Maths - Propositional Logic

A proposition is a statement which is either true or false, but it cannot be both true and false at the same time.
A question, exclamation or a command are all not propostions.

Logical Operators:

Unary Operator (Works with one value)   NOT  ¬    e.g      ¬P
^ AND
˅ OR
͢ ͢    Conditional (Always false when Q is false (When P͢  Q)) otherwise it is always true

<-> Only true if both are true or if both are false.



Implications relating to P -> Q

Implication: P->Q
If a student is female, then the student has brown hair.

Converse: Q->P
If a student has brwn hair, then they are female.

Inverse: ¬P->¬Q
If a student is not female, then they do not have brown hair

Contrapositive: ¬Q->¬P
If a student is not brown haired, they are not female.

TRUTH TABLES

P                Q                   P->Q                 ¬Q                  ¬P              ¬Q->¬P

T                T                      T                        F                    F                    T
T                F                      F                        T                     F                   F
F                T                      T                        F                     T                   T
F                F                      T                        T                     T                   T







Argument Validity

An argument is a set of compound propositions called premises together with another compound proposition that is suggested follows from the premises.
This is called a conclusion at the end.

The general form of an argument is  p1^p2^p3...pN) = Conclusion.

The conjunction of premises (and gate) is only true if all the premises are to be true. The argument is only valid when each premise is true!!

p1    p2    p3          C
T      T      T           T
at least one F      T or F


Example:
P - You read alot                                    Q - You are a brilliant conversationalist
R - You have many friends

The argument put forward is:
'If you read alot' = P, 'you will become a brilliant conversationalist' = Q. If you become a brilliant conversationalist, 'you will have many friends' = R. Therefore if you don't have many friends, you dont read alot.

The premises taken from the above arguments are:
p1 - If you read alot, you will become a brilliant conversationalist.    P->Q
p2 - if you are a brilliant conversationalist, you will have many friends.    Q->R
C - If you dont have many friends, you dont read alot.    ¬R->¬Q

p1^p2 = C
or
(p->q)^(q->r) = (¬r->¬p)


TRUTH TABLE


Where premises are true.
                                                    p1                      p2                                      C
P           Q              R                 P->Q                Q->R       ¬R     ¬P          ¬R->¬P

T           T             T                        T                    T             F          F               T
F          T             T                         T                    T             F         T               T
F           F              T                       T                     T           F          T                T
F          F               F                       T                    T             T         T                 T

To see if an argument is valid we must look at the rows of the truth table for which the premises are all true, and when all the conclusions are also true. If this happens (as above) we know that the argument is valid.





Contingency
e.g. (P v Q) -> ¬P

Propostions that are sometimes true and sometimes false, depending on P and Q values. With at least one T and at least one F appearing.
(P  v  Q) -> ¬P
      F
      F
      T
      T
Contains both True and False values in the conclusion.

Taughtology :
A taughtology is when all values in conclusion are True
e.g. P->( P  v  Q)
                    T
                    T
                    T
                    T
All final value ares true.


Contradiction:
 Is when all the final values are false.
P ^ (¬P  ^  Q)
            F
            F
            F
            F

Each final value is false.