Formalizing Propositional Logic

Statements in human language can be ambiguous. Formalization seeks to remove this ambiguity.

For example, if your boss says "you may take Thursday or Friday off", can we take both days off?

Another example, "if a request occurs then it will be acknowledged or the process does not make progress". This one is ambiguous, it could be or it could be , so we need to ask the person who wrote it

Example

If the train arrives late and there are no taxis at the station, then John is late for the meeting.
John is not late for his meeting.
The train did arrive late.
Therefore, there are taxis at the station.

Letter Declarative Sentence
p the train is late
q there are taxis at the station
r John is late for his meeting

Argument:
If and negation , then .
Not .
.
Therefore,
.

Steps

  1. Identify statements you can answer with "true" or "false" without connectives. These are your propositions
    • A "prime proposition" is a statement that cannot be broken down (i.e no connectives)
  2. Identify connectives using Nissanke's rules
    • Weird one: unless means

List

Nissanke's rules

    • not
    • does not hold
    • is it not the case that
    • is false
    • and
    • but
    • not only but
    • while
    • despite
    • yet
    • although
    • or
    • or or both
    • "and/or"
    • unless
    • if then
    • if
    • only if
    • when
    • is sufficient for
    • if
    • only if
    • when
    • is sufficient for
    • is necessary for
    • implies
    • if and only if ( iff )
    • is necessary and sufficient for
    • is exactly if
    • is equivalent to
Example

You graduate only if you work hard

g means "you graduate"
h means "you work hard"

Examples

c means it's cold, s means it's snowing

  1. It is cold but it is not snowing
  2. It is neither snowing nor cold
    • Do not apply de-morgans law, we are formalizing the sentence and there's an "or" in the sentence
  3. It is cold if it is snowing
  4. It is snowing only if it is cold
Example

It is not the case that a student passing the assignments is sufficient for a student to pass SE212

a means "a student passes the assignments"
p means "a student passes SE212"

a p intuitive
T T F T F
T F T T T
F T F T F
F F F F F
Warning

Watch out for vacuous truths, which would cause "false implies everything".
Example: if an animal moos, then it is a cow

Animal does not moo: could be anything
Animal does moo: is a cow