conversations for policy-making can be structured and analyzed, using Controlled Natural
Languages, First Order Logic reasoners, Semantic Wikis, and argumentation frameworks.
Adam Wyner and Tom van Engers. A Framework for Enriched, Controlled On-line Discussion Forums for e-Government Policy-making. EGOVIS 2010.
Adam Wyner, Tom van Enger, and Kiavash Bahreini. From Policy-making Statements to First-order Logic. Electronic Government and Electronic Participation 2010.
Adam Wyner and Tom van Enger. Towards Web-based Mass Argumentation in Natural Language. (long version of this EKAW 2010 poster).
The document discusses the requirements and components of expert systems. It notes that expert systems must have a narrow problem area requiring significant human expertise. They must perform at an expert level and be able to explain their reasoning. Structurally, they require a knowledge base containing domain expertise separated from inference procedures, as well as a working memory, inference engine, explanation facility, and user interface. The knowledge base stores facts and rules, the inference engine applies rules to derive new facts, and working memory stores known facts.
This document provides an introduction to propositional logic and first-order logic. It defines propositional logic, including propositional variables, connectives like conjunction and disjunction, and the laws of propositional logic. It then introduces first-order logic, which adds quantifiers, variables, functions, and predicates to represent objects, properties, and relations in a domain. First-order logic allows for more expressive statements about individuals and generalizations than propositional logic alone.
FINAL PROJECT, MATH 251, FALL 2015[The project is Due Mond.docxvoversbyobersby
FINAL PROJECT, MATH 251, FALL 2015
[The project is Due Monday after the thanks giving recess]
.NAME(PRINT).________________ SHOW ALL WORK. Explain and
SKETCH (everywhere anytime and especially as you try to comprehend the prob-
lems below) whenever possible and/or necessary. Please carefully recheck your
answers. Leave reasonable space between lines on your solution sheets. Number
them and print your name.
Please sign the following. I hereby affirm that all the work in this project was
done by myself ______________________.
1) i) Explain how to derive the representation of the Cartesian coordinates x,y,z
in terms of the spherical coordinates ρ, θ, φ to obtain
(0.1) r =< x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), z = ρcos(φ) > .
What are the conventional ranges of ρ, θ, φ?
ii) Conversely, explain how to express ρ, sin(θ), cos(θ), cos(φ), sin(φ) as
functions of x,y,z.
iii) Consider the spherical coordinates ρ,θ, φ. Sketch and describe in your own
words the set of all points x,y,z in x,y,z space such that:
a) 0 ≤ ρ ≤ 1, 0 ≤ θ < 2π, 0 ≤ φ ≤ π b) ρ = 1, 0 ≤ θ < 2π, 0 ≤ φ ≤ π,
c) 0 ≤ ρ < ∞, 0 ≤ θ < 2π, φ = π
4
, d) ρ = 1, 0 ≤ θ < 2π, φ = π
4
,
e) ρ = 1, θ = π
4
, 0 ≤ φ ≤ π. f) 1 ≤ ρ ≤ 2, 0 ≤ θ < 2π, π
6
≤ φ ≤ π
3
.
iv) In a different set of Cartesian Coordinates ρ, θ, φ sketch and describe in your
own words the set of points (ρ, θ, φ) given above in each item a) to f). For example
the set in a) in x,y,z space is a ball with radius 1 and center (0,0,0). However, in
the Cartesian coordinates ρ, θ, φ the set in a) is a rectangular box.
2) [Computation and graphing of vector fields]. Given r =< x,y,z > and the
vector Field
(0.2) F(x,y,z) = F(r) =< 1 + z,yx,y >,
1
FINAL PROJECT, MATH 251, FALL 2015 2
i) Draw the arrows emanating from (x,y,z) and representing the vectors F(r) =
F(x,y,z) . First draw a 2 raw table recording F(r) versus (x,y,z) for the 4 points
(±1,±2,1) . Afterwards draw the arrows.
ii) Show that the curve
(0.3) r(t) =< x = 2cos(t), y = 4sin(t), z ≡ 0 >, 0 ≤ t < 2π,
is an ellipse. Draw the arrows emanating from (x(t),y(t),z(t)) and representing
the vector values of dr(t)
dt
, F(r(t)) = F(x(t),y(t),z(t)) . Let θ(t) be the angle
between the arrows representing dr(t)
dt
and F(r(t)) . First draw a 5 raw table
recording t, (x(t),y(t),z(t)), dr(t)
dt
, F(r(t)), cos(θ(t)) for the points (x(t),y(t),z(t))
corresponding to t = 0,π
4
, 3π
4
, 5π
4
, 7π
4
. Then draw the arrows.
iii) Given the surface
r(θ,φ) =< x = 2sin(φ)cos(θ), y = 2sin(φ)sin(θ), z = 2cos(φ) >,0 ≤ θ < 2π, 0 ≤ φ ≤ π,
in parametric form. Use trigonometric formulas to show that the following iden-
tity holds
x2(θ,φ) + y2(θ,φ) + z2(θ,φ) ≡ 22.
iv) Draw the arrows emanating from (x(θ,φ),y(θ,φ),z(θ,φ)) and representing the
vectors ∂r(θ,φ)
∂θ
× ∂r(θ,φ)
∂φ
, F(r(θ,φ)) = F(x(θ,φ),y(θ,φ),z(θ,φ)) . Let α(θ,φ) be
the angle between the arrows representing ∂r(θ,φ)
∂θ
× ∂r(θ,φ)
∂φ
and F(r(θ,φ)) . First
draw a table with raws and columns recording (θ,φ),(x(θ,φ),y ...
This document provides instructions for a 150-minute mathematics scholarship test consisting of 45 multiple-choice questions across three sections: Algebra, Analysis, and Geometry. The instructions specify that candidates should answer each question in the provided answer booklet and not on the question paper. Various mathematical notations and concepts are defined for reference in answering the questions.
00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).pptDediTriLaksono1
The document defines various concepts related to fuzzy sets and fuzzy logic. It defines the support, core, normality, crossover points, and other properties of fuzzy sets. It also defines operations on fuzzy sets like union, intersection, complement, and algebraic operations. It discusses the extension principle for mapping fuzzy sets through functions. It provides examples of applying the extension principle and compositions of fuzzy relations. Finally, it discusses linguistic variables and modifiers like hedges, negation, and connectives that are used to modify terms in a linguistic variable.
This document discusses two applications of tangent lines: differentials and linear approximation, and finding the tangent line T(b) at a nearby point b. It explains that the tangent line T(x) at point (a, f(a)) is given by T(x) = f'(a)(x - a) + f(a). The slope f'(a) is identified with the derivative dy/dx. There are two ways to find T(b): directly using T(x), or by finding the differential ΔT = dy and using ΔT + f(a) = T(b).
There are infinitely many maximal primitive positive clones in a diagonalizable algebra M* that serves as an algebraic model for provability logic GL. The paper constructs M* as the subalgebra of infinite binary sequences generated by the zero element (0,0,...). It defines primitive positive clones as sets of operations closed under existential definitions, and shows there are infinitely many maximal clones K1, K2, etc. that preserve the relations x = ¬Δi, x = ¬Δ2, etc. in M*. This provides a simple example of infinitely many maximal primitive positive clones.
Here the concept of "TRUE" is defined according to Alfred Tarski, and the concept "OCCURING EVENT" is derived from this definition.
From here, we obtain operations on the events and properties of these operations and derive the main properties of the CLASSICAL PROB-ABILITY. PHYSICAL EVENTS are defined as the results of applying these operations to DOT EVENTS.
Next, the 3 + 1 vector of the PROBABILITY CURRENT and the EVENT STATE VECTOR are determined.
The presence in our universe of Planck's constant gives reason to\linebreak presume that our world is in a CONFINED SPACE. In such spaces, functions are presented by Fourier series. These presentations allow formulating the ENTANGLEMENT phenomenon.
Global Journal of Science Frontier Research: FMathematics and Decision Sciences Volume 18 Issue 2 Version 1.0 Year 2018
From moments to sparse representations, a geometric, algebraic and algorithmi...BernardMourrain
Tensors (10-14 September 2018, Polytechnico di Torino) - From moments to sparse representations, a geometric, algebraic and algorithmic viewpoint. Tensors (10-14 September 2018, Polytechnico di Torino) - From moments to sparse representations, a geometric, algebraic and algorithmic viewpoint. Part 1.
The document provides an overview of the Chase Algorithm, which is used to determine if a dependency D follows from a given set of dependencies S. It begins with a recap of relevant concepts from first-order logic for database theory, including formulas, models/instances, semantics, and dependencies. It then introduces the Chase Algorithm, which rewrites D as much as possible using rules in S, checking if the result D' is a tautology. If so, D follows from S. The document also discusses embeddings of formulas into instances and the satisfiability and termination of the Chase.
Existence Theory for Second Order Nonlinear Functional Random Differential Eq...IOSR Journals
This document presents an existence theory for solutions to second order nonlinear functional random differential equations in Banach algebras. It begins by introducing the type of random differential equation being studied and defining relevant function spaces. It then states several theorems and lemmas from previous works that will be used to prove the main results. The paper goes on to prove that under certain Lipschitz conditions and boundedness assumptions on the operators defining the equation, the random differential equation has at least one random solution in the given function space. It also shows that the set of such random solutions is compact. The results generalize previous existence theorems to the random case.
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
This document summarizes Chris Swierczewski's general exam presentation on computational applications of Riemann surfaces and Abelian functions. The presentation covered the geometry and algebra of Riemann surfaces, including bases of cycles, holomorphic differentials, and period matrices. Applications discussed include using Riemann theta functions to find periodic solutions to integrable PDEs like the Kadomtsev–Petviashvili equation. The talk also discussed linear matrix representations of algebraic curves and the constructive Schottky problem of realizing a Riemann matrix as the period matrix of a curve.
Interval valued intuitionistic fuzzy homomorphism of bf algebrasAlexander Decker
This document discusses interval-valued intuitionistic fuzzy homomorphisms of BF-algebras. It begins with introducing BF-algebras and defining interval-valued intuitionistic fuzzy sets. It then defines interval-valued intuitionistic fuzzy ideals of BF-algebras and provides an example. Interval-valued intuitionistic fuzzy homomorphisms of BF-algebras are introduced and some properties are investigated, including showing that the image of an interval-valued intuitionistic fuzzy ideal under a homomorphism is also an interval-valued intuitionistic fuzzy ideal if it satisfies the "sup-inf" property.
The document summarizes the brachistochrone problem from calculus of variations. It introduces the brachistochrone curve as the curve of fastest descent under gravity between two points. The problem is then solved using tools from calculus of variations, arriving at the Euler-Lagrange equation. This equation shows that the brachistochrone curve between two points is a cycloid. Additionally, the document discusses that the cycloid is a tautochronic curve, meaning an object will take the same amount of time to slide from any point on it to the lowest point.
Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic...Ali Ajouz
Jacobi forms of lattice index, whose theory can be viewed as extension of the theory of classical Jacobi forms, play an important role in various theories, like the theory of orthogonal modular forms or the theory of vertex operator
algebras. Every Jacobi form of lattice index has a theta expansion which implies, for index of odd rank, a connection to half integral weight modular forms and then via Shimura lifting to modular forms of integral weight, and implies a direct connection to modular forms of integral weight if the rank is
even. The aim of this thesis is to develop a Hecke theory for Jacobi forms of lattice index extending the Hecke theory for the classical Jacobi forms, and to study how the indicated relations to elliptic modular forms behave under Hecke operators. After defining Hecke operators as double coset operators,
we determine their action on the Fourier coefficients of Jacobi forms, and we determine the multiplicative relations satisfied by the Hecke operators, i.e. we study the structural constants of the algebra generated by the Hecke operators. As a consequence we show that the vector space of Jacobi forms
of lattice index has a basis consisting of simultaneous eigenforms for our Hecke operators, and we discover the precise relation between our Hecke algebras and the Hecke algebras for modular forms of integral weight. The
latter supports the expectation that there exist equivariant isomorphisms between spaces of Jacobi forms of lattice index and spaces of integral weight modular forms. We make this precise and prove the existence of such liftings
in certain cases. Moreover, we give further evidence for the existence of such liftings in general by studying numerical examples.
Solution to schrodinger equation with dirac comb potential slides
This document summarizes solving the Schrödinger equation for a Dirac comb potential. The potential is an infinite series of Dirac delta functions spaced periodically. Floquet theory is used to solve the time-independent Schrödinger equation for this potential. Boundary conditions are applied and the resulting equations are solved graphically. Allowed energy bands are determined and plotted versus wave vector for both attractive and repulsive delta function potentials.
The document discusses vector spaces and related concepts:
1) It defines a vector space as a set V with vector addition and scalar multiplication operations that satisfy certain properties. Examples of vector spaces include R2, the plane in R3, and the space of real polynomials.
2) A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication and thus forms a vector space with the inherited operations. Examples given include the x-axis in Rn and solution spaces of linear differential equations.
3) The span of a set of vectors is the smallest subspace that contains those vectors, consisting of all possible linear combinations of the vectors in the set.
1. Write an equation in standard form of the parabola that has th.docxKiyokoSlagleis
1.
Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = 2x
2
, but with the given point as the vertex (5, 3).
A. f(x) = (2x - 4) + 4
B. f(x) = 2(2x + 8) + 3
C. f(x) = 2(x - 5)
2
+ 3
D. f(x) = 2(x + 3)
2
+ 3
2 of 20
5.0 Points
Find the coordinates of the vertex for the parabola defined by the given quadratic function.
f(x) = 2(x - 3)
2
+ 1
A. (3, 1)
B. (7, 2)
C. (6, 5)
D. (2, 1)
3 of 20
5.0 Points
Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the following rational function.
g(x) = x + 3/x(x + 4)
A. Vertical asymptotes: x = 4, x = 0; holes at 3x
B. Vertical asymptotes: x = -8, x = 0; holes at x + 4
C. Vertical asymptotes: x = -4, x = 0; no holes
D. Vertical asymptotes: x = 5, x = 0; holes at x - 3
4 of 20
5.0 Points
"Y varies directly as the n
th
power of x" can be modeled by the equation:
A. y = kx
n
.
B. y = kx/n.
C. y = kx
*n
.
D. y = kn
x
.
5 of 20
5.0 Points
40 times a number added to the negative square of that number can be expressed as:
A.
A(x) = x
2
+ 20x.
B. A(x) = -x + 30x.
C.
A(x) = -x
2
- 60x.
D.
A(x) = -x
2
+ 40x.
6 of 20
5.0 Points
The graph of f(x) = -x
3
__________ to the left and __________ to the right.
A. rises; falls
B. falls; falls
C. falls; rises
D. falls; falls
Solve the following formula for the specified variable:
V = 1/3 lwh for h
7 of 20
Write an equation that expresses each relationship. Then solve the equation for y.
x varies jointly as y and z
A. x = kz; y = x/k
B. x = kyz; y = x/kz
C. x = kzy; y = x/z
D. x = ky/z; y = x/zk
8 of 20
8 times a number subtracted from the squared of that number can be expressed as:
A. P(x) = x + 7x.
B.P(x) = x
2
- 8x.
C. P(x) = x - x.
P(x) = x
2
+ 10x.
9of 20
Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
f(x) = x
4
- 9x
2
A. x = 0, x = 3, x = -3; f(x) crosses the x-axis at -3 and 3; f(x) touches the x-axis at 0.
B. x = 1, x = 2, x = 3; f(x) crosses the x-axis at 2 and 3; f(x) crosses the x-axis at 0.
C. x = 0, x = -3, x = 5; f(x) touches the x-axis at -3 and 5; f(x) touches the x-axis at 0.
D. x = 1, x = 2, x = -4; f(x) crosses the x-axis at 2 and -4; f(x) touches the x-axis at 0.
10 of 20
Find the domain of the following rational function.
f(x) = x + 7/x
2
+ 49
A. All real numbers < 69
B. All real numbers > 210
C. All real numbers ≤ 77
D. All real numbers
11 of 20
Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = 3x
2
or g(x) = -3x
2
, but with the given maximum or minimum.
Minimum = 0 at x = 11
A. f(x) = 6(x - 9)
B. f(x) = 3(x - 11)
2
C. f(x) = 4(x + 10)
D. f(x) = 3(x
2
- 15)
2
12 of 20
Solve the following polynomial inequality.
3x
2
+ 10x - 8 ≤ 0
A. [6, 1/3]
B. [-4, 2/3]
C. [-9, 4/5]
D. [8, 2/7]
13 of 20
Find the coordinate.
This document summarizes research on the existence of best proximity points for mappings between subsets of metric spaces.
The paper introduces the concepts of proximal intersection property and diagonal property for pairs of subsets, and proves that every pair of subsets in a Hilbert space satisfies the diagonal property. It establishes the existence of best proximity points for contractive mappings between subsets that satisfy the proximal intersection property or diagonal property.
Similar to Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard isomorphism (20)
The Graph Minor Theorem: a walk on the wild side of graphsMarco Benini
The document summarizes the Graph Minor Theorem, which states that the set of all finite graphs forms a well-quasi ordering under the graph minor relation. It discusses Robertson and Seymour's 500-page proof of this theorem. It then outlines Nash-Williams' technique for proving that a quasi-order is a well-quasi ordering and describes an attempted proof of the Graph Minor Theorem using this technique. However, the attempt fails due to issues with the "coherence" of embeddings between decomposed components of graphs. Maintaining coherence of embeddings under the graph minor relation is identified as the key challenge in finding a simpler proof of the Graph Minor Theorem.
The famous Kruskal's tree theorem states that the collection of finite trees labelled over a well quasi order and ordered by homeomorphic embedding, forms a well quasi order. Its intended mathematical meaning is that the collection of finite, connected and acyclic graphs labelled over a well quasi order is a well quasi order when it is ordered by the graph minor relation.
Oppositely, the standard proof(s) shows the property to hold for trees in the Computer Science's sense together with an ad-hoc, inductive notion of embedding. The mathematical result follows as a consequence in a somewhat unsatisfactory way.
In this talk, a variant of the standard proof will be illustrated explaining how the Computer Science and the graph-theoretical statements are strictly coupled, thus explaining why the double statement is justified and necessary.
The Graph Minor Theorem: a walk on the wild side of graphsMarco Benini
The Graph Minor Theorem says that the collection of finite graphs
ordered by the minor relation is a well quasi order. This apparently
innocent statement hides a monstrous proof: the original result by
Robertson and Seymour is about 500 pages and twenty articles, in which a
new and deep branch of Graph Theory has been developed.
The theorem is famous and full of consequences both on the theoretical side
of Mathematics and in applications, e.g., to Computer Science. But there
is no concise proof available, although many attempts have been made.
In this talk, arising from one such failed attempts, an analysis of the
Graph Minor Theorem is presented. Why is it so hard?
Assuming to use the by-now standard Nash-Williams's approach to prove it,we will
illustrate a number of methods which allow to solve or circumvent some
of the difficulties. Finally, we will show that the core of this line of
thought lies in a coherence question which is common to many parts of
Mathematics: elsewhere it has been solved, although we were unable to
adapt those solutions to the present framework. So, there is hope for a
short proof of the Graph Minor Theorem but it will not be elementary.
This document discusses how to present negative results in research in a positive manner. It notes that while the goal of research is to find positive answers, negative results are also useful for understanding what does not work and setting limits. The document provides examples of reframing negative results positively, such as precisely defining where a procedure fails instead of just saying it failed, or stating positive theorems about the limitations of methods. It also addresses how to present inherent negative results like impossibility theorems and how to address "dead ends" in research.
The document discusses variations on Higman's Lemma, which states that a set of finite sequences ordered by embedding forms a well quasi-order if and only if the underlying set ordered does. The document defines well quasi-orders and related concepts. It then examines properties of products, coproducts, equalizers, coequalizers and exponentiation in categories of well quasi-orders and related categories. It proves Higman's Lemma using categorical concepts like equalizers and coequalizers. It also examines properties of categories of descending chains of a quasi-order and how these relate to properties of the original quasi-order.
Proof-Theoretic Semantics: Point-free meaninig of first-order systemsMarco Benini
This document summarizes a talk on providing a semantics for first-order logical theories using logical categories. The semantics interprets formulae as objects in a category and proofs as morphisms, without assuming elements exist. Quantifiers are interpreted using stars and costars. A logical category is a prelogical category where stars and costars exist to interpret all formulae. This semantics is sound and complete - a formula is true if a proof morphism exists. The semantics can interpret many other approaches and inconsistent theories have "trivial" models.
A talk I gave at the Yonsei University, Seoul in July 21st, 2015.
The aim was to show my background contribution to the CORCON (Correctness by Construction) research project.
I have to thank Prof. Byunghan Kim and Dr Gyesik Lee for their kind hospitality.
Numerical Analysis and Epistemology of InformationMarco Benini
The slides of my presentation at the workshop "Philosophical Aspects of Computer Science", European Centre for Living Technology, University “Ca’ Foscari”, Venice, March 2015.
L'occhio del biologo: elementi di fotografiaMarco Benini
The slides of the course "L'occhio del biologo", Alta Formazione, Università degli Studi dell'Insubria.
It is a small course on the fundamentals of photography oriented towards the scientific photography in a biological laboratory.
Marie Skłodowska Curie Intra-European FellowshipMarco Benini
A brief report of my experience as a Marie Curie Research Fellow in Leeds to illustrate to my colleagues what means to participate in such a program.
I have to acknowledge the kind invitation of the Research Office of the Università degli Studi dell'Insubria and the Rector delegate to research, Prof. Umberto Piarulli.
This document discusses representing data types and programming in an abstract way that hides the concrete representation of data. It presents an approach where programs operate on abstract representations of data rather than concrete implementations, allowing computations to be performed without inspecting the output. As an example, it shows an abstract implementation of list concatenation that computes correctly without knowing the concrete list representation. This approach ensures correctness while preventing inspection of results.
By analysing the explanation of the classical heapsort algorithm via the method of levels of abstraction mainly due to Floridi, we give a concrete and precise example of how to deal with algorithmic knowledge. To do so, we introduce a concept already implicit in the method, the ‘gradient of explanations’. Analogously to the gradient of abstractions, a gradient of explanations is a sequence of discrete levels of explanation each one refining the previous, varying formalisation, and thus providing progressive evidence for hidden information. Because of this sequential and coherent uncovering of the information that explains a level of abstraction—the heapsort algorithm in our guiding example—the notion of gradient of explanations allows to precisely classify purposes in writing software according to the informal criterion of ‘depth’, and to give a precise meaning to the notion of ‘concreteness’.
This talk aims at introducing, through a very simple example, a way to represent data types in the λ-calculus, and thus, in functional programming languages, so that the structure of the data types itself becomes a parameter.
This very simple technical trick allows to reconsider programming as a way to express morphisms between models of a logical theory. As an application, it allows to realise a way to perform anonymous computations.
From a philosophical point of view, the presented approach shows how it is possible to conceive a real programming system where properties like correctness of programs can be proved, but data cannot be inspected, not even in principle.
In this talk, logically distributive categories are introduced to provide a sound and complete semantics to multi-sorted, first-order, intuitionistic-based logical theories. The peculiar aspect is that no universe is required to interpret terms, making the semantics really point-free.
CORCON2014: Does programming really need data structures?Marco Benini
This talk tries to suggest how computer programming can be conceptually simplified by using abstract mathematics, in particular categorical semantics, so to achieve the 'correctness by construction' paradigm paying no price in term of efficiency.
Also, it introduces an alternative point of view on what is a program and how to conceive data structures, namely as computable morphisms between models of a logical theory.
CORCON2014: Does programming really need data structures?
Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard isomorphism
1. Intuitionistic First-Order Logic
Categorical semantics via the Curry-Howard isomorphism
Marco Benini
M.Benini@leeds.ac.uk
Department of Pure Mathematics
University of Leeds
14th November 2012
2. Introduction
An observation: in part D of P. Johnstone’s Sketches of an Elephant, there
is a categorical semantics for the simply typed λ-calculus. In the very same
class of models, one can give a semantics to the corresponding fragment of
propositional logic.
The problem:
is it possible to do the same for full first-order intuitionistic logic?
2 of 32
3. Introduction
Johstone’s account comes from Lambek and Scott’s Introduction to
Higher-Order Categorical Logic. The considered propositional logic is
minimal logic limited to conjunction and implication.
In Taylor, Practical Foundation of Mathematics, one finds that the treatment
of disjunction requires distributive categories in order to follow the same
pattern as the previous works.
To my knowledge, no categorical semantics appears in literature which
models the full first-order intuitionistic logic AND the corresponding
λ-calculus.
3 of 32
4. The λ-calculus
Definition 1 (Lambda signature)
A λ-signature Σ = 〈S , F , R , Ax〉 is a structure where
1. 〈S , F , R 〉 is a logical signature, i.e.,
1.1 a set S of sort symbols;
1.2 a set F of function symbols, each one decorated as f : s1 × · · · × sn → s0 ,
with s0 , . . . , sn ∈ S;
1.3 a set R of relation symbols, each one decorated as r : s1 × · · · × sn , with
s1 , . . . , sn ∈ S;
2. Ax is the set of axiom symbols, each one decorated as a : A → B where
A, B ∈ λTypes(Σ) and FV(A → B ) = .
We call LTerms(Σ) the collection of logical terms constructed from the
signature Σ, assuming to have a denumerable set of variables Vs for each
s ∈ S.
4 of 32
5. The λ-calculus
Definition 2 (Lambda type)
Fixed a λ-signature Σ, the λ-types on Σ are inductively defined along with
their free variables as follows:
1. 0, 1 ∈ λTypes(Σ) and FV(0) = FV(1) = ;
2. if p : s1 × · · · × sn ∈ R and t1 : s1 , . . . , tn : sn ∈ LTerms(Σ), then
p(t1 , . . . , tn ) ∈ λTypes(Σ) and FV(p(t1 , . . . , tn )) = n=1 FV(ti : si );
i
3. if A, B ∈ λTypes(Σ) then A × B , A + B , A → B ∈ λTypes(Σ) and
FV(A × B ) = FV(A + B ) = FV(A → B ) = FV(A) ∪ FV(B );
4. if x ∈ Vs and A ∈ λTypes(Σ) then ∀x : s. A, ∃x : s. A ∈ λTypes(Σ) and
FV(∀x : s. A) = FV(∃x : s. A) = FV(A) {x : s}.
5 of 32
6. The λ-calculus
Definition 3 (Lambda term)
Fixed a λ-signature Σ = 〈S , F , R , Ax〉, for each type t ∈ λTypes(Σ), we
assume there is a denumerable set Wt of (typed) variables.
A λ-term is inductively defined together with its free variables as:
1. if x ∈ Wt then x : t ∈ λTerms(Σ) and FV(x : t ) = {x : t };
2. if f : A → B ∈ Ax and t : A ∈ λTerms(Σ) then f (t ): B ∈ λTerms(Σ) and
FV(f (t ): B ) = FV(t : A);
3. if s : A, t : B ∈ λTerms(Σ) then 〈s, t 〉 : A × B ∈ λTerms(Σ) and
FV(〈s, t 〉 : A × B ) = FV(s : A) ∪ FV(t : B );
4. if t : A × B ∈ λTypes(Σ) then fst(t ): A ∈ λTerms(Σ),
snd(t ): B ∈ λTerms(Σ) and
FV(fst(t ): A) = FV(snd(t ): B ) = FV(t : A × B ); →
6 of 32
7. The λ-calculus
→ (Lambda term)
5. if t : A ∈ λTerms(Σ) then inlB (t ): A + B ∈ λTerms(Σ),
inrB (t ): B + A ∈ λTerms(Σ) and
FV(inlB (t ): A + B ) = FV(inrB (t ): B + A) = FV(t : A);
6. if s : A + B , t : A → C , r : B → C ∈ λTerms(Σ) then
when(s, t , r ): C ∈ λTerms(Σ) and
FV(when(s, t , r ): C ) = FV(s : A + B ) ∪ FV(t : A → C ) ∪ FV(r : B → C );
7. if x ∈ WA and t : B ∈ λTerms(Σ) then (λx : A. t ): A → B ∈ λTerms(Σ) and
FV((λx : A. t ): A → B ) = FV(t : B ) {x : A};
8. if s : A → B , t : A ∈ λTerms(Σ) then s · t : B ∈ λTerms(Σ) and
FV(s · t : B ) = FV(s : A → B ) ∪ FV(t : A);
9. ∗ : 1 ∈ λTerms(Σ) and FV(∗ : 1) = ;
10. FA : 0 → A ∈ λTerms(Σ) and FV(FA : 0 → A) = ; →
7 of 32
8. The λ-calculus
→ (Lambda term)
11. if x ∈ Vs and t : A ∈ λTerms(Σ) where x : s ∈ FV∗ (t : A), then
allI(λx : s. t ):(∀x : s. A) ∈ λTerms(Σ) and
FV(allI(λx : s. t ):(∀x : s. A)) = FV(t : A);
12. if t :(∀x : s. A) ∈ λTerms(Σ) and r : s ∈ LTerms(Σ) then
allE(t , r ):(A[r /x ]) ∈ λTerms(Σ) and
FV(allE(t , r ):(A[r /x ])) = FV(t :(∀x : s. A));
13. if x ∈ Vs , r : s ∈ LTerms(Σ) and t :(A[r /x ]) ∈ λTerms(Σ) then
exIx (t ):(∃x : s. A) ∈ λTerms(Σ) and
FV(exIx (t ):(∃x : s. A)) = FV(t :(A[r /x ]));
14. if t :(∃x : s. A), ∈ λTerms(Σ) and r : A → B ∈ λTerms(Σ) where
x : s ∈ FV∗ (r : A → B ), then exE(t , (λx : s. r )): B ∈ λTerms(Σ) and
FV(exE(t , (λx : s. r )): B ) = FV(t :(∃x : s. A)) ∪ FV(r : A → B ).
In the previous definition, x : s ∈ FV∗ (t : A) if and only if there is
r ∈ λTypes(Σ) and y ∈ Wr such that x : s ∈ FV(r ) and y : r ∈ FV(t : A).
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9. The λ-calculus
Definition 4 (Lambda calculus)
A derivation is inductively defined by the following inference rules, whose
antecedents and consequents are equalities-in-context within a fixed
λ-signature Σ:
(eq0 ) x : A. s =C t y : B . s[r1 /x1 , . . . , rn /xn ] =C t [r1 /x1 , . . . , rn /xn ] where, for
any 1 ≤ i ≤ n, y : B . ri : Ai is a term-in-context;
(x : A. s1 =B1 t1 )
.
.
(eq1 ) . x : A. r [s/y ] =C r [t /y ];
(x : A. sm =Bm tm )
(eq2 ) x : A. x =A x;
(eq3 ) x : A, y : A. x =A y x : A, y : A. y =A x;
(x : A, y : A, z : A. x =A y )
(eq4 ) x : A, y : A, z : A. x =A z;
(x : A, y : A, z : A. y =A z )
→
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10. The λ-calculus
→ (Lambda calculus)
(eq5 ) x : A. s =C t x : A. (λy : B . s) =B →C (λy : B . t );
(eq6 ) x : A. r =C t x : A. allI(λy : s. r ) =(∀y : s. C ) allI(λy : s. t );
(eq7 ) x : A. u =C v x : A. exE(t , (λy : s. u )) =C exE(t , (λy : s. v ));
(×0 ) x : 1. x =1 ∗;
(×1 ) x : A, y : B . fst(〈x , y 〉) =A x;
(×2 ) x : A, y : B . snd(〈x , y 〉) =B y ;
(×3 ) z : A × B . 〈fst(z ), snd(z )〉 =A×B z;
(+0 ) x : A. when(inlB (a), t , s) =C t · a;
(+1 ) x : A. when(inrD (b), t , s) =C s · b;
→
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11. The λ-calculus
→ (Lambda calculus)
(+2 ) when y : A1 ∈ FV(x1 : A1 + A2 ) ∪ FV(x3 : B1 → C ) ∪ FV(x4 : B2 → C ) and
y : A2 ∈ FV(x2 : A1 + A2 ) ∪ FV(x3 : B1 → C ) ∪ FV(x4 : B2 → C )
x0 : A1 + A2 , x1 : A1 → (B1 + B2 ), x2 : A2 → (B1 + B2 ),
x3 : B1 → C , x4 : B2 → C .
when(when(x0 , x1 , x2 ), x3 , x4 ) =C
=C when(x0 , (λy : A1 . when(x1 · y , x3 , x4 )),
(λy : A2 . when(x2 · y , x3 , x4 ))) ;
(+3 ) x : A, y : 0. FA ·y =A x;
(→0 ) x : A. (λy : C . s) · t =B s[t /y ];
(→1 ) x : A. (λy : C . t · y ) =C →B t where y : C ∈ FV(t : C → B );
(∀0 ) x : A. allE(allI(λz : s. t ), r ) =B [r /z ] t [r /z ];
(∀1 ) x : A. allE(u , r ) =B allE(v , r ) r : s∈LTerms(Σ) x : A. u =(∀z : s. B ) v ;
→
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12. The λ-calculus
→ (Lambda calculus)
(∃0 ) x : A. exE(exIz (t ), (λz : s. v )) =B (v [r /z ]) · t;
(∃1 ) x : A. exE(u , (λz : s. r )) =B exE(u , (λz : s. t )) x : A. r =C →B t where
FV(r : C → B ) = FV(t : C → B );
(∃2 ) v :(∃y : s. A). w =B exE(v , (λy : s. (λz : A. w [exIy (z )/v ]))) with
z : A ∈ FV(w : B );
(∃3 ) x : A. exE(exE(a, (λy : s. (λz : D . b))), (λy : s. c )) =C
=C exE(a, (λy : s. (λz : D . exE(b, (λy : s. c )))));
(∃4 ) x : A. exE(a, (λy : s. (λz : C . b[exIy (z )/w ]))) =B b[a/w ] with
z : C ∈ FV(b : B ).
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13. Semantics
Definition 5 (Logically distributive category)
Fixed a λ-signature Σ = 〈S , F , R , Ax〉, a category C together with a map
M : λTypes(Σ) → Obj C is said to be logically distributive if it satisfies the
following seven conditions:
1. C has finite products;
2. C has finite co-products;
3. C has exponentiation;
4. C is distributive, i.e., for every A, B , C ∈ Obj C, the arrow
∆ = [1A × ι1 , 1A × ι2 ]: (A × B ) + (A × C ) → A × (B + C ) has an inverse,
where [_, _] is the co-universal arrow of the (A × B ) + (A × C ) co-product,
_ × _ is the product arrow, 1A is the identity arrow on A, and
ι1 : B → B + C, ι2 : C → B + C are the canonical injections of the B + C
co-product.
→
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14. Semantics
→ (Logically distributive category)
For every s ∈ S, A ∈ λTypes(Σ), and x ∈ Vs , let
ΣA (x : s): LTerms(Σ)(s) → C be the functor from the discrete category
LTerms(Σ)(s) = t : s | t : s ∈ LTerms(Σ) to C defined by t : s → M (A[t /x ]).
Also, for every s ∈ S, A ∈ λTypes(Σ), and x ∈ Vs , let C(∀x : s. A) be the
subcategory of C whose objects are the vertices of the cones on ΣA (x : s)
such that they are of the form MB for some B ∈ λTypes(Σ) and
x : s ∈ FV(B ). Moreover, the arrows of C(∀x : s. A) , apart identities, are the
arrows in the category of cones over ΣA (x : s) having the objects of
C(∀x : s. A) as domain and M (∀x : s. A) as co-domain. →
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15. Semantics
→ (Logically distributive category)
Finally, for every s ∈ S, A ∈ λTypes(Σ), and x ∈ Vs , let C(∃x : s. A) be the
subcategory of C whose objects are the vertices of the co-cones on
ΣA (x : s) such that they are of the form MB for some B ∈ λTypes(Σ) and
x : s ∈ FV(B ). Moreover, the arrows of C(∃x : s. A) , apart identities, are the
arrows in the category of co-cones over ΣA (x : s) having the objects of
C(∃x : s. A) as co-domain and M (∃x : s. A) as domain.
5. All the subcategories C(∀x : s. A) have terminal objects, and all the
subcategories C(∃x : s. A) have initial objects;
→
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16. Semantics
→ (Logically distributive category)
6. The M map is such that
6.1 M (0) = 0, the initial object of C;
6.2 M (1) = 1, the terminal object of C;
6.3 M (A × B ) = MA × MB, the binary product in C;
6.4 M (A + B ) = MA + MB, the binary co-prooduct in C;
6.5 M (A → B ) = MB MA , the exponential object in C;
6.6 M (∀x : s. A) is the terminal object in the subcategory C(∀x : s. A) ;
6.7 M (∃x : s. A) is the initial object in the subcategory C(∃x : s. A) ;
7. For every x ∈ Vs , A, B ∈ λTypes(Σ) with x : s ∈ FV(A), MA × M (∃x : s. B )
is an object of C(∃x : s. A×B ) since, if M (∃x : s. B ), {δt }t : s∈LTerms(Σ) is a
co-cone over ΣB (x : s), and there is one by condition (5), then
MA × M (∃x : s. B ), {1MA ×δt }t : s∈LTerms(Σ) is a co-cone over ΣA×B (x : s).
Thus, there is a unique arrow !: M (∃x : s. A × B ) → MA × M (∃x : s. B ) in
C(∃x : s. A×B ) . Our last condition requires that the arrow ! has an inverse.
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17. Semantics
Definition 6 (Σ-structure)
Given a λ-signature Σ = 〈S , F , R , Ax〉, a Σ-structure is a triple 〈C, M , MAx 〉
such that C together with M forms a logically distributive category and MAx
is a map from Ax such that MAx (a : A → B ) ∈ HomC (MA, MB ).
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18. Semantics
Definition 7 (λ-terms semantics)
Fixed a Σ-structure 〈C, M , MAx 〉, let A ≡ A1 × · · · An , and let
x ≡ x1 : A1 , . . . , xn : An be a context. The semantics of a term-in-context
x . t : B, notation x . t : B , is an arrow in HomC (MA, MB ) inductively defined
as follows:
1. x . xi : Ai = πi , the i-th projector of the product MA = MA1 × · · · × MAn ;
2. if a : C → B ∈ Ax then x . a(t ): B = MAx a ◦ x . t : C ;
3. x . 〈s, t 〉 : B × C = ( x . s : B , x . t : C ) where (_, _) is the universal arrow
of the product MB × MC;
4. x . fst(t ): B = π1 ◦ x . t : B × C where π1 is the first canonical projector
of the product MA × MB;
5. x . snd(t ): C = π2 ◦ x . t : B × C where π2 is the second canonical
projector of the product MA × MB;
→
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19. Semantics
→ (λ-terms semantics)
6. x . (λz : C . t ): C → B is the exponential transpose of
x , z : C . t : B : MA × MC → MB;
7. x . s · t : B = ev ◦ ( x . s : C → B , x . t : C ) where ev is the exponential
evaluation arrow;
8. x . inlB (t ): C + B = ι1 ◦ x . t : C with ι1 the first canonical injection of the
co-product MC + MB;
9. x . inrC (t ): C + B = ι2 ◦ x . t : B with ι2 the second canonical injection of
the co-product MC + MB;
→
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20. Semantics
→ (λ-terms semantics)
10. calling [_, _] the co-universal arrow of (MA × MC1 ) + (MA × MC2 ), (_, _)
the universal arrow of MA × (MC1 + MC2 ), and noticing that the arrow
∆ : (MA × MC1 ) + (MA × MC2 ) → MA × (MC1 + MC2 ) has an inverse
because C with M is logically distributive
x . when(t , u , v ): B = [ev ◦ ( x . u : C1 → B × 1MC1 ) ,
ev ◦ ( x . v : C2 → B × 1MC2 )] ◦
◦ ∆−1 ◦ (1MA , x . t : C1 + C2 ) ;
11. x . ∗ : 1 =!: MA → 1, the universal arrow of the terminal object;
12. x . FB : 0 → B is the exponential transpose of
(!: 0 → MB ) ◦ (πn+1 : MA × 0 → 0);
→
20 of 32
21. Semantics
→ (λ-terms semantics)
13. x . allI(λz : s. t ):(∀z : s. B ) = β ◦ α where
α ≡ 1MAi1 × · · · × 1MAik : MA → MA with A ≡ Ai1 × · · · × Aik , where
x ≡ {xi1 : Ai1 , . . . , xik : Aik } = FV(t : B ), and β : MA → M (∀z : s. B ) is the
universal arrow from MA to the terminal object in C∀z : s. B ;
14. x . allE(t , r ): B [r /z ] = pr ◦ x . t :(∀z : s. B ) where
pr : M (∀z : s. B ) → M (B [r /z ]) is the r -th projector of the unique cone on
ΣB (z : s) whose vertex is M (∀z : s. B ).
It is worth noticing that pr = w :(∀z : s. B ). allE(w , r ): B [r /z ] ;
15. x . exIz (t ):(∃z : s. B ) = jr ◦ x . t : B [r /z ] where
jr : M (B [r /z ]) → M (∃z : s. B ) is the r -th injection of the unique co-cone
on ΣB (z : s) whose vertex is M (∃z : s. B ).
It is worth noticing that jr = w : B [r /z ]. exIz (w ):(∃z : s. B ) ;
→
21 of 32
22. Semantics
→ (λ-terms semantics)
16. x . exE(t , (λz : s. r )): B = γ ◦ β−1 ◦ (α, x . t :(∃z : s. C ) ) where
16.1 α ≡ 1MAi1 × · · · × 1MAik : MA → MA with A ≡ Ai1 × · · · × Aik , where
x ≡ {xi1 : Ai1 , . . . , xik : Aik } = FV(t :(∃z : s. C )) ∪ FV(r : C → B );
16.2 β : M (∃z : s. A × C ) → MA × M (∃z : s. C ) is the co-universal arrow in the
subcategory C∃z : s. A ×C ;
16.3 γ : M (∃z : s. A × C ) → MB is the co-universal arrow in C∃z : s. A ×C .
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23. Soundness
Definition 8 (Validity)
An equality-in-context x . s =A t is valid in the λ-theory T , a set of
equalities-in-context, when, in every logically distributive category C, each
model M of T is also a model of x . s =A t.
A Σ-structure M in C is a model of a theory T when it is a model of each φ
in T .
Finally, M is a model of an equality-in-context x . t =A s if x . t : A = x . s : A .
Theorem 9 (Soundness)
If an equation-in-context x . s =A t is derivable from a λ-theory T , then
x . s =A t is valid in each model of T in every logically distributive category.
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24. Completeness
Definition 10 (Syntactical equivalence)
Given a λ-theory T , the syntactical equivalence of two terms-in-context is
defined by fixing the generated equivalence classes. Precisely, the
equivalence class [x : A. t : B ] is defined as the minimal set, composed by
terms-in-context, such that
1. x : A. t : B ∈ [x : A. t : B ]—reflexivity;
2. if T y : D . s =C r , where y : D . s =C r is an equality-in-context, and
y : D . s : C ∈ [x : A. t : B ], then y : D . r : C ∈ [x : A. t : B ]—closure under
provable equivalence;
→
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25. Completeness
→ (Syntactical equivalence)
3. if y : D . s : C is a term-in-context and, for some 1 ≤ i < m and
z : Di × Di +1 ∈ FV(s : C ) ∪ y1 : D1 , . . . , ym : Dm , it happens that
y1 : D1 , . . . , yi −1 : Di −1 , z : Di × Di +1 , yi +1 : Di +2 ,
. . . , ym : Dm . s[fst(z )/yi ][snd(z )/yi +1 ]: C ∈ [x : A. t : B ] ,
then y : D . s : C ∈ [x : A. t : B ]—closure under associativity in contexts;
4. if y : D . s : C is a term-in-context and, for some 1 ≤ i < m and
z : Di +1 × Di ∈ FV(s : C ) ∪ y1 : D1 , . . . , ym : Dm , it happens that
y1 : D1 , . . . , yi −1 : Di −1 , z : Di +1 × Di , yi +1 : Di +2 ,
. . . , ym : Dm . s[snd(z )/yi ][fst(z )/yi +1 ]: C ∈ [x : A. t : B ] ,
then y : D . s : C ∈ [x : A. t : B ]—closure under commutativity in contexts;
→
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26. Completeness
→ (Syntactical equivalence)
5. if y : D . s : C ∈ [x : A. y : B ] and z : Di ∈ FV(s : C ) ∪ y1 : D1 , . . . , ym : Dm for
some 1 ≤ i ≤ m, then
y1 : D1 , . . . , yi −1 : Di −1 , z : Di , yi +1 : Di +1 , . . . , ym : Dm . s[z /yi ]: C
is in [x : A. t : B ]—closure under α-renaming in contexts.
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27. Completeness
Definition 11 (Syntactical category)
Given a λ-theory T , the syntactical category CT has λTypes(Σ) as objects,
where Σ is the λ-signature of T , and the equivalence classes
[x : A. t : B ]: A → B as arrows.
Identities are given by the classes [x : A. x : A]: A → A for each λ-type A,
and composition is given by substitution:
[y : B . s : C ] ◦ [x : A. t : B ] = [x : A. s[t /y ]: C ] .
Moreover, the map MT : λTypes(Σ) → Obj CT is defined as MT A = A.
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28. Completeness
Proposition 12
The CT category is logically distributive.
Proposition 13
Given a λ-theory T on the Σ signature, the Σ-structure 〈CT , MT , MAx 〉 on
the corresponding syntactical category is defined by MAx which maps
f : A → B ∈ Ax to [x : A. f (x ): B ].
This Σ-structure is a model for T and, moreover, it satisfies exactly those
equalities-in-context which are provable in T .
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29. Completeness
Proposition 14
For every logically distributive category C, there is a biijection between
equivalence classes, modulo natural equivalences, of structure-preserving
functors CT → C and equivalence classes, modulo isomorphisms, of
models of T in C, induced by the map F → F (MT ).
Theorem 15 (Completeness)
If x . s =A t is an equality-in-context valid in every model for T in each
logically distributive category, then T x . s =A t.
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30. Soundness and Completeness in Logic
Definition 16 (Valid type)
A λ-type A is valid in the model N = 〈N, N , NAx 〉 when there exists an
arrow 1 → NA in N.
A λ-type A is a logical consequence in the model N of the λ-types
B1 , . . . , Bn when there exists N (B1 × · · · × Bn ) → NA in N.
A λ-type A is a logical consequence of B1 , . . . , Bn when it is a logical
consequence of B1 , . . . , Bn in every model in every logically distributive
category.
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31. Soundness and Completeness in Logic
Proposition 17
A λ-type A is a logical consequence of B1 , . . . , Bn if and only if there exists a
term-in-context x : B1 × · · · × Bn . t : A.
Corollary 18
A λ-type A is a logical consequence of B1 , . . . , Bn if and only if there is a
proof of A from the hypotheses B1 , . . . , Bn , when λ-types are interpreted as
logical formulae and λ-terms as logical proofs, according to the
Curry-Howard isomorphism.
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