புதன், 18 டிச., 2024 at PM 1:46

Introduction to Logic

SEE

Dictionary

abduction
analogy
and elimination
and introduction
antecedent
biconditional
biconditional elimination
biconditional introduction
completeness
conclusion
conjunct
conjunction
consequent
Consistency Theorem
contingency
contrapositive
converse
deduction
Deduction Theorem
disjunct
disjunction
Equivalence Theorem
falsifiability
Fitch system
implication
implication elimination
implication introduction
induction
instance
inverse
linear proof
logical constant
logical consistency
logical entailment
logical equivalence
metavariable
negation
negation elimination
negation introduction
operator precedence
or elimination
or introduction
premise
proof
proposition
proposition constant
propositional language
propositional logic
propositional sentence
propositional vocabulary
provability
reiteration
relational logic
rule of inference
satisfaction
satisfiability
schema
sentence
soundness
structured proof
target
truth assignment
truth table
unsatisfiability
Unsatisfiability Theorem
validity

1. abduction

2. analogy

3. and elimination

4. and introduction

5. antecedent

6. biconditional

7. biconditional elimination

8. biconditional introduction

9. completeness

10. conclusion

11. conjunct

12. conjunction

13. consequent

14. Consistency Theorem

15. contingency

16. contrapositive

17. converse

18. deduction

19. Deduction Theorem

20. disjunct

21. disjunction

22. Equivalence Theorem

23. falsifiability

24. Fitch system

25. implication

26. implication elimination

27. implication introduction

28. induction

29. instance

30. inverse

31. linear proof

32. logical constant

33. logical consistency

34. logical entailment

35. logical equivalence

36. metavariable

37. negation

38. negation elimination

39. negation introduction

40. operator precedence

41. or elimination

42. or introduction

43. premise

44. proof

45. proposition

46. proposition constant

47. propositional language

48. propositional logic

49. propositional sentence

50. propositional vocabulary

51. provability

52. reiteration

53. relational logic

54. rule of inference

55. satisfaction

56. satisfiability

57. schema

58. sentence

59. soundness

60. structured proof

61. target

62. truth assignment

63. truth table

64. unsatisfiability

65. Unsatisfiability Theorem

66. validity

Here’s the list categorized based on semantical relation, i.e., how these concepts relate to meaning, truth, or interpretation within logical systems.

Semantical Relation: Truth and Meaning

1. antecedent

2. consequent

3. biconditional

4. contrapositive

5. converse

6. implication

7. negation

8. inverse

Semantical Relation: Truth Values

1. contingency

2. falsifiability

3. logical consistency

4. logical entailment

5. logical equivalence

6. satisfaction

7. satisfiability

8. unsatisfiability

9. validity

Semantical Relation: Propositional Structure

1. proposition

2. propositional language

3. propositional logic

4. propositional sentence

5. propositional vocabulary

6. schema

7. sentence

Semantical Relation: Interpretation and Assignment

1. completeness

2. provability

3. soundness

4. truth assignment

Soundness and truth are related but distinct concepts in logic. Here's how they diverge:

Soundness

Definition: A logical system is sound if every statement that can be proven within the system is true in all interpretations (or models) of the system.

Focus: Ensures that the rules of inference do not lead to false conclusions from true premises.

Scope: Concerns the relationship between provability and truth.

Example: If a logical system proves , and is true in the real world or under the given semantics, the system is sound.

Truth

Definition: Truth is a property of a statement that corresponds to reality or is satisfied under a particular interpretation.

Focus: It is independent of whether the statement is provable within a formal system.

Scope: Concerns the correspondence to reality or model semantics.

Example: A statement like "It is raining" is true if it corresponds to actual weather conditions, regardless of whether it's provable within a logical framework.

Where They Diverge

1. Provability vs. Reality:

Soundness depends on a system's internal rules correctly reflecting the semantics of truth.

Truth exists independently of the system—it depends on the external model or reality.

1. Application:

A logical system can prove something (provability) that is not true in reality if it is unsound.

Conversely, a true statement may not be provable within a particular logical system if the system is incomplete.

1. Example of Divergence:

Inconsistent Systems: An inconsistent system could "prove" both and , but one of them must be false under a truthful interpretation, violating soundness.

Gödel's Incompleteness: Some true statements about arithmetic are unprovable in a formal system, showing truth extends beyond provability.

Thus, while soundness ensures that provable statements are true, truth itself is broader and doesn't rely on the provability within a specific formal system.

The concepts of logics, sets, relations, functions, and logic gates are interrelated and form the foundation of mathematics, computer science, and digital systems. Here's a structured explanation of their relationships:

1. Logics and Sets

Logic: The study of reasoning and truth. In formal systems, logic evaluates the truth of propositions (statements) based on rules.

Set Theory: The study of collections of objects (sets) and their relationships.

Relation: Logic uses sets to define truth assignments. For example:

A proposition in logic can be seen as a function mapping elements (interpretations) from a set to a truth value ().

Logical connectives () can be modeled as operations on sets, like union (), intersection (), and complement.

Example:

in logic corresponds to the intersection of sets and .

1. Logics and Relations

A relation is a subset of a Cartesian product of sets. It defines how elements from one set relate to elements of another.

In predicate logic, a predicate can be interpreted as a relation. For instance:

: A binary relation between and .

Logical inference rules often rely on relations to establish connections between variables or truth values.

Example:

The truth table of a logical statement defines a relation between inputs (propositions) and outputs (truth values).

1. Sets and Relations

Relations are constructed from sets. Given two sets and , a relation is a subset of the Cartesian product .

Relations can express logical concepts like:

Equivalence relations (reflexive, symmetric, transitive) in logic are tied to concepts like logical equivalence.

Order relations (partial, total) relate to propositional precedence.

Example:

: A relation defining a "greater-than" property.

1. Relations and Functions

A function is a specific type of relation where every element in the domain is associated with exactly one element in the codomain.

Logical expressions can often be represented as functions mapping inputs (propositions) to outputs (truth values).

Example:

The logical NOT operator () is a function mapping and .

1. Logics and Logic Gates

Logic gates are physical implementations of logical operations in digital systems.

Each gate (AND, OR, NOT, etc.) corresponds to a basic logical connective.

AND (): Outputs 1 (True) if both inputs are 1.

OR (): Outputs 1 if at least one input is 1.

NOT (): Inverts the input.

Relation to Logic:

Logic gates use Boolean logic, which operates on sets of truth values ().

1. Functions and Logic Gates

Logic gates implement Boolean functions.

Each gate represents a function mapping input combinations to an output.

AND gate:

OR gate:

NOT gate:

Example:
A half-adder in circuits uses XOR (exclusive OR) and AND gates to compute sum and carry, which are Boolean functions.

Summary of Relationships

1. Logic ↔ Sets: Logic relies on set operations for modeling truth values and logical connectives.

2. Logic ↔ Relations: Predicates in logic are interpreted as relations between elements.

3. Sets ↔ Relations: Relations are subsets of Cartesian products of sets.

4. Relations ↔ Functions: Functions are a special type of relation with unique mappings.

5. Logic ↔ Logic Gates: Logic gates are physical representations of Boolean logic.

6. Functions ↔ Logic Gates: Logic gates compute Boolean functions based on logical operations.

These interconnections form the theoretical foundation for both abstract reasoning and practical computation.