# Addition – a logical implication

P => (P v Q) meaning, P implies (P or Q)

If the compound statement P=>(P v Q) is given and is true (it is always true) then we can conclude P by just taking P.

If P = the traffic light is green
and Q = the traffic light is red.

Then in English, P=>(P v Q ) will be “If the traffic light is green then the traffic light is green or (otherwise) the traffic light is red”. As mentioned, this compound statement is always true.

So, If the compound expression (P => (P v Q)) is given, we know it is true, so we can always conclude P by just taking P.

Suppose that, it is false that (P) the traffic light is green,
Suppose that, it is true that (Q) the traffic light is red,

Then we can conclude that it is false that the traffic light is green, because P is false.

Suppose that, it is true that (P) the traffic light is green,
Suppose that, it is false that (Q) the traffic light is red,

Then we can conclude that it is true that the traffic light is green, because P is true.

Suppose that it is true that (P) the traffic light is green,
Suppose that it is true that (Q) the traffic light is red,

Then normally we would conclude that the traffic light is broken, but in this case, logically we would conclude that the traffic light is green because P is true.

Suppose that it is true that (P) the traffic false is green,
Suppose that it is true that (Q) the traffic false is red,

Again, normally we would conclude that the traffic light is broken, but in this case, logically we would conclude that it is false that the traffic light is green because P is false.

In the truth table we can see the tautology in P=>(P v Q) – the compound statement is always true – which proves what we have said in the examples.

# Modus tollens – a logical implication

((P => Q) ^ not Q) => not P named Modus tollens meaning the way that denies by denying – denying the consequent Q.

If the compound statement  (P => Q) ^ not Q) => not P is given and is true (it is always true) and P => Q is true then not Q must be true – therefore not P must be true (see truth table to follow the reasoning).

```P => Q (P implies Q true) ^ not Q (true) ======================== then not P is True```

In English

P = an intruder is detected
Q = the alarm goes off

(P=>Q) If an intruder is detected, then the alarm goes off.
(^ not Q) and The alarm does not go off.
(=> not P) then, no intruder is detected

said a little more coherently:

If an intruder is detected, the alarm will go off, and when the alarm does not go off, then, no intruder is detected.

see Wikipedia

# Modus ponens – a logical implication

(P ^ (P => Q)) => Q named Modus ponens meaning affirms by affirming – affirming the antecedent/premise P

If the compound statement (P ^ (P => Q)) => Q is given and is true (it is always true) and P => Q is true then P must be true – therefore Q must be true (see truth table to follow the reasoning).

``` P => Q (If P => Q is true then...) P (must be true) therefore Q must be true.```

In English:
P = today is Tuesday
Q = I will go to work

If today is Tuesday, then I will go to work.
Today is Tuesday.
Therefore, I will go to work.

In instances of modus ponens we assume as premises that p => q is true and p is true. Only one line of the truth table – the last – satisfies these two conditions. On this line, q is also true. Therefore, whenever p => q is true and p is true, q must also be true

see Wikipedia

# Income tax evasion.

Brown, Jones and Smith are suspected of income tax evasion. They testify under oath as follows:

Brown: Jones is guilty and Smith is innocent

Jones: If Brown is guilty, then so is smith.

Smith: I am innocent, but at least one of the others is guilty.

Assuming everyone told the truth. Who is/are guilty/innocent.

Assuming the innocent told the truth, who are/is guilty/innocent.

We have to transfer the statements into logical notation and solve it. The first part is easy. Assuming that everyone is telling the truth and looking at the first statement we can infer that Brown is guilty and smith is innocent.

Looking at the second statement: “if Brown is guilty then so is smith”  the contrapositive of that is: “If smith is not guilty then brown is not guilty”. Therefore Brown is innocent.

So the answer to the first portion is that:

Brown is innocent,

Jones is guilty,

Smith is innocent.

To answer the second part of the question we can use logical notation:

B = Brown is innocent

not B = Brown is quilty

J = Jones is innocent

not J = Jones is guilty

S = Smith is innocent

not S = Smith is guilty

What are the statements they are making:

Brown is making the following statement:

B: not J ^ S

J: not B => not S

S: S ^ (not B v not J)

To answer the second part of the question: “Assuming the innocent told the truth, who are/is guilty/innocent”

# Free drinks in the pub.

A tourist is enjoying an afternoon refreshment in a local pub in England when the bartender says to him: “Do yu see those 3 men over there? One is Mr. X who always tells the truth, one is Mr Y who always lies and one is Mr. Z who sometimes lies by randomly answering questions with yes or no.

You may ask them 3 yes or no questions indication who should answer. If after asking these questions you can identify Mr. X, Mr. Y and Mr.Z, they will buy you a drink. What yes or no questions should the thirsty tourist ask?

The first question is important. It should be asked in such a ways so that you can eliminate Mr.Z. It is easier to deal with a person who always tells the truth or who always lies.

# Murder in a train

Five persons A,B,C,D,E are in a compartment in a train. A,C,E are men and B and D are women. The train passes through a tunnel and when it emerges, it is found that E is murdered. An enquiry is held. A, B, C, D make the following statements.

• A: I am innocent; B was talking to E when the train was passing through the tunnel.
• B: I am innocent; I was not talking to E when the train was passing through the tunnel.
• C: I am innocent; D committed the murder.
• D: I am innocent; one of the men committed the murder.

Four of these 8 statements are true and four are false. Assuming only one person committed the murder, who did it?

So, there are 8 statements.

A is making statement 1 and statement 2

B is making statement 1 and statement 2

C is making statement 1 and statement 2

D is making statement 1 and statement 2

one of the 1st statements is false and three are true. three of the 2dn statements are false and one is true.

This is given because four of the statements are true and four of them are false.

Now look at the second statement of A and B:

A says, “B was talking to E when the train was passing through the tunnel”

B says “I was not talking to E when the train was passing through the tunnel”

One statement is the negation of the the other.

If what A says is true then what B says is false

If what A says is false then waht B says is true.

Therefore one of these statements is true and one of them is false. If one of these statements are true, then the others must be false (this is because we we mentioned above, one of these statements is true and the other 3 are false.So the second statement of C and D are false.

Looking at the 2nd statement C made: “D committed the murder” is therefore false. So D did not commit the murder.

Looking at the 2nd statement of D: “one of the men committed the murder”. That is also false. So this means that a women commited the murder. Therefore A and C did not commit the murder.

This leaves us with B. So B must have committed the murder.

# Liars in a Country

A certain country is inhabited by only by people that tell the truth or who tell lies and who respond to questions with yes or no.

A tourist comes to a fork in the road where one branch leads to the capital and the other does not. There is no sign indicating which branch to take, but there is an inhabitant, Mr. Z, standing at the fork. What single question should the tourist ask to determine which branch to take.

This is the situation:

The tourist walks up to the person at the fork in the road who maybe a truth teller or a liar. (Liars always tell lies and truth tellers always say the truth).

Either the left fork leads to the the capitol, or the right fork leads to the capitol.

Either the person (Mr z) is a truth teller or a liar.

So What are the possibilities.

• The person maybe a truth teller (TT) and the left road (LEFT) leads to the capitol.
• The person maybe a liar (LIAR) and the left road (LEFT) leads to the capitol.
• The person maybe a truth teller (TT) and the right road (RIGHT) leads to the capitol.
• The person maybe a liar (LIAR) and the and the right road (RIGHT) leads to the capitol.

So there are 4 possibilities and a single question with a yes or no answer should take care of all theses things.

Because the tourist does not know if the man sitting at the fork of the road is a truth teller or a liar there are four possibilities like this:

The left road may leat to the capital or the right road may lead to the capital.

The man at the fork is a liar or a truth teller.

Now, if the answer is yes, the tourist has to take the left road,

If the answer is no he has to take the left road.

But because the liar always lies, if the answer is yes, he will say no, and if the answer is no, he will say yes. So the correct answers will look like this.

So the question that the tourist needs to ask will be something like this:

“Is it true that you are a truth teller and the left road leads to the capital or  you are a liar and the right road leads to the capital.” This is a single yes or no answer question.

This question is a combination of the two yes cases in the above picture.

If Mr.z is a truth teller and he says yes, then the tourist can take the left road.

If Mr.Z is a liar and the right road leads to the capital then the correct answer will be yes, but Mr.Z will say no so the tourist can take the right road.

Another way of asking the question could be this:

“Is it true, If I ask you whether the left road leads to the capital, you will say yes, won’t you?”

So again you look at the four possibilities : If the man is a truth teller, then the left road leads to the capital, if he is a liar then the right road leads to the capital.

If the answer is yes then the man must be a truth teller and the left road must lead to the capital. The mans answer will be a combination of yes and yes.  If I ask you whether the left road leads to the capital, you will say yes = YES and won’t you? = YES. In this case the tourist will take the left road.

If he is a liar then he has to answer the first part of the question: “If I ask you whether the left road leads to the capital, you will say yes” he will have to answer NO because he is a liar – and he will also have to answer the second part of the question: “won’t you” with NO again, because he is a liar. In this case the Tourist will take the right road.