Classical Propositional Operators: An Exercise in the Foundations of Logic

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Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. The Mathematics of GIS 1. Contents 0. The Structure of Mathematics Propositional Logic Predicate Logic Logical Inference Set Theory Relations and Functions Coordinate Systems and Transformations Algebraic Structures III 9. Ordered Sets Graph Theory Fuzzy Logic and GIS IV Spatial Modeling Solutions of Exercises References and Bibliography List of Figures Figure 1.

Sub-disciplines of mathematics and their relationships Raster Calculator with logical connectors VENN diagram Non-commutativity of the Cartesian product Sample relations Composition of relations Functions and relations Topological relations Spatial relations derived from topological invariants Map projections with singularities Cartesian coordinate system in the plane Cartesian coordinate system in 3-D space Polar coordinate system in the plane Spherical coordinate system Conversion between Cartesian and polar coordinates in the plane Conversion between Cartesian coordinates and spherical coordinates Geographic coordinate system Point vector Cross product of two vectors Scalar triple product Manual digitizing setup Raster calculator interface Neighborhood axioms Continuous function Example of a homeomorphic function Equivalent approaches to the definition of a topological space, open sets and neighborhoods and related theorems Interior upper left , boundary upper right , closure lower left and exterior lower right of an open set Separation axioms T0, T1, and T Separation axioms T3 and T Relationship between separation characteristics of topological spaces VI Figure Simplexes of dimension 0, 1, 2, and Valid simplicial complex left and invalid simplicial complex right Unit balls and cells Cell decomposition and skeletons Construction of a CW complex Two-dimensional spatial data set as cell complex Topological mapping Closed polygon boundary check Node consistency check Poset and corresponding diagram Lower bounds Normal completion lattice Normal completion Geometric interpretation of new lattice elements Complete graphs Isomorphic graphs Connected G and disconnected H graph Directed graphs Planar graph Dual graph Undirected and directed graph Linear membership function Sinusoidal membership function Gaussian membership function Set inclusion Fuzzy set union operators Fuzzy set intersection Fuzzy set and its complement Law of the excluded middle and law of contradiction for fuzzy set Average.

Membership function for Tall and Not Very Tall Membership function for Tall and Slightly Tall VII Figure Fuzzy sets of the rules Fuzzy inference step Fuzzy inference final result Simplified Method Membership functions for flat and steep slope Membership functions for favorable and unfavorable aspect. Analysis with a fuzzy logic approach left and a crisp approach right Platonic solids as building blocks of matter Spatial modeling is a structure preserving mapping from the real world to a spatial model.

Data modeling from the real world to a database digital landscape model , and from there to digital cartographic models and analogue products for visualization Two data layers in a field-based model Layers in an object based model List of Tables Table 1. Logical Connectors Truth tables for logical operators Logical identities Logical implications Logical relationships involving quantifiers Rules of inference Rules of inference involving predicates and quantifiers Rules for set operations ArcInfo overlay commands Properties of the Cartesian product Properties of interior, closure, and boundary of a set Arc table for the arc-node structure Special elements and the closure operator in the normal completion Characteristic function for height classes Membership values for the height classes Rules for set operations valid for crisp and fuzzy sets Rules valid only for crisp sets Operators for hedges Hedges and their models Mathematics is written in the language of logic, and set theory is the very fundament on which all mathematical theories are built.

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Logic is not only the language of mathematics. It appears also in programming languages as syntactic constructs to express propositions, predicates, and to infer conclusions from given or assumed facts. The purpose of this book is to provide the reader with the mathematical knowledge needed when they have to deal with spatial information systems. Readers are expected to have a general knowledge of high school mathematics. The use of computers and software for the handling and processing of spatial data requires new contents such as discrete mathematics and topology.

The book is structured into 13 chapters Chapter 1 gives a brief overview of the structure of mathematics and how the different mathematical disciplines are built on top of more fundamental ones. The next three chapters deal with mathematical logic, the language and foundation of mathematics. Propositional and predicate logic are presented as well as logical inference, the methods of drawing logical conclusions from given facts. Chapter 5 and 6 are an introduction into the basic notions of sets, set operations, relations, and mappings. These two chapters together with the three chapters on logic represent the foundation for the subsequent chapters dealing with mathematical structures.

The next chapter on coordinate systems and transformations builds the bridge between the foundation and the more advanced chapters on mathematical structures. Much of chapter 7 would normally be considered to belong either to analytical geometry or to linear algebra. Chapters 8 to 11 present the highly relevant subjects of algebra, topology, ordered sets, and graph theory. These chapters address the mathematical core of many GIS functions from data storage, consistency to spatial analysis.

Uncertainty plays an increasingly important role in GIS. Chapter 12 addresses fuzzy logic and its applications in GIS. It shows how vague concepts can be formalized in mathematical language and how they are applied to spatial decision making. XI It shows that spatial modeling is built on solid mathematics as well as that there are challenging and interesting philosophical questions as to how to represent models of spatial features.

The book can be read in several ways as illustrated by the horizontal blocks in the following diagram. For someone with a particular interest in more advanced structures chapters 8 to 11 will be of interest. Chapters 7, 12, and 13 can be read individually without losing too much of the context. The best way, of course, is to read all the text from chapter 1 to This book is work in progress and not every chapter or section is complete. The author appreciates any comments and hints that might help to improve the text or its appearance.

Wolfgang Kainz Vienna, August The understanding of what mathematics is has changed over the centuries. In the beginning, mathematics was mainly devoted to practical calculations related to trade and land surveying. Over the centuries, mathematics has become a scientific discipline with many applications in all domains of life. This chapter gives a brief history of mathematics and explains how the different theories and branches of mathematics are rooted in logic and set theory.

In the beginning, mathematics was always related to practical problems of commerce, trading and surveying.

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This is the reason why the ancient cultures mainly developed practical solutions for arithmetic and geometric problems. In the fifth century before Christ, the ancient Greeks started to do mathematics for its own sake, and to focus the scientific attention to mathematics as a science. The concept of axioms and logical deduction was developed then. The Indians and Arabs further developed the number concept and trigonometry.

In the 17th and 18th century, the concepts of calculus and analytical geometry were developed as a consequence of the intensive studies in physics and natural sciences. In the 19th century, mathematicians began to establish an axiomatic foundation of mathematical theories.

Lecture 1 - Propositional Logic

Starting from a minimal set of axioms statements theorems can be derived whose validity can be formally established proof. This axiomatic approach has been applied since then to formalize mathematics. Logic and set theory play an important role as the language and foundation principle, respectively. It defines rules how to derive new statements from existing ones, and provides methods to prove their validity. Set theory deals with sets, the fundamental building block of mathematical structures, and the operations defined on them.

The notation of set theory is the basic tool to describe structures and operations in mathematical disciplines. Relations define relationships among elements of a set or several sets. These relationships allow for instance the classification of elements into equivalence classes or the comparison of elements with regard to certain attributes.

Functions or mappings are a special kind of relations. Sets whose elements are in certain relationships to each other or follow certain operations are mathematical structures. We distinguish between three major structures in mathematics, algebraic, order, and topologic structures. In sets with an algebraic structure we can do arithmetic, sets with an order structure allow the comparison of elements, and sets with a topologic structure allow to introduce concepts of convergence and continuity. Calculus is based on topology. Often, sets carry more than one structure.

The real numbers, for instance, carry an algebraic, an order, and a topologic structure. Results from algebraic topology are used in the theory of geographic information systems GIS. Figure 1 shows the sub-disciplines and their position in a general concept of mathematics and the fundamental building blocks. Sub-disciplines of mathematics and their relationships On top of the different structures and mixed structures, we find the many mathematical disciplines such as calculus, algebra, and analytical geometry.

The classical theories of great importance in spatial data handling are analytical geometry, linear algebra, and calculus. With the introduction of digital technologies of GIS other branches of mathematics became equally important, such as topology, graph theory, and the investigation of non-continuous discrete sets and their operations. The latter two fall under the domain that is usually called finite or discrete mathematics that plays an important role in computer science and its applications.

Such statements are called propositions. Any other statements for which we cannot establish whether they are true or false are not the subject of logic. This chapter explains the principles of propositional logic by introducing the concepts of proposition, propositional variable, propositional form, and logical operators.

The translation of natural language into propositions and the establishment of their truth-values with the help of truth tables are shown as well. Here, we will only deal with a two-valued logic. This is the logic on which most of the mathematical disciplines are built, and which is used in computing a bit can only assume two states, on or off, one or zero.

Definition 1 Assertion and Proposition. An assertion is a statement. If an assertion is either true or false, but not both1, we call it a proposition.

If a proposition is true, it has a truth-value of true; if it is false, it has a truth-value of false. Truth-values are usually written as true, false, or T, F, or 1, 0. In the following sections, we will use the notation for truth-values. Example 1. The following statements illustrate the concept of assertion, proposition and truthvalues. Proposition 3 is false, and 4 is true. Their truth-value depends on the value of the variables x and y.

Only when we replace the variables with some values, the assertion becomes a proposition. Often we have to be more general in writing down assertions. For this, we use Definition 2 Propositional Variable. A propositional variable is an arbitrary proposition whose truth-value is unspecified. We use upper case letters P, Q, R,… for propositional variables.

We can combine propositions and propositional variables to form new assertions. Example 2. The law of the excluded middle characterized a two-valued logic. Definition 3 Propositional Form. A propositional form is an assertion that contains at least one propositional variable. When we substitute propositions for the propositional variables of a propositional form, we get a proposition. When we use logical connectives to derive new propositions from old ones, the truth-value of the new proposition depends on the logical connective and the truth-values of the old propositions.

Example 3. Logical operators are used to combine propositions or propositional variables. Table 1 shows the most common operators. Table 1. This is done using truth tables that are defined for every operand. Table 2 shows the truth tables for the most common logical operators. Negation is a unary operator, i. The other operators apply to two operands.

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The conjunction or logical and is only true if both operands are true. The disjunction or inclusive or is true whenever at least one of the operands is true.

Modal Logic

The exclusive or is only true if either one or the other operand is true, but never both. It usually follows from the context what we mean. In mathematics, we cannot operate in this way. Therefore, we must make a distinction between inclusive and exclusive or. I can go to work and I can be tired at the same time.

A Gentle Introduction to Abstract Algebraic Logic

A person cannot be alive and dead at the same time. We exclude here the possibility of being a zombie, a state of existence the living dead that appears frequently in horror movies. Example 4. In natural language, the implication expresses a causal or inherent relationship between a premise and a conclusion.

In propositional logic, there need not be any relationship between the premise and the conclusion of an implication. We have to keep this in mind in order not to get confused by some propositions. Example 5. The implication is true because P is false and Q is true. According to the truth table for implications, anything either a true or a false statement can follow from a false proposition. Two propositions that have the same truth-values are said to be logically equivalent.

Whenever there are n propositional variables involved in a propositional form, we have 2n possible combinations of true and false to investigate. Example 6. Definition 4 Tautology, Contradiction, Contingency. A propositional form whose truth-value is true for all possible truth-values of its propositional variables is called a tautology. A contradiction or absurdity is a propositional form that is always false. A contingency is a propositional form that is neither a tautology nor a contradiction. The following examples illustrate the concepts of tautology, contradiction, and contingency.

Example 7. Such equivalence is also called a logical identity. We can replace one propositional form with its equivalent form. This helps often to simplify logical expressions. Table 3 lists the most important logical identities. Table 3. Logical identities 1. Since it is complete on its own, all other connectives can be expressed using only the stroke. In the following the IF Because all four rows under "taut" are 1's, the equivalence indeed represents a tautology. An arbitrary propositional formula may have a very complicated structure. It is often convenient to work with formulas that have simpler forms, known as normal forms.

Some common normal forms include conjunctive normal form and disjunctive normal form. Any propositional formula can be reduced to its conjunctive or disjunctive normal form. Reduction to normal form is relatively simple once a truth table for the formula is prepared. But further attempts to minimize the number of literals see below requires some tools: reduction by De Morgan's laws and truth tables can be unwieldy, but Karnaugh maps are very suitable a small number of variables 5 or less. Some sophisticated tabular methods exist for more complex circuits with multiple outputs but these are beyond the scope of this article; for more see Quine—McCluskey algorithm.

A string of literals connected by ANDs is called a term. A string of literals connected by OR is called an alterm. In the same way that a 2 n -row truth table displays the evaluation of a propositional formula for all 2 n possible values of its variables, n variables produces a 2 n -square Karnaugh map even though we cannot draw it in its full-dimensional realization. Each Karnaugh-map square and its corresponding truth-table evaluation represents one minterm. Any propositional formula can be reduced to the "logical sum" OR of the active i. When in this form the formula is said to be in disjunctive normal form.

But even though it is in this form, it is not necessarily minimized with respect to either the number of terms or the number of literals. In the following table, observe the peculiar numbering of the rows: 0, 1, 3, 2, 6, 7, 5, 4, 0. The first column is the decimal equivalent of the binary equivalent of the digits "cba", in other words:.

This numbering comes about because as one moves down the table from row to row only one variable at a time changes its value. Gray code is derived from this notion. This notion can be extended to three and four-dimensional hypercubes called Hasse diagrams where each corner's variables change only one at a time as one moves around the edges of the cube. Hasse diagrams hypercubes flattened into two dimensions are either Veitch diagrams or Karnaugh maps these are virtually the same thing. When working with Karnaugh maps one must always keep in mind that the top edge "wrap arounds" to the bottom edge, and the left edge wraps around to the right edge—the Karnaugh diagram is really a three- or four- or n-dimensional flattened object.

Veitch improved the notion of Venn diagrams by converting the circles to abutting squares, and Karnaugh simplified the Veitch diagram by converting the minterms, written in their literal-form e. Produce the formula's truth table. Number its rows using the binary-equivalents of the variables usually just sequentially 0 through n-1 for n variables.

However, this formula be reduced both in the number of terms from 4 to 3 and in the total count of its literals 12 to 6. Use the values of the formula e. If values of "d" for "don't care" appear in the table, this adds flexibility during the reduction phase. Minterms of adjacent abutting 1-squares T-squares can be reduced with respect to the number of their literals , and the number terms also will be reduced in the process. Two abutting squares 2 x 1 horizontal or 1 x 2 vertical, even the edges represent abutting squares lose one literal, four squares in a 4 x 1 rectangle horizontal or vertical or 2 x 2 square even the four corners represent abutting squares lose two literals, eight squares in a rectangle lose 3 literals, etc.

One seeks out the largest square or rectangles and ignores the smaller squares or rectangles contained totally within it. This process continues until all abutting squares are accounted for, at which point the propositional formula is minimized. For example, squares 3 and 7 abut. These two abutting squares can lose one literal e. One seeks out the largest square or rectangles. This process continues until all abutting squares are accounted for, at which point the propositional formula is said to be minimized.

Example: The map method usually is done by inspection. The following example expands the algebraic method to show the "trick" behind the combining of terms on a Karnaugh map:. The following uses brackets [ and ] only to keep track of the terms; they have no special significance:. Then assign the variable "s" to the left-most sentence "This sentence is simple". The second sentence can be expressed as:. So their conjunction AND is a falsehood.

This is an example of the paradoxes that result from an impredicative definition —that is, when an object m has a property P, but the object m is defined in terms of property P. Engineers, on the other hand, put them to work in the form of propositional formulas with feedback.

The notion of a propositional formula appearing as one of its own variables requires a formation rule that allows the assignment of the formula to a variable. In general there is no stipulation either axiomatic or truth-table systems of objects and relations that forbids this from happening. The simplest case occurs when an OR formula becomes one its own inputs e.

Observe that q's "definition" depends on itself "q" as well as on "s" and the OR connective; this definition of q is thus impredicative. Either of two conditions can result: [24] oscillation or memory.

Classical Propositional Operators: An Exercise In The Foundations Of Logic

It helps to think of the formula as a black box. Without knowledge of what is going on "inside" the formula-"box" from the outside it would appear that the output is no longer a function of the inputs alone. That is, sometimes one looks at q and sees 0 and other times 1. To avoid this problem one has to know the state condition of the "hidden" variable p inside the box i. When this is known the apparent inconsistency goes away. To understand [predict] the behavior of formulas with feedback requires the more sophisticated analysis of sequential circuits.

Propositional formulas with feedback lead, in their simplest form, to state machines; they also lead to memories in the form of Turing tapes and counter-machine counters. From combinations of these elements one can build any sort of bounded computational model e. Turing machines , counter machines , register machines , Macintosh computers , etc. Oscillation with delay : If an delay [25] ideal or non-ideal is inserted in the abstract formula between p and q then p will oscillate between 1 and 0: If either of the delay and NOT are not abstract i.

Analysis requires a delay to be inserted and then the loop cut between the delay and the input "p". The delay must be viewed as a kind of proposition that has "qd" q-delayed as output for "q" as input. This new proposition adds another column to the truth table. The inconsistency is now between "qd" and "p" as shown in red; two stable states resulting:.

Without delay, inconsistencies must be eliminated from a truth table analysis. After "breaking" the feed-back, [26] the truth table construction proceeds in the conventional manner. But afterwards, in every row the output q is compared to the now-independent input p and any inconsistencies between p and q are noted i. Rows revealing inconsistencies are either considered transient states or just eliminated as inconsistent and hence "impossible".

About the simplest memory results when the output of an OR feeds back to one of its inputs, in this case output "q" feeds back into "p". This behavior, now time-dependent, is shown by the state diagram to the right of the once-flip. The next simplest case is the "set-reset" flip-flop shown below the once-flip. The formula known as "clocked flip-flop" memory "c" is the "clock" and "d" is the "data" is given below.

When c goes from 1 to 0 the last value of the data remains "trapped" at output "q". The state diagram is similar in shape to the flip-flop's state diagram, but with different labelling on the transitions. Bertrand Russell lists three laws of thought that derive from Aristotle : 1 The law of identity : "Whatever is, is. The use of the word "everything" in the law of excluded middle renders Russell's expression of this law open to debate. If restricted to an expression about BEING or QUALITY with reference to a finite collection of objects a finite "universe of discourse" -- the members of which can be investigated one after another for the presence or absence of the assertion—then the law is considered intuitionistically appropriate.

See more at Venn diagram. Although a propositional calculus originated with Aristotle, the notion of an algebra applied to propositions had to wait until the early 19th century. In an adverse reaction to the year tradition of Aristotle's syllogisms , John Locke 's Essay concerning human understanding used the word semiotics theory of the use of symbols.

By Richard Whately had critically analyzed the syllogistic logic with a sympathy toward Locke's semiotics. Gottlob Frege 's massive undertaking resulted in a formal calculus of propositions, but his symbolism is so daunting that it had little influence excepting on one person: Bertrand Russell. First as the student of Alfred North Whitehead he studied Frege's work and suggested a famous and notorious emendation with respect to it around the problem of an antinomy that he discovered in Frege's treatment cf Russell's paradox.

Russell's work led to a collatoration with Whitehead that, in the year , produced the first volume of Principia Mathematica PM. It is here that what we consider "modern" propositional logic first appeared. From Wikipedia, the free encyclopedia. Hamilton p. Kleene p. When a value falls outside the defined range s the value becomes "u" -- unknown; e. Most philosophers and mathematicians just accept the material definition as given above. But some do not, including the intuitionists ; they consider it a form of the law of excluded middle misapplied.

Tarski comments on the use of quotes in his " Identity of things and identity of their designations; use of quotation marks" p. Bender and Williamson p. Kleene ranks all 11 symbols. Minsky presents a state machine that will do the job, and by use of induction recursive definition Minsky proves the "method" and presents a theorem as the result. A fully generalized "parenthesis grammar" requires an infinite state machine e.

Wickes offers a good example of 8 of the 2 x 4 3-variable maps and 16 of the 4 x 4 4-variable maps. But Kleene went on to assert that the problem has not been solved satisfactorily and impredicative definitions can be found in analysis. He gives as example the definition of the least upper bound l. Given a Dedekind cut of the number line C and the two parts into which the number line is cut, i. Thus the definition of u , an element of C , is defined in terms of the totality C and this makes its definition impredicative. Kleene asserts that attempts to argue this away can be used to uphold the impredicative definitions in the paradoxes.

Kleene In abstract idealized mathematical systems adequate loop gain is not a problem. Barwise, H. Keisler and K. Kunen, eds. Gandy considered this to be the most important of his principles: "Contemporary physics rejects the possibility of instantaneous action at a distance" p. Gandy was Alan Turing 's student and close friend.

McCluskey p. Mathematical logic. Formal system Deductive system Axiomatic system Hilbert style systems Natural deduction Sequent calculus. Propositional calculus and Boolean logic. Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Many-valued logic.