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Introduction to First-Order Logic — Explore equations

Introduction to First-Order Logic

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Equation and object guide

All 115 stable reader expression records, two formal objects, and eight references are indexed here. Two forward references are retained as inert, explicitly unavailable targets.

115 expression records

Expression 4

s

Conventional reading: variable assignment s

Meaning here: The variable assignment s, a function assigning a domain element to every individual variable.

10 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/satisfaction.tex, line 61, column 48
  2. Occurrence 2: content/first-order-logic/introduction/satisfaction.tex, line 65, column 15
  3. Occurrence 3: content/first-order-logic/introduction/satisfaction.tex, line 66, column 14
  4. Occurrence 4: content/first-order-logic/introduction/satisfaction.tex, line 71, column 65
  5. Occurrence 5: content/first-order-logic/introduction/satisfaction.tex, line 93, column 6
  6. Occurrence 6: content/first-order-logic/introduction/satisfaction.tex, line 104, column 27
  7. Occurrence 7: content/first-order-logic/introduction/semantic-notions.tex, line 14, column 33
  8. Occurrence 8: content/first-order-logic/introduction/semantic-notions.tex, line 18, column 51
  9. Occurrence 9: content/first-order-logic/introduction/semantic-notions.tex, line 21, column 50
  10. Occurrence 10: content/first-order-logic/introduction/semantic-notions.tex, line 22, column 55

Expression 10

P(a)

Conventional reading: P of a

Meaning here: The atomic sentence asserting that the object denoted by constant a has property P.

8 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/syntax.tex, line 21, column 52
  2. Occurrence 2: content/first-order-logic/introduction/syntax.tex, line 23, column 14
  3. Occurrence 3: content/first-order-logic/introduction/syntax.tex, line 36, column 43
  4. Occurrence 4: content/first-order-logic/introduction/formulas.tex, line 38, column 26
  5. Occurrence 5: content/first-order-logic/introduction/formulas.tex, line 54, column 33
  6. Occurrence 6: content/first-order-logic/introduction/satisfaction.tex, line 35, column 9
  7. Occurrence 7: content/first-order-logic/introduction/satisfaction.tex, line 45, column 6
  8. Occurrence 8: content/first-order-logic/introduction/substitution.tex, line 16, column 15

Expression 14

a

Conventional reading: constant a

Meaning here: The sole individual constant a in the chapter's simplified first-order language.

8 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/syntax.tex, line 20, column 11
  2. Occurrence 2: content/first-order-logic/introduction/syntax.tex, line 22, column 34
  3. Occurrence 3: content/first-order-logic/introduction/formulas.tex, line 26, column 14
  4. Occurrence 4: content/first-order-logic/introduction/formulas.tex, line 30, column 61
  5. Occurrence 5: content/first-order-logic/introduction/satisfaction.tex, line 17, column 25
  6. Occurrence 6: content/first-order-logic/introduction/satisfaction.tex, line 18, column 63
  7. Occurrence 7: content/first-order-logic/introduction/satisfaction.tex, line 59, column 42
  8. Occurrence 8: content/first-order-logic/introduction/substitution.tex, line 19, column 48

Expression 22

M,s⊨A

Conventional reading: structure M under assignment s satisfies formula A

Meaning here: Formula A is true in structure M relative to variable assignment s.

6 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/satisfaction.tex, line 72, column 11
  2. Occurrence 2: content/first-order-logic/introduction/satisfaction.tex, line 80, column 40
  3. Occurrence 3: content/first-order-logic/introduction/satisfaction.tex, line 81, column 41
  4. Occurrence 4: content/first-order-logic/introduction/semantic-notions.tex, line 13, column 50
  5. Occurrence 5: content/first-order-logic/introduction/semantic-notions.tex, line 21, column 25
  6. Occurrence 6: content/first-order-logic/introduction/semantic-notions.tex, line 22, column 21

Expression 23

x

Conventional reading: variable x

Meaning here: The individual variable x, whose free and bound occurrences are tracked in the surrounding definitions.

18 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/formulas.tex, line 47, column 62
  2. Occurrence 2: content/first-order-logic/introduction/sentences.tex, line 35, column 36
  3. Occurrence 3: content/first-order-logic/introduction/sentences.tex, line 38, column 47
  4. Occurrence 4: content/first-order-logic/introduction/sentences.tex, line 39, column 25
  5. Occurrence 5: content/first-order-logic/introduction/sentences.tex, line 40, column 66
  6. Occurrence 6: content/first-order-logic/introduction/sentences.tex, line 41, column 61
  7. Occurrence 7: content/first-order-logic/introduction/sentences.tex, line 44, column 71
  8. Occurrence 8: content/first-order-logic/introduction/sentences.tex, line 45, column 66
  9. Occurrence 9: content/first-order-logic/introduction/sentences.tex, line 48, column 6
  10. Occurrence 10: content/first-order-logic/introduction/sentences.tex, line 49, column 40
  11. Occurrence 11: content/first-order-logic/introduction/sentences.tex, line 49, column 67
  12. Occurrence 12: content/first-order-logic/introduction/sentences.tex, line 50, column 68
  13. Occurrence 13: content/first-order-logic/introduction/substitution.tex, line 21, column 14
  14. Occurrence 14: content/first-order-logic/introduction/substitution.tex, line 26, column 11
  15. Occurrence 15: content/first-order-logic/introduction/substitution.tex, line 26, column 59
  16. Occurrence 16: content/first-order-logic/introduction/substitution.tex, line 32, column 5
  17. Occurrence 17: content/first-order-logic/introduction/substitution.tex, line 33, column 5
  18. Occurrence 18: content/first-order-logic/introduction/models-theories.tex, line 44, column 62

Expression 25

|M|

Conventional reading: the domain of structure M

Meaning here: The nonempty set of objects over which the first-order structure M interprets its language.

6 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/satisfaction.tex, line 16, column 40
  2. Occurrence 2: content/first-order-logic/introduction/satisfaction.tex, line 21, column 26
  3. Occurrence 3: content/first-order-logic/introduction/satisfaction.tex, line 59, column 17
  4. Occurrence 4: content/first-order-logic/introduction/satisfaction.tex, line 62, column 34
  5. Occurrence 5: content/first-order-logic/introduction/substitution.tex, line 42, column 16
  6. Occurrence 6: content/first-order-logic/introduction/models-theories.tex, line 48, column 52

Expression 26

∀v0P(v0,v0)∀v0∀v1∀v2((P(v0,v1)∧P(v1,v2))→P(v0,v2))

Conventional reading: First axiom: for every v sub zero, P relates v sub zero to itself. Second axiom: for every v sub zero, v sub one, and v sub two, if P relates v sub zero to v sub one and P relates v sub one to v sub two, then P relates v sub zero to v sub two

Meaning here: The two first-order sentences asserting that P is reflexive and transitive, so its interpretation is a preorder.

1 occurrence
  1. Occurrence 1: content/first-order-logic/introduction/models-theories.tex, line 40, column 1

Expression 30

M

Conventional reading: structure M

Meaning here: The first-order structure M used as a mathematically precise case for evaluating formulas and sentences.

20 occurrences
  1. Occurrence 1: content/first-order-logic/first-order-logic.tex, line 14, column 27
  2. Occurrence 2: content/first-order-logic/introduction/first-order-logic.tex, line 46, column 27
  3. Occurrence 3: content/first-order-logic/introduction/first-order-logic.tex, line 47, column 43
  4. Occurrence 4: content/first-order-logic/introduction/satisfaction.tex, line 20, column 45
  5. Occurrence 5: content/first-order-logic/introduction/satisfaction.tex, line 30, column 48
  6. Occurrence 6: content/first-order-logic/introduction/satisfaction.tex, line 31, column 40
  7. Occurrence 7: content/first-order-logic/introduction/satisfaction.tex, line 32, column 37
  8. Occurrence 8: content/first-order-logic/introduction/satisfaction.tex, line 35, column 49
  9. Occurrence 9: content/first-order-logic/introduction/satisfaction.tex, line 37, column 36
  10. Occurrence 10: content/first-order-logic/introduction/satisfaction.tex, line 38, column 16
  11. Occurrence 11: content/first-order-logic/introduction/satisfaction.tex, line 39, column 41
  12. Occurrence 12: content/first-order-logic/introduction/satisfaction.tex, line 40, column 16
  13. Occurrence 13: content/first-order-logic/introduction/satisfaction.tex, line 40, column 55
  14. Occurrence 14: content/first-order-logic/introduction/satisfaction.tex, line 45, column 46
  15. Occurrence 15: content/first-order-logic/introduction/satisfaction.tex, line 48, column 4
  16. Occurrence 16: content/first-order-logic/introduction/satisfaction.tex, line 55, column 45
  17. Occurrence 17: content/first-order-logic/introduction/satisfaction.tex, line 71, column 16
  18. Occurrence 18: content/first-order-logic/introduction/satisfaction.tex, line 83, column 50
  19. Occurrence 19: content/first-order-logic/introduction/models-theories.tex, line 18, column 47
  20. Occurrence 20: content/first-order-logic/introduction/models-theories.tex, line 46, column 57

Expression 35

P

Conventional reading: predicate symbol P

Meaning here: The sole predicate symbol P in the chapter's simplified first-order language.

7 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/syntax.tex, line 22, column 51
  2. Occurrence 2: content/first-order-logic/introduction/formulas.tex, line 25, column 40
  3. Occurrence 3: content/first-order-logic/introduction/satisfaction.tex, line 18, column 1
  4. Occurrence 4: content/first-order-logic/introduction/satisfaction.tex, line 19, column 55
  5. Occurrence 5: content/first-order-logic/introduction/substitution.tex, line 19, column 38
  6. Occurrence 6: content/first-order-logic/introduction/models-theories.tex, line 36, column 50
  7. Occurrence 7: content/first-order-logic/introduction/models-theories.tex, line 51, column 20

Expression 36

∀x(A(x)→B(x))

Conventional reading: for every x, if A of x then B of x

Meaning here: The universally quantified premise saying that every object satisfying A also satisfies B.

2 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/first-order-logic.tex, line 69, column 25
  2. Occurrence 2: content/first-order-logic/introduction/first-order-logic.tex, line 82, column 52

Expression 39

Γ

Conventional reading: Gamma

Meaning here: The set Gamma of first-order sentences used as premises, axioms, or a theory in the surrounding discussion.

17 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/semantic-notions.tex, line 34, column 41
  2. Occurrence 2: content/first-order-logic/introduction/models-theories.tex, line 18, column 22
  3. Occurrence 3: content/first-order-logic/introduction/models-theories.tex, line 19, column 43
  4. Occurrence 4: content/first-order-logic/introduction/models-theories.tex, line 20, column 6
  5. Occurrence 5: content/first-order-logic/introduction/models-theories.tex, line 39, column 5
  6. Occurrence 6: content/first-order-logic/introduction/models-theories.tex, line 47, column 39
  7. Occurrence 7: content/first-order-logic/introduction/models-theories.tex, line 49, column 28
  8. Occurrence 8: content/first-order-logic/introduction/models-theories.tex, line 52, column 62
  9. Occurrence 9: content/first-order-logic/introduction/models-theories.tex, line 53, column 59
  10. Occurrence 10: content/first-order-logic/introduction/models-theories.tex, line 56, column 56
  11. Occurrence 11: content/first-order-logic/introduction/soundness-completeness.tex, line 16, column 1
  12. Occurrence 12: content/first-order-logic/introduction/soundness-completeness.tex, line 21, column 22
  13. Occurrence 13: content/first-order-logic/introduction/soundness-completeness.tex, line 37, column 24
  14. Occurrence 14: content/first-order-logic/introduction/soundness-completeness.tex, line 39, column 1
  15. Occurrence 15: content/first-order-logic/introduction/soundness-completeness.tex, line 40, column 40
  16. Occurrence 16: content/first-order-logic/introduction/soundness-completeness.tex, line 45, column 6
  17. Occurrence 17: content/first-order-logic/introduction/soundness-completeness.tex, line 46, column 1

Expression 49

v0

Conventional reading: variable v sub zero

Meaning here: The first indexed individual variable in the chapter's official variable vocabulary.

10 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/formulas.tex, line 32, column 50
  2. Occurrence 2: content/first-order-logic/introduction/satisfaction.tex, line 58, column 4
  3. Occurrence 3: content/first-order-logic/introduction/satisfaction.tex, line 59, column 63
  4. Occurrence 4: content/first-order-logic/introduction/satisfaction.tex, line 66, column 61
  5. Occurrence 5: content/first-order-logic/introduction/satisfaction.tex, line 89, column 22
  6. Occurrence 6: content/first-order-logic/introduction/satisfaction.tex, line 89, column 60
  7. Occurrence 7: content/first-order-logic/introduction/satisfaction.tex, line 93, column 44
  8. Occurrence 8: content/first-order-logic/introduction/satisfaction.tex, line 94, column 1
  9. Occurrence 9: content/first-order-logic/introduction/satisfaction.tex, line 104, column 13
  10. Occurrence 10: content/first-order-logic/introduction/substitution.tex, line 19, column 62

Expression 50

∀xA(x)

Conventional reading: for every x, formula A of x

Meaning here: The universal quantification of A of x with respect to x, asserting A of every domain object without claiming that other free variables are closed.

4 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/substitution.tex, line 24, column 33
  2. Occurrence 2: content/first-order-logic/introduction/substitution.tex, line 35, column 6
  3. Occurrence 3: content/first-order-logic/introduction/substitution.tex, line 37, column 6
  4. Occurrence 4: content/first-order-logic/introduction/substitution.tex, line 39, column 33

Expression 54

A

Conventional reading: formula A

Meaning here: The metavariable A denotes an arbitrary formula.

33 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/first-order-logic.tex, line 27, column 33
  2. Occurrence 2: content/first-order-logic/introduction/first-order-logic.tex, line 28, column 14
  3. Occurrence 3: content/first-order-logic/introduction/first-order-logic.tex, line 30, column 15
  4. Occurrence 4: content/first-order-logic/introduction/first-order-logic.tex, line 35, column 61
  5. Occurrence 5: content/first-order-logic/introduction/first-order-logic.tex, line 36, column 31
  6. Occurrence 6: content/first-order-logic/introduction/first-order-logic.tex, line 37, column 21
  7. Occurrence 7: content/first-order-logic/introduction/first-order-logic.tex, line 37, column 58
  8. Occurrence 8: content/first-order-logic/introduction/first-order-logic.tex, line 38, column 44
  9. Occurrence 9: content/first-order-logic/introduction/first-order-logic.tex, line 42, column 48
  10. Occurrence 10: content/first-order-logic/introduction/first-order-logic.tex, line 46, column 6
  11. Occurrence 11: content/first-order-logic/introduction/first-order-logic.tex, line 47, column 19
  12. Occurrence 12: content/first-order-logic/introduction/syntax.tex, line 46, column 19
  13. Occurrence 13: content/first-order-logic/introduction/formulas.tex, line 41, column 39
  14. Occurrence 14: content/first-order-logic/introduction/formulas.tex, line 44, column 21
  15. Occurrence 15: content/first-order-logic/introduction/formulas.tex, line 47, column 37
  16. Occurrence 16: content/first-order-logic/introduction/formulas.tex, line 78, column 41
  17. Occurrence 17: content/first-order-logic/introduction/satisfaction.tex, line 32, column 16
  18. Occurrence 18: content/first-order-logic/introduction/satisfaction.tex, line 37, column 53
  19. Occurrence 19: content/first-order-logic/introduction/satisfaction.tex, line 39, column 58
  20. Occurrence 20: content/first-order-logic/introduction/sentences.tex, line 30, column 14
  21. Occurrence 21: content/first-order-logic/introduction/sentences.tex, line 35, column 56
  22. Occurrence 22: content/first-order-logic/introduction/sentences.tex, line 38, column 12
  23. Occurrence 23: content/first-order-logic/introduction/sentences.tex, line 40, column 12
  24. Occurrence 24: content/first-order-logic/introduction/sentences.tex, line 44, column 12
  25. Occurrence 25: content/first-order-logic/introduction/sentences.tex, line 47, column 12
  26. Occurrence 26: content/first-order-logic/introduction/sentences.tex, line 48, column 13
  27. Occurrence 27: content/first-order-logic/introduction/semantic-notions.tex, line 15, column 52
  28. Occurrence 28: content/first-order-logic/introduction/semantic-notions.tex, line 16, column 56
  29. Occurrence 29: content/first-order-logic/introduction/semantic-notions.tex, line 20, column 27
  30. Occurrence 30: content/first-order-logic/introduction/semantic-notions.tex, line 34, column 65
  31. Occurrence 31: content/first-order-logic/introduction/substitution.tex, line 20, column 31
  32. Occurrence 32: content/first-order-logic/introduction/substitution.tex, line 32, column 14
  33. Occurrence 33: content/first-order-logic/introduction/soundness-completeness.tex, line 16, column 29

Expression 58

R

Conventional reading: R

Meaning here: The relation denoted by R.

5 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/models-theories.tex, line 33, column 10
  2. Occurrence 2: content/first-order-logic/introduction/models-theories.tex, line 37, column 59
  3. Occurrence 3: content/first-order-logic/introduction/models-theories.tex, line 45, column 10
  4. Occurrence 4: content/first-order-logic/introduction/models-theories.tex, line 46, column 10
  5. Occurrence 5: content/first-order-logic/introduction/models-theories.tex, line 47, column 52

Expression 74

Γ⊨A

Conventional reading: Gamma semantically entails A

Meaning here: Every structure that satisfies all assumptions in Gamma also satisfies formula A.

6 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/semantic-notions.tex, line 33, column 25
  2. Occurrence 2: content/first-order-logic/introduction/soundness-completeness.tex, line 22, column 19
  3. Occurrence 3: content/first-order-logic/introduction/soundness-completeness.tex, line 25, column 32
  4. Occurrence 4: content/first-order-logic/introduction/soundness-completeness.tex, line 27, column 18
  5. Occurrence 5: content/first-order-logic/introduction/soundness-completeness.tex, line 29, column 10
  6. Occurrence 6: content/first-order-logic/introduction/soundness-completeness.tex, line 31, column 4

Expression 80

∀x(A(x)→B(x)),∃xA(x)⊨∃xB(x)

Conventional reading: for every x, if A of x then B of x; and, there exists x such that A of x; together semantically entail: there exists x such that B of x

Meaning here: Every structure in which all A objects are B objects and at least one A object exists also contains at least one B object.

1 occurrence
  1. Occurrence 1: content/first-order-logic/introduction/first-order-logic.tex, line 51, column 27

Expression 85

∀v0P(v0)

Conventional reading: for every v sub zero, P of v sub zero

Meaning here: The universal sentence asserting that predicate P holds of every domain object.

1 occurrence
  1. Occurrence 1: content/first-order-logic/introduction/substitution.tex, line 16, column 44

Expression 86

Γ⊢A

Conventional reading: Gamma derives formula A

Meaning here: There is a derivation of A from the set of premises Gamma in the derivation system under discussion.

6 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/soundness-completeness.tex, line 16, column 35
  2. Occurrence 2: content/first-order-logic/introduction/soundness-completeness.tex, line 22, column 57
  3. Occurrence 3: content/first-order-logic/introduction/soundness-completeness.tex, line 25, column 4
  4. Occurrence 4: content/first-order-logic/introduction/soundness-completeness.tex, line 26, column 60
  5. Occurrence 5: content/first-order-logic/introduction/soundness-completeness.tex, line 29, column 39
  6. Occurrence 6: content/first-order-logic/introduction/soundness-completeness.tex, line 31, column 30

Expression 91

B

Conventional reading: formula B

Meaning here: The metavariable B denotes an arbitrary formula.

14 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/first-order-logic.tex, line 27, column 16
  2. Occurrence 2: content/first-order-logic/introduction/first-order-logic.tex, line 30, column 24
  3. Occurrence 3: content/first-order-logic/introduction/first-order-logic.tex, line 35, column 33
  4. Occurrence 4: content/first-order-logic/introduction/first-order-logic.tex, line 36, column 48
  5. Occurrence 5: content/first-order-logic/introduction/first-order-logic.tex, line 36, column 63
  6. Occurrence 6: content/first-order-logic/introduction/first-order-logic.tex, line 38, column 11
  7. Occurrence 7: content/first-order-logic/introduction/first-order-logic.tex, line 38, column 22
  8. Occurrence 8: content/first-order-logic/introduction/formulas.tex, line 44, column 30
  9. Occurrence 9: content/first-order-logic/introduction/formulas.tex, line 78, column 51
  10. Occurrence 10: content/first-order-logic/introduction/satisfaction.tex, line 40, column 34
  11. Occurrence 11: content/first-order-logic/introduction/sentences.tex, line 42, column 11
  12. Occurrence 12: content/first-order-logic/introduction/sentences.tex, line 42, column 63
  13. Occurrence 13: content/first-order-logic/introduction/sentences.tex, line 46, column 11
  14. Occurrence 14: content/first-order-logic/introduction/sentences.tex, line 51, column 14

Expression 95

∃xA(x)

Conventional reading: there exists x such that A of x

Meaning here: The existential sentence asserting that at least one object satisfies A.

4 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/first-order-logic.tex, line 70, column 1
  2. Occurrence 2: content/first-order-logic/introduction/first-order-logic.tex, line 73, column 1
  3. Occurrence 3: content/first-order-logic/introduction/first-order-logic.tex, line 82, column 1
  4. Occurrence 4: content/first-order-logic/introduction/first-order-logic.tex, line 83, column 18

Expression 100

M⊨A

Conventional reading: structure M satisfies formula A

Meaning here: Sentence A is true in first-order structure M, independently of variable assignments.

4 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/first-order-logic.tex, line 46, column 55
  2. Occurrence 2: content/first-order-logic/introduction/semantic-notions.tex, line 21, column 1
  3. Occurrence 3: content/first-order-logic/introduction/semantic-notions.tex, line 27, column 32
  4. Occurrence 4: content/first-order-logic/introduction/semantic-notions.tex, line 29, column 46

Expression 105

t

Conventional reading: term t

Meaning here: An arbitrary first-order term t used in universal instantiation and substitution.

8 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/substitution.tex, line 20, column 46
  2. Occurrence 2: content/first-order-logic/introduction/substitution.tex, line 26, column 18
  3. Occurrence 3: content/first-order-logic/introduction/substitution.tex, line 26, column 66
  4. Occurrence 4: content/first-order-logic/introduction/substitution.tex, line 32, column 22
  5. Occurrence 5: content/first-order-logic/introduction/substitution.tex, line 32, column 41
  6. Occurrence 6: content/first-order-logic/introduction/substitution.tex, line 41, column 31
  7. Occurrence 7: content/first-order-logic/introduction/substitution.tex, line 42, column 46
  8. Occurrence 8: content/first-order-logic/introduction/substitution.tex, line 44, column 11

Expression 106

A(t)

Conventional reading: formula A of t

Meaning here: The result of substituting term t for the relevant free occurrences of x in A of x.

5 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/substitution.tex, line 25, column 14
  2. Occurrence 2: content/first-order-logic/introduction/substitution.tex, line 27, column 48
  3. Occurrence 3: content/first-order-logic/introduction/substitution.tex, line 34, column 60
  4. Occurrence 4: content/first-order-logic/introduction/substitution.tex, line 37, column 35
  5. Occurrence 5: content/first-order-logic/introduction/substitution.tex, line 39, column 62

Expression 109

∃xB(x)

Conventional reading: there exists x such that B of x

Meaning here: The existential conclusion asserting that at least one object satisfies B.

2 occurrences
  1. Occurrence 1: content/first-order-logic/introduction/first-order-logic.tex, line 70, column 39
  2. Occurrence 2: content/first-order-logic/introduction/first-order-logic.tex, line 84, column 8

Expression 114

aM=1∈{1,2}=PM

Conventional reading: the interpretation of constant a in structure M equals one, which is an element of the set containing one and two, which equals the interpretation of predicate P in M

Meaning here: The chain verifies that a denotes one and that one belongs to the extension of P, so P of a is true in M.

1 occurrence
  1. Occurrence 1: content/first-order-logic/introduction/satisfaction.tex, line 46, column 1

Two formal objects

  1. Inductive definition of formulas in the simplified first-order language — content/first-order-logic/introduction/formulas.tex, line 35.
  2. First-order axioms for a reflexive and transitive relation — content/first-order-logic/introduction/models-theories.tex, line 40.

Eight references

  1. (P. D. Magnus and Tim Button and J. Robert Loftis and Aaron Thomas-Bolduc and Robert Trueman and Richard Zach, 2021) — content/first-order-logic/introduction/first-order-logic.tex, line 16.
  2. the atomic-formula clause — content/first-order-logic/introduction/formulas.tex, line 54.
  3. the negation clause — content/first-order-logic/introduction/formulas.tex, line 59.
  4. the atomic-formula clause — content/first-order-logic/introduction/formulas.tex, line 61.
  5. the existential-quantifier clause — content/first-order-logic/introduction/formulas.tex, line 61.
  6. the limiting clause — content/first-order-logic/introduction/formulas.tex, line 63.
  7. the future section on inductive definitions (not yet available) — content/first-order-logic/introduction/formulas.tex, line 73.
  8. the future section on structural induction (not yet available) — content/first-order-logic/introduction/formulas.tex, line 74.