Previously, we get that p
~(~p)
. Negation of the other rules we already know is p ^ ~ q ~ (p -> q), which is the negation of a conditional. Whether we can find a formula that is identical, ie, the negation, in another compound statement, ie, disjunction and conjunctions? The answer is yes, the logic put forward by the British mathematician Augustus de Morgan.
De Morgan's rules, so the formulas of negation and disjunction of conjunctions is called, is as follows.
- The negation of a conjunction is given by ~(p ^ q) ~p v ~q
- The negation of a disjunction is given by ~(p v q) ~p ^ ~q
When you buy a car, the car can be new or old. Sales will say "of course not true this car is not a new car." This compound sentence has a one-sentence statement (This car is new) and two negation.
"Absolutely not true" "This car is not a new car"
~ ~ p
Does this mean a new car? To find out the truth, we can construct the truth table for logic expression ~ (~ p) and compare it with the p. Since there is only one single statement, then we only need 2 lines in the table. We will have a column p, ~ p and ~ (~ p). ~ p gives the value opposite p. Medium ~ (~ p) gives the value in contrast to ~ p. So the table would be as follows.
Note that the third column of ~ (~ p) has a truth value is identical to p. If like this, then ~ (~ p) is said is equivalent to p. That is, meaning they are the same statement. An equivalent expression tends to be a symbolic expression that has an identical truth value to each other.
Expression p
q
read p is equivalent to q or p and q equivalent. From the above table we find that ~ (~ p) is equivalent to p, can we write p
~(~p).
EXAMPLE
Is the statement "If my landlord, who pays my property tax" and the statement "I am the owner of the house and I do not pay property tax" equivalent?
COMPLETION
We begin by giving a symbolic representation of the statements above.
P: My landlord
q: I pay property tax
p -> q: if I am the owner of the house, I pay property tax.
p ^ ~ p: I am the owner of the house and I do not pay property tax.
Truth table for this case will contain four lines. The table we will use as below.
| p | q | ~q | p^~q | p -> q |
| B | B | | | |
| B | S | | | |
| S | B | | | |
| S | S | | | |
Now we just give the truth value corresponding to each column. Column ~ q, will we fill with the opposite truth value from column q. The next column of the contents of B in the second row and the S on the other line, since conjunction requires the truth value of B for each factor. Meanwhile in the last column, because the conditional is only false if p is true and q is false, then fill the second row with S and B on the other line.
| p | q | ~q | p^~q | p -> q |
| B | B | S | S | B |
| B | S | B | B | S |
| S | B | S | S | B |
| S | S | B | S | B |
Because the truth values in the column p ^ ~ q is not equal to p -> q, the two statements are not equivalent. Note that p ^ ~ q and p -> q has the truth values of the exact opposite. When this happens in two statements, one compound statement is the negation of another compound statement. Consequently, p ^ ~ q is the negation of p -> q. This relationship can be expressed by p ^ ~ q ~ (p -> q). The negation of a conditional premise is equivalent to conjunction and negation conclusion.
The statement that looks different in reality may have the same purpose. When we have two statements are equivalent, we can replace each other without changing the meaning. Emotional factors that influence the selection of our statement in practice. Not on its meaning.
Conditional is a compound statement in the form 'if p then q' is denoted p -> q. What is truth conditional? Notice the phrase 'If you give Rp. 20.000, - to me, I bought concert tickets going for you. "Symbolic representation of the sentence is as follows:
p: You gave me Rp. 20.000, -
q: I bought concert tickets for you.
p -> q: If you gave me Rp 20,000, -, I'll buy you a concert ticket.
Conditional can be viewed as a promise. Assuming you really gave me 20 thousand, then I will have two options, whether buying or not. When I bought (q = B) then the statement is true. But if I do not buy (q = S) then the sentence is false. This situation can be described as follows.
| p | q | p->q |
| B | B | B |
| B | S | S |
| S | B | ? |
| S | S | ? |
Then what if you do not give me 20 thousand? p = S? Of course if I would give you a ticket or not, p -> q is not wrong. Because the promise is only fulfilled if p is true. Since p -> q is false, then if p either directly make the truth value of p -> q is true. Table complete conditional is as follows.
| p | q | p->q |
| B | B | B |
| B | S | S |
| S | B | B |
| S | S | B |
Disjunction of two statements combined with logical conjunctions 'or'. Compound statement "Burhan is a doctor or a cadre of the Golkar Party" is a disjunction (or inclusive) with the symbolic presentation:
p: Burhan a doctor
q: Burhan Golkar Party cadres
pvq: Burhan a doctor or a cadre of the Golkar Party.
Despite the fact that a doctor is not Burhan Golkar Party cadres, disjunction pvq above remain true. Disjunction is true if there are at least one constituent that statement is true. Disjunction of one if the truth value of each constituent wrong.
| p | q | pvq |
| B | B | B |
| B | S | B |
| S | B | B |
| S | S | S |
Conjunctions combine two statements with logical conjunctions "and." Compound sentence "Burhan a doctor and the Golkar Party cadres" is a symbolic representation sebagia conjunction with the following.
p: Burhan a doctor
q: Burhan democrat party cadres
p ^ q: Burhan a doctor and democrat party cadres.
The value of truth of a statement depends on the truth value of compound statements that constitute them. How many rows are needed in conjunction truth table for p ^ q? Since p has two possibilities, as well as q, the combination of the possibilities that there are 4 (2.2).
A conjunction p ^ q is true if each of the statements that constitute true, a combination other than that is false. Symbols p and q can represent any statement. Compound statement p and q depend on the truth value of each p and q. For example conjunction p ^ q wrong when p false and q true. Here is the truth table conjunction p ^ q more.
| p | q | p^q |
| B | B | B |
| B | S | S |
| S | B | S |
| S | S | S |
