Logical Statements
Statement
A statement is a sentence that is either True or False.
This statement is True. 5 is greater than 2.
This is a False statement, it is not wrongly written, it is a VALID False statement
This is NOT a statement, since we don't know what
When the hypothesis is false, the statement is vacuously true, but true in a sense of unimportant, or uninteresting, or vacuous sense
New Statements
Example
"My shirt is gray but my shorts are not"
- "but" is logically the same thing as "and"
So the sentence above can be logically traduced as the logical form:
Conditional Statement
Denoted as
| T | T | T | F | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |
The implications
The implication
If I study hard, then I will pass
Either I don't study hard, or I pass
Contrapositive
If I study hard, then I will pass.
Either I didn't study hard, or I pass.
If I don't pass, then I didn't study hard.
Either I pass, or I didn't study hard.
All these four statements are logical equivalents.
Negating a conditional
So the only way an implication can be false, is when the hypothesis is True but the conclusion is False
Converse & Inverse
The Converse of a statement
The Inverse of a statement
Biconditional Statements
The biconditional
If I study hard, then I will pass AND if I pass, then I studied hard:
I will pass if and only if I study hard:
These both are logically equivalent
Necessary & Sufficient Conditions
- "
- "
- If
If we want a Rectangle
It is sufficient to have a Square
But necessary to have a Quadrilateral
Argument
A valid argument is a list of premises(statements) from which the conclusion follows
Modus Ponens
Modus Ponens is an argument of the form:
| Variables | Variables | | | Premises | Premises | | | Conclusion |
|---|---|---|---|---|---|---|
| | | ||||||
| | | ||||||
| T | T | | | T | T | T | |
| T | F | | | F | T | (don't care) | |
| F | T | | | T | F | (don't care) | |
| F | F | | | T | F | (don't care) |
We don't really care of the cases where the conclusion is not true because in those scenarios the premises are not satisfied. In this logical argument I am assuming that both of my premises are true, and if they are not true I don't care
Modus Tollens
Modus Tollens is an argument of the form:
If
Therefore,
If I'm the USA President then I'm an American citizen.
I'm not an American citizen.
Therefore, I'm not the USA President.
Generalization
Generalization is an argument of the form:
Therefore,
It is basically the idea that if I have some true statement, the disjunction with any other statement will always be true.
I'm a Canadian.
Therefore, I'm a Canadian or I'm a unicorn.
Specialization
Opposite of generalization, specialization is an argument of the form:
Therefore,
I'm a Canadian and I have a PhD.
Therefore, I'm a Canadian.
Contradiction
Contradiction is an argument of the form:
Therefore,
If I assume my premise to be false, and this implies a contradiction, then the premise is true.
.
Tautology
A tautology
| T | T | T |
| T | F | T |
"A dog is a mammal"
Contradiction
A contradiction
| F | T | F |
| F | F | F |
"A dog is a reptile"