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Halmos P.R., Givant S. — Logic as Algebra
Halmos P.R., Givant S. — Logic as Algebra

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Название: Logic as Algebra

Авторы: Halmos P.R., Givant S.

Аннотация:

This book is based on the notes for a course in logic given by Paul Halmos. This book retains the spirit and purpose of those notes, which was to show that logic can (and perhaps should) be viewed from an algebraic perspective.

Propositional logic and monadic predicate calculus—predicate logic with a single quantifier—are the principal topics treated. The connections between logic and algebra are carefully explained. The key notions and the fundamental theorems are elucidated from both a logical and algebraic perspective.

The final section gives a unique and illuminating algebraic treatment of the theory of syllogisms—perhaps the oldest branch of logic, and a subject that is neglected in most modern logic texts.

The presentation is aimed at a broad audience—mathematics amateurs, students, teachers, philosophers, linguists, computer scientists, engineers, and professional mathematicians. All that is required of the reader is an acquaintance with some of the basic notions encountered in a first course in modern algebra. In particular, no prior knowledge of logic is assumed. The book could serve equally well as a fireside companion and as a course text.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 152

Добавлена в каталог: 30.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
2-valued functional monadic algebra      112 119 122 126
2-valued homomorphism      61 75
A-valued functional monadic algebra      112
A-valued homomorphism      61
Absorption laws for conjunction      34 39 42
Absorption laws for disjunction      27 39 42
Algebra      18
Algebra of sentences      30
Algebraic identities      35
Algebraic logic      134
Algebraic structure      18 28 41
ALL      107
Alphabet      4 9 10 13 20 79
Alphabetic conventions      42
AND      9 31 32
Antisymmetry      28 49
Aristotelean logic      127
Aristotle      129 131
Associative law      35 95 101 103 104
Associative law for conjunction      37 42
Associative law for disjunction      35 36 42
Associative operation      20
Associativity of composition      63
Automorphism      61
Axiom      6 10 20 22 23 66 89
Axiom of Choice      73
Axiom Schema      21
Axiom schemata      20
Axioms of the propositional calculus      22
B-algebra      80 91
Barbara      131 132
Baroco      131
Basic data      6 10 11
Basic word      5
Bass      124
Belongs to      9
Bemalip      131
Bi-implication      32 54
Biconditional      32
Binary operation      18
Binary rejection      54
Bocardo      131
Boolean addition      117
Boolean algebra      42 46 91 96
Boolean congruence      80 81
Boolean endomorphism      133
Boolean filter      69 70
Boolean homomorphism      60
Boolean ideal      67 120 121
Boolean logic      83 91
Boolean polynomial      47 61
Boolean subalgebra      59
Boolean sum      53 54
Boolean unit      49
Boolean zero      49
Calemes      131
CAM      131
Camestres      131
Canonical homomorphism      74
Cardinality      56
Cartesian Plane      114
Celarent      131
Cesare      131
Characteristic      54
Characteristic function      62 96 97 108
Classification problem      130 131
Co-countable set      56
Co-finite set      56
Collision of variables      22
Combinatorial method      3
Combinatorial proof      1
Commutative diagram      64
Commutative law      35 95 102 104 111
Commutative law for conjunction      34 35 42
Commutative law for disjunction      27 35 42
Commutativity      46
Compactness Theorem      93
Complement      47
Complementation      43 47 51 80 96
Complemented distributive lattice      96
Complemented lattice      95
complete      7 8 11
Complete logic      84
Completeness      126
Composite      63
Composition of homomorphisms      63
Concatenation      18 20
Conclusion      130
Concrete model      126
Congruence relation      30—32 70 79 118
Congruence relation induced by an ideal      71
Conjunction      31 32 41 80
Conjunctive normal form      89
Connective      15
Consequence      90
Consistency      45 84 126
Consistent      7 8
Consistent logic      46 84
Constant      133 134
Constant function      108 133
Constant propositional function      109 133
Contradiction      46 85
Contradictory      129
Contradual      47
Contraposition      129
Contrary      129
Contravalid      85 127
Contravalid element      85 127
CONVERSE      129
Coset      71
Countable set      56
Darapti      131 132
Datisi      131
de Morgan’s laws      32 52 102
Deduction Theorem      91—93
Derived axiom schema      24
Derived rule of inference      25
Derived theorem schema      26
Dimatis      131
Direct power of an algebra      108
Direct product of algebras      122 123
Disamis      131
Discrete quantifier      117
Disjoint      55
Disjunction      15 18 32 41 80
Disjunctive normal form      89
Distributive lattice      95 100
Distributive law      18 35 100 101 104 111 113 115 117
Distributive law for conjunction over disjunction      39 42
Distributive law for disjunction over conjunction      37 38 42
Distributive operation      110
Dual      33 47
Duality      46
Duality principle      33
Duality principle for entailment      33
Duality principle for implication      33
Duality principle for logical equivalence      33
Dyadic algebra      123
Electronic typewriter      4
Elementary number theory      11
Elementary real number theory      11
Elementary theory of addition      11
empty function      19
Empty sequence      20
Empty set      56
Endomorphism      61 82 83
Entailment      28
Epimorphism      61 74
EQUIVALENCE      29
Equivalence class      41
Equivalence relation      29 41 70
Examples of Boolean algebras      55
Examples of homomorphisms      62
Examples of ideals      67 68
Examples of monadic algebras      117
Existence of free algebras      65
Existence theorem      75—77 121
Existential quantifier      114 115 117
FALSE      85 127
Falsity      43
Felapton      131
Ferio      131
Feriso      131
Fesapo      131
Festino      131
Field of sets      56 72
figure      130
Filter      69 70 79 81 118
Filter generated by a set      70 90
Finite function      108
Finite sequence      13 19
Finite set      56
Finite-valued function      109
Finite-valued propositional function      109
For all      9
Formal language      3
Formal proof      23 24 26
Formula      13
free      20
Free algebra      63
Free B-algebra      83
Free Boolean algebra      64 66 77 107 123
Free generators      64
Free monadic algebra      123 128
Free semigroup      20
Freely generated algebra      65
Fresison      131
Fully invariant congruence      83
Function algebra      109
Functional algebra      113
Functional monadic algebra      111 112 117 122
Functional quantifier      113
Generators      60
Genetic description      7 10
Godel incompleteness theorem      11
Gottschalk      47
Grammar      5
Greatest lower bound      51
Group of integers      123
Half-closed interval      56
Hasse diagram      97
Hilbert — Ackermann      40 131
Homomorphic image      61
Homomorphism      60 82 118
Homomorphism theorem      74 118
Horizontal set      57
Ideal      67 74 118
Ideal generated by a set      68 69
Idempotence law      48 95 104 115
Idempotence laws for conjunction      34
Idempotence laws for disjunction      27
Idempotent operation      110
identities      35
Identity element for addition      54
Identity element for conjunction      50
Identity element for disjunction      50
Identity element for multiplication      54
Identity mapping      63 64 117
If and only if      32
Image      62
Immediate ancestor      44
Implication      16 28 53
Implies      16
Improper ideal      68
Improper subalgebra      59
Inconsistency      46
Inconsistent      11
Increasing operation      110 113 114 125
Induced homomorphism      63
Infimum      51 95 102
Infinite associative law      103 104
Infinite cardinal      56
Infinite commutative law      102 104
Infinite De Morgan laws      102
Infinite distributive law      104 113
Infinite idempotence law      104
Infinite operation      101
Instruction      6 10 11
Interpretation      43 44 66 84 126
Intersection      56
Intersection of algebras      60
Intersection of ideals      68
Intuitionistic logic      96
Invariant congruence      83
Inverse image      62
Isomorphic algebras      57 61 113
Isomorphic free algebras      65
Isomorphism      52 61 113
Join      43 47 80 95
Kernel      66 74 118
Kernel of a congruence relation      71
Kernel of a homomorphism      66
Klein four-group      47
Language      5 18
Lattice      95
Law of contraposition      27
Laws of double negation      27
Laws of the excluded middle      27
Least upper bound      51
Left half-closed interval      57
letter      13
Logic      1
Logical equivalence      29 35 41
Logical symbol      9
Logistic method      3
Lukasiewicz      17 18
Major premise      130
Major term      130
Many-valued logic      96
Maximal congruence      84
Maximal ideal      71—73 118 120 121
Maximal ideal theorem      72 76 77 121 126
Maybe      125
Meet      43 47 80 95
Metatheorem      25
Middle term      130
Minor premise      130
Minor term      130
Mnemonic word      129—131
Modal logic      96 124 125
Model      126
Modular lattice      100
Modular law      100 101 111
Modular operation      111 113 114 125
modus ponens      23 89 90
Monadic algebra      112 117
Monadic filter      118 125 130
Monadic homomorphism      118 126
Monadic ideal      118 120 121
Monadic logic      125—127
Monadic predicate calculus      107
Monadic quotient algebra      118
Monadic subalgebra      118
Monomial sentence      42
Monomorphism      61 67
Monotony law      115 117
Monotony law for conjunction      50
Monotony law for disjunction      27 50
Mood      130
Negation      15 18 32 41 80
1 2
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