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Introduction to Mathematical Logic
One of the pioneers of mathematical logic in the twentieth century was Alonzo Church. He introduced such concepts as the lambda calculus, now an essential tool of computer science, and was the founder of the Journal of Symbolic Logic. In Introduction to Mathematical Logic, Church presents a masterful overview of the subject--one which should be read by every researcher and student of logic. The previous edition of this book was in the Princeton Mathematical Series.
378 pages, Kindle Edition
First published January 1, 1956
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July 8, 2024
A “TEXTBOOK” FOR A BEGINNING COURSE IN MATHEMATICAL LOGIC
Alonzo Church (1903-1995) was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science.
He wrote in the Preface, “This is a revised and much enlarged edition … [of the Introduction] which was published in 1944 as one of the Annals of Mathematics Studies. In spite of extensive additions, it remains an introduction rather than a comprehensive treatise. It is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work. As a textbook it offers a beginning course in mathematical logic, but presupposes some substantial mathematical background.”
He states in the Introduction, “it is desirable or practically necessary for purposes of logic to employ a specially devised language, a FORMALIZED LANGUAGE as we shall call it, which shall reverse the tendency of the natural languages and shall follow or reproduce the logical form---at the expense, where necessary, of brevity and facility of communication. To adopt a particular formalized language thus involves adopting a particular theory or system of logical analysis.” (Pg. 2-3)
But he acknowledges, “There is not yet a theory of the meaning of proper names upon which general agreement has been reached as the best. But it is necessary to outline briefly the theory which will be adopted here, due in its essentials to Gottlob Frege.” (Pg. 3-4) Later, he explains, “The notations which we use as sentence connectives---and those which we use as quantifiers---are adaptations of those in Whitehead and Russell’s Principia Mathematica (some of which were taken from Peano). Various other notations are in use, and the student who would compare the treatments of different authors must learn a certain facility in shifting from one system of notation to another.” (Pg. 38-39)
He explains, “In setting up a formalized language we first employ as meta-language a certain portion of English. We shall not attempt to delimit precisely this portion of the English language, but describe it approximately by saying that it is just sufficient to enable us to give general directions for the manipulation of concrete physical objects… It is thus a language which deals with matters of everyday human experience, going beyond such matters only in that no finite upper limit is imposed on the number of objects that may be involved in any particular case, or on the time that may be required for their manipulation according to instructions. Those additional portions of English are excluded which would be used in order to treat of infinite classes or of various like abstract objects which are an essential part of the subject matter of mathematics. Our procedure is not to define the new language merely by means of translations of its expressions… into corresponding English expressions, because in this way it would hardly be possible to avoid carrying over into the new language the logically unsatisfactory features of the English language. Rather, we begin by setting up, in abstraction from all considerations of meaning, the purely formal part of the language, so obtaining an uninterpreted calculus or LOGISTIC SYSTEM.” (Pg. 47-48)
He outlines, “In this book we will be concerned with the task of formalizing an object language, and theoretical syntax will be treated informally, presupposing in any connection such general knowledge of mathematics as is necessary for the work at hand. Thus we do not apply even the informal axiomatic method to our treatment of syntax. But the reader must always understand that syntactical discussions are carried out in a syntax language whose formalization is ultimately contemplated, and distinctions based upon such formalization may be relevant to the discussion.” (Pg. 59)
He observes, “The notion of CONSISTENCY of a logistic system is semantical in motivation, arising from the requirement that nothing which is logically absurd or self-contradictory in meaning shall be a theorem, or that there shall not be two theorems of which one is the negation of the other. But we seek to modify this originally semantical notion in such a way as to make it syntactical in character (and therefore applicable to a logistic system independent of the interpretation adopted for it). This may be done by defining ‘relative consistency with respect to’ any transformation by which each sentence or propositional form ‘A’ is transformed into a sentence or propositional form ‘A-prime’…. Or we may define ‘absolute consistency’ by the condition that not every sentence and propositional form whatever would be proved… Or, following Hilbert, we might in the case of a particular system select an appropriate particular sentence and define the system as being consistent if that particular sentence is not a theorem… Or if the system has propositional variables, we may define it as being ‘consistent in the sense of Post’ if a ‘wff’ consisting of a propositional variable alone is not a theorem.” (Pg. 108)
He points out historically, “The logistic method was first applied by Frege in , 1879]… However, Frege’s work received little recognition or understanding until long after its publication, and the propositional calculus continued development from the older point of view, as may be seen in the work of C.S. Peirce, Ernst Schröder, Guiseppe Peano, and others. The beginnings of a change (though not yet the logistic method) appear in the work of Peano and his school. And from this course A.N. Whitehead and Bertrand Russell derived much of their earlier inspiration; later they became acquainted with the more profound work of Frege and were perhaps the first to appreciate its significance. After Frege, the earliest treatments of propositional calculus by the linguistic method are by Russell. Some indications of such a treatment may be found in The Principles of Mathematics (1903). It is extended propositional calculus which is there contemplated rather than propositional calculus; but by making certain changes in the light of later developments, it is possible to read into Russell’s discussion the following axioms for a partial system … of propositional calculus with implication and conjunction as primitive connectives…” (Pg. 156)
He notes, “abstract algebra is thus formalized within one of the pure functional calculi, and in this sense we may say if we like that it has been reduced to a branch of pure logic. Many other branches of mathematics are customarily treated in a similar way, so that their formalization brings them entirely within one of the pure functional calculi. And though it is more natural or more usual in some cases than other, it seems clear that every branch of mathematics might be treated in this way if we chose… Thus it is possible to say that all of mathematics is reducible to pure logic, and to maintain that logic and mathematics should be characterized, not as different subjects, but as elementary and advanced parts of the same subject.” (Pg. 332)
As a “textbook” for beginning students, this book is obviously somewhat “behind the times”; but serious students of logic wanting to follow the history and development of their subject may find this important and influential work by a “giant” in the field well worth studying.
Alonzo Church (1903-1995) was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science.
He wrote in the Preface, “This is a revised and much enlarged edition … [of the Introduction] which was published in 1944 as one of the Annals of Mathematics Studies. In spite of extensive additions, it remains an introduction rather than a comprehensive treatise. It is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work. As a textbook it offers a beginning course in mathematical logic, but presupposes some substantial mathematical background.”
He states in the Introduction, “it is desirable or practically necessary for purposes of logic to employ a specially devised language, a FORMALIZED LANGUAGE as we shall call it, which shall reverse the tendency of the natural languages and shall follow or reproduce the logical form---at the expense, where necessary, of brevity and facility of communication. To adopt a particular formalized language thus involves adopting a particular theory or system of logical analysis.” (Pg. 2-3)
But he acknowledges, “There is not yet a theory of the meaning of proper names upon which general agreement has been reached as the best. But it is necessary to outline briefly the theory which will be adopted here, due in its essentials to Gottlob Frege.” (Pg. 3-4) Later, he explains, “The notations which we use as sentence connectives---and those which we use as quantifiers---are adaptations of those in Whitehead and Russell’s Principia Mathematica (some of which were taken from Peano). Various other notations are in use, and the student who would compare the treatments of different authors must learn a certain facility in shifting from one system of notation to another.” (Pg. 38-39)
He explains, “In setting up a formalized language we first employ as meta-language a certain portion of English. We shall not attempt to delimit precisely this portion of the English language, but describe it approximately by saying that it is just sufficient to enable us to give general directions for the manipulation of concrete physical objects… It is thus a language which deals with matters of everyday human experience, going beyond such matters only in that no finite upper limit is imposed on the number of objects that may be involved in any particular case, or on the time that may be required for their manipulation according to instructions. Those additional portions of English are excluded which would be used in order to treat of infinite classes or of various like abstract objects which are an essential part of the subject matter of mathematics. Our procedure is not to define the new language merely by means of translations of its expressions… into corresponding English expressions, because in this way it would hardly be possible to avoid carrying over into the new language the logically unsatisfactory features of the English language. Rather, we begin by setting up, in abstraction from all considerations of meaning, the purely formal part of the language, so obtaining an uninterpreted calculus or LOGISTIC SYSTEM.” (Pg. 47-48)
He outlines, “In this book we will be concerned with the task of formalizing an object language, and theoretical syntax will be treated informally, presupposing in any connection such general knowledge of mathematics as is necessary for the work at hand. Thus we do not apply even the informal axiomatic method to our treatment of syntax. But the reader must always understand that syntactical discussions are carried out in a syntax language whose formalization is ultimately contemplated, and distinctions based upon such formalization may be relevant to the discussion.” (Pg. 59)
He observes, “The notion of CONSISTENCY of a logistic system is semantical in motivation, arising from the requirement that nothing which is logically absurd or self-contradictory in meaning shall be a theorem, or that there shall not be two theorems of which one is the negation of the other. But we seek to modify this originally semantical notion in such a way as to make it syntactical in character (and therefore applicable to a logistic system independent of the interpretation adopted for it). This may be done by defining ‘relative consistency with respect to’ any transformation by which each sentence or propositional form ‘A’ is transformed into a sentence or propositional form ‘A-prime’…. Or we may define ‘absolute consistency’ by the condition that not every sentence and propositional form whatever would be proved… Or, following Hilbert, we might in the case of a particular system select an appropriate particular sentence and define the system as being consistent if that particular sentence is not a theorem… Or if the system has propositional variables, we may define it as being ‘consistent in the sense of Post’ if a ‘wff’ consisting of a propositional variable alone is not a theorem.” (Pg. 108)
He points out historically, “The logistic method was first applied by Frege in , 1879]… However, Frege’s work received little recognition or understanding until long after its publication, and the propositional calculus continued development from the older point of view, as may be seen in the work of C.S. Peirce, Ernst Schröder, Guiseppe Peano, and others. The beginnings of a change (though not yet the logistic method) appear in the work of Peano and his school. And from this course A.N. Whitehead and Bertrand Russell derived much of their earlier inspiration; later they became acquainted with the more profound work of Frege and were perhaps the first to appreciate its significance. After Frege, the earliest treatments of propositional calculus by the linguistic method are by Russell. Some indications of such a treatment may be found in The Principles of Mathematics (1903). It is extended propositional calculus which is there contemplated rather than propositional calculus; but by making certain changes in the light of later developments, it is possible to read into Russell’s discussion the following axioms for a partial system … of propositional calculus with implication and conjunction as primitive connectives…” (Pg. 156)
He notes, “abstract algebra is thus formalized within one of the pure functional calculi, and in this sense we may say if we like that it has been reduced to a branch of pure logic. Many other branches of mathematics are customarily treated in a similar way, so that their formalization brings them entirely within one of the pure functional calculi. And though it is more natural or more usual in some cases than other, it seems clear that every branch of mathematics might be treated in this way if we chose… Thus it is possible to say that all of mathematics is reducible to pure logic, and to maintain that logic and mathematics should be characterized, not as different subjects, but as elementary and advanced parts of the same subject.” (Pg. 332)
As a “textbook” for beginning students, this book is obviously somewhat “behind the times”; but serious students of logic wanting to follow the history and development of their subject may find this important and influential work by a “giant” in the field well worth studying.
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