Manual Mathematical Logic

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To evaluate a sentence to determine whether or not it is biconditional.

  • Mathematical Logic.
  • Welcome to the Department for Mathematical Logic and Foundational Research.
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Tautologies To identify the individual parts of a compound statement. To construct a truth table for a compound statement to determine whether or not it is a tautology. Equivalence To define logical equivalence. To construct a truth table for several compound statements to determine which two are logically equivalent. To recognize that the biconditional of two equivalent statements is a tautology. Practice Exercises To complete 10 additional exercises as practice with mathematical logic.

Includes interactive truth tables.

To assess students' understanding of all concepts in this unit. Challenge Exercises To solve 10 additional problems that challenge students' understanding of mathematical logic. To hone students' problem-solving skills. Solutions To review complete solutions to all exercises presented in this unit. Includes the problem, step-by-step solutions, and final answer for each exercise.

Mathematical Logic. To identify a statement as true, false or open. To define logical connector, compound statement, and conjunction. To define disjunction. Conditional Statements. To identify the hypothesis and conclusion of a conditional. Compound Statements. To evaluate sentences represented by compound statements with logical connectors.

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Biconditional Statements. To list a biconditional in symbolic and in sentence form.

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To identify the individual parts of a compound statement. To define logical equivalence. Practice Exercises. To complete 10 additional exercises as practice with mathematical logic. Afterwards, this language was perfected in the joint work of B. Russell and A. Whitehead, Principia Mathematica , in which they attempted to reduce the whole of mathematics to logic. But this attempt was not crowned with success, since it turned out to be impossible to deduce the existence of infinite sets from purely logical axioms.

Although the logistic program of Frege—Russell on the foundations of mathematics never achieved its major aim, the reduction of mathematics to logic, in their papers they created a rich logical apparatus without which the appearance of mathematical logic as a valuable mathematical discipline would have been impossible.

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At the turn of the 19th century into the 20th century, antinomies cf. Antinomy related to the fundamental ideas of set theory were found. The strongest impression at that time was made by the Russell paradox. Let be the set containing exactly those sets which are not an element of itself. It is easy to convince oneself that is an element of itself if and only if is not an element of itself.

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Certainly one may circumvent this contradiction by stating that such a set cannot occur. However, if a set consisting precisely of all the elements satisfying some clearly defined condition, such as that given above in the definition of , need not exist, then where is the guarantee that in everyday work one will not meet with sets which also need not exist? And, in general, what conditions must the definition of a set satisfy so that the set does exist? One thing was clear: Cantor's theory of sets must somehow be restricted. Brouwer opposed the application of rules of classical logic to infinite sets.

Mathematics | Mathematical Logic | Amherst College

In his intuitionistic program it was suggested that the abstraction of actual infinity be removed from the discussion, that is, remove infinite sets as complete collections. While admitting the existence of arbitrarily large natural numbers, the intuitionist comes out against the consideration of the natural numbers as a complete set. They believe that in mathematics every proof concerning the existence of an object must be constructive, that is, must be accompanied by a construction of that object.

If the premise that the object sought for does not exist results in a contradiction, then this, in the opinion of intuitionists, cannot be considered as a proof of its existence. Particular criticism on the part of intuitionists is aimed at the law of the excluded middle. Since the law was initially considered in association with finite sets, and taking into account the fact that many properties of finite sets are not satisfied by infinite sets for example, that each proper part is less than the whole , intuitionists regard the application of this law to infinite sets as inadmissible.

A brief history of logic: from Leibniz to Boole - Math Foundations 254 - N J Wildberger

For example, in order to assert that Fermat's problem has a positive or a negative solution, the intuitionist must give the corresponding solution. So long as Fermat's problem is unsolved, this disjunction is regarded as illegitimate. The same requirement is imposed on the meaning of each disjunction.

This requirement of intuitionists may create difficulties even in the consideration of problems connected with finite sets. Imagine that, with eyes closed, a ball is taken from an urn in which there are three black and three white balls, and that the ball is then put back. If no one sees the ball, then one cannot possibly know what colour it was.

However, it is doubtful whether one can seriously dispute the certainty of the assertion that the ball was either black or white in colour. Intuitionists have constructed their own mathematics, with interesting distinctive peculiarities, but it has turned out to be more complicated and cumbersome than classical mathematics. The positive contribution of intuitionists to the investigation of questions in the foundations of mathematics is that they have once more decisively stressed the distinction between the constructive and the non-constructive in mathematics; they have made a careful analysis of the many difficulties which have been encountered in the development of mathematics, and, by the same token, they have contributed to overcoming them.

Hilbert see the Appendices VII—X in [9] planned another way to overcome the difficulties arising in the foundations of mathematics at the turn of the 19th century into the 20th century. This route, based on the application of the axiomatic method in the discussion of formal models of interesting mathematics, and in the investigation of questions of consistency of such models by reliable finitary means, was given the name Hilbert's finitism in mathematics. Recognizing the unreliability of geometrical intuition, Hilbert first of all undertook a careful review of Euclidean geometry, liberating it from the appeal to intuition.

As a result of this revision his Grundlagen der Geometrie appeared, [9]. Questions of consistency of various theories were essentially considered even before Hilbert. Thus the projective model of the non-Euclidean Lobachevskii geometry constructed by F. Klein reduced the question of consistency of Lobachevskii's geometry to the consistency of Euclidean geometry.

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The consistency of Euclidean geometry can similarly be reduced to the consistency of analysis, that is, the theory of real numbers. However, it was not clear how it would be possible to construct models of analysis and arithmetic for these consistency proofs. The merit of Hilbert is that he gave a direct way for the investigation of this question.

Consistency of a given theory means that one cannot obtain a contradiction in it, that is, it is not possible to prove both an assertion and its negation.

Mathematical Logic

Hilbert suggested representing the theory under discussion as a formal axiomatic system, in which those and only those assertions are derivable which are theorems of that theory. Then for the proof of consistency it suffices to establish the non-derivability in the theory of certain assertions. Thus a mathematical theory whose consistency one wishes to prove becomes an object of study in a mathematical science which Hilbert called metamathematics, or proof theory.

Hilbert wrote that the paradoxes of set theory have arisen not from the law of the excluded middle but "rather that mathematicians have used inadmissible and meaningless formations of ideas which in my proof theory are excluded …. To remove from mathematicians the law of the excluded middle is the same as taking the telescope away from astronomers or forbidding a boxer to use his fists" see [9]. Hilbert suggested distinguishing between "real" and "ideal" assumptions of classical mathematics.

The first have a genuine meaning but the second need not. Assumptions corresponding to the use of the actual infinite are ideal. Ideal assumptions can be added to the real ones in order that the simple results of logic be applicable even to arguments about infinite sets. This essentially simplifies the structure of the whole theory in much the same way as the addition of the line at infinity intersecting any two parallel lines in the projective geometry of the plane. The program suggested by Hilbert for the foundation of mathematics, and his enthusiasm for it, inspired his contemporaries into an intensive development of the axiomatic method.

Thus the formation of mathematical logic as an independent mathematical discipline is linked with Hilbert's initiative, at the start of 20th century, and the subsequent development of proof theory based on the logical language developed by Frege, Peano and Russell.

The objective of modern mathematical logic is diverse. First of all one must mention the investigation of logical and logico-mathematical calculi founded on classical predicate calculus. In K. This theorem showed that predicate calculus is a logical system on the basis of which mathematics can be formulated. Based on predicate calculus various logico-mathematical theories have been constructed see Logico-mathematical calculus , representing the formalization of interesting mathematical theories: arithmetic, analysis, set theory, group theory, etc.

Side-by-side with elementary theories cf. Elementary theory , higher-order theories were also considered. In these one also admits quantifiers over predicates, predicates over predicates, etc. The traditional questions studied in these formal logical systems were the investigation of the structure of the deductions in the system, derivability of various formulas, and questions of consistency and completeness. According to this theorem, if a formal system containing arithmetic is consistent, then the assertion of its consistency expressed in the system cannot be proved by formalization within it.

This means that with questions on foundations of mathematics the matter is not as simple as first desired or believed by Hilbert. Similar proofs of the consistency of arithmetic were obtained by G.