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Understanding Conjunction and Disjunction in Formal Logic, Study notes of Mathematics

University of Hawaii (UH)Mathematics

The concepts of conjunction and disjunction in formal logic through examples and their significance in making logical statements. It also discusses the reasons for formalizing logic and considerations in establishing a formal system.

Typology: Study notes

Pre 2010

Uploaded on 04/12/2010

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9.3 Conjunction, disjunction
Suppose Aand Bare propositions. The following
are all different ways of saying the same thing:
Aand Bare true.
Both Aand Bare true.
Ais true and Bis true.
Aand B.
A∧B.
A&B.
A∧Bis true when both Aand Bare true, and
false if either of them is false. This connective is
called conjunction.
Examples
1.1+1=2and2+2=4
2.1+1=2and2+2=6
3. Abraham Lincoln was the President of the United
States and a famous cosmonaut.
30
pf3
pf4

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9.3 Conjunction, disjunction

Suppose A and B are propositions. The following are all different ways of saying the same thing:

A and B are true. Both A and B are true. A is true and B is true. A and B. A ∧ B. A&B.

A ∧ B is true when both A and B are true, and false if either of them is false. This connective is called conjunction.

Examples

  1. 1 + 1 = 2 and 2 + 2 = 4
  2. 1 + 1 = 2 and 2 + 2 = 6
  3. Abraham Lincoln was the President of the United States and a famous cosmonaut.

Suppose A and B are propositions. The following are all different ways of saying the same thing:

A or B are true. Either A or B are true. Either A is true or B is true or both are true. A is true or B is true. A or B. A ∨ B.

A∨B is true when A is true or B is true or both, and false if both of them is false. This connective is called disjunction.

Examples

  1. 1 + 1 = 2 or 2 + 2 = 4
  2. 1 + 1 = 2 or 2 + 2 = 6
  3. 1 + 1 = 3 or 2 + 2 = 6
  4. Abraham Lincoln was the President of the United States or a famous cosmonaut.

Considerations in establishing a formal system:

  • Should not be unnecessary cumbersome, diffi- cult to use, or difficult to typeset (Examples: Lull et al; Lewis Carroll; Frege; Principia Mathematica; RPN; JSL)
  • Should correspond to intuition.
  • Should be extensive enough to cover all sita- tions of interest.
  • Should be small enough to be tractable.
  • (Propositional logic; Predicate Logic [aka First Order Logic]; Higher order logics and infinitary logics.)
  • Should allow for a distinction between syntac- tic and semantic argument. (more later)
  • Should be amenable to metamathematical anal- ysis, for example, mathematical proofs about statements about the logic.