Workshop 2 - LOGIC and LANGUAGE

Workshop 2 - LOGIC and LANGUAGE.

In this part of the course, we shall investigate how normal language is governed by logic rules, which although may not be apparent to us, nevertheless govern our language, and hence, the way wethink. You have seen some examples of sentences which we shall not consider, due to bad grammar, ambiguity and so on; we think about only DECLARATIVE SENTENCES. We also throw out sentences of the type;

I think that music is great!

It is asserted that Astrology is bunk.

and are left with sentences like

If you come home early, you can watch TV.

Steve is cheerful and brave.

These sentences are declarative sentences and can be enclosed in special brackets like this;

Holmes and Watson were on the case.

2.1 LOGIC Connectives. 1. AND

Sentences refer to other sentences, and are therefore connected to other sentences. Think of the complex sentence;

This can be written as two simpler sentences

We can make life a little easier by writing a symbol for each sentence, like this;

P = <Henry was a king>

Q = <Henry has six wives>

so the sentence may be written, in symbols, as;P Q

You see that the symbol " " means AND. It’s the AND Connective.

The behaviour of all connectives is studied and summarized

using TRUTH TABLES. In these tables

0 represents FALSE

1 represents TRUE

Now, if we have one sentence, it can have a value 0 or 1 ; in other words it can be false or true. In this example wehave two sentences, P and Q. So there are 4 possible combinations;

P false / Q false
P false / Q true
P true / Q false
P true / Q true

This is what the truth table summarizes, on its left side.

P / Q
0 / 0
0 / 1
1 / 0
1 / 1

What goes in the right hand column? You must decide. Look at the second line, (0 1). Pis false and Q is true, so what about P Q … is that true or false? Well, if Henry was a king is false AND Henry had two wives is true, then Henry was a king and had two wives is false. So you must write a 0 there.

Thinking like this, fill in the truth table for the AND connective.

P / Q / P Q
0 / 0
0 / 1
1 / 0
1 / 1

Now you've done that, let's do it again, but this time using the electronic AND – Gate. Remember that a light ON is 1 is true and a light OFF is 0 is false. Make up this circuit; (using Multimedia Logic)

and using it, fill in the truth table for the AND connective.

2.2 The OR Connective.

In an advertisement for a job as entertainments manager at a well-known holiday camp in Worcester, the following sentence appeared;

"The applicant must be either a sportsman or a bishop".

So we have two atomic sentences;

P = The applicant is a sportsman

Q =The headmaster is a bishop.

Now, think out the truth table for the OR P v Q connective

P / Q / P Q
0 / 0
0 / 1
1 / 0
1 / 1

The last line might give you some trouble; it's a well-known problem, and we'll discuss it soon.

Now investigate the behavior of the electronic OR - Gate by building this circuit;

and finding the truth table;

P / Q / P Q
0 / 0
0 / 1
1 / 0
1 / 1

And check out that this agrees with the truth table you suggested above.

2.3 The NOT Connective.

Here we need only one sentence, e.g., P = Cows can fly. When preceded by the symbol ¬ the sentence is negated. So ¬ P means ‘it's not true that cows can fly’ or more simply stated, 'Cows cannot fly'. Work out by thinking the truth table for NOT;

P / ¬ P
0 / 1
1 / 0

and investigate the electronic NOT- Gate using this wiring;

and get the truth table

P / ¬ P
0
1

2.4 So can Cows Fly ?

Consider the sentence P = <Cows can fly

write down, in English, the meaning of the sentence P ~ P.

Now work out the truth table for it;

P / P ~ P
0
1

and check it by building up this circuit

andfrom this finding the truth table

P / P ~P
0
1

Now write down the meaning of the set of sentences and construct its truth table. You should recognize this as the tautology mentioned in class.

P / P ~P
0
1

Summary so far...You should have found that for each connective AND,OR, NOT, the truth tables you got by thinking about the meaning of the English sentences are identical to the tables got from the corresponding electronic chips. That's quite fantastic; it means electronic logic circuits may be used to make models of language.

2.5 De Morgan satisfies your English Teacher.

How often have you heard that you should avoid “double negatives”? See what I mean? Consider the sentences:

P = De Quincey was an author

Q = De Quincey smoked opium

Then the compound sentence < ¬P ¬Q> would read:

It is NOT true that De Quincey was an author> ANDIt is NOT true that he smoked opium>

or, more simply

De Morgan’ s theorems help us tosimplify this mess. He said

Using P and Q as given above write down the English for ¬

Now let’s test using electronic circuits if his theorem is OK. First build up ¬P ¬Q> and get its truth table;

P / Q / ¬P¬Q>
0 / 0
0 / 1
1 / 0
1 / 1

and compare the truth table with the one that you get from this circuit for ¬

P / Q / ¬ 
0 / 0
0 / 1
1 / 0
1 / 1

Aha! So they are identical. De Morgan had another theorem which said that

Choose twosentences P and Q;

and now write down the meanings of the sentences;

¬P ¬Q>

and

¬

Now check that the theorem is OK by building the two circuits using your simulator and comparing their truth tables;

First ¬P ¬Q>

P / Q / ¬P ¬Q>
0 / 0
0 / 1
1 / 0
1 / 1

Now ¬

P / Q / ¬
0 / 0
0 / 1
1 / 0
1 / 1

2.6 The Conditional - IF

One of the most fundamental truth functors in our daily language is the “IF”. Just think how many times in a day we say or think this “IF”. How on earth can we get this “IF” into digital electronics and formal logic. Well, let’s see.

Let’s take for example, the sentence set “If Fred. is at home, then he is asleep”. We may divide it into two atomic sentences;

P = <Fred is at home

Q = Fred is asleep

and we shall represent the IFfunctor by the arrow; P  Q

Now, you work out the truth table for the ‘IF’ by thinking, and discussing!

P / Q / P  Q
0 / 0
0 / 1
1 / 0
1 / 1

Here are some hints:

Line 4.No problem. Here Fred is at home and Fred is asleep.

Line 3.OK but vital. Fred is at home and Fred is not asleep.

Lines 1 and 2. In both, Fred is not at home. In one case he is asleep (perhaps on a park bench)

and on the other day he is not asleep. The pointis, could P  Qbe true here?

Linguists agree with this definition. It is false only in one case, and the definition leads to some interesting language paradoxes as you will see if there is time.

But how do we realize the IF with electronics? There is no 'if-Gate' defined in electronics. The way it is done is to look at the correct IF truth table, shown here. (I hope you got close to this).

P / Q / P  Q
0 / 0 / 1
0 / 1 / 1
1 / 0 / 0
1 / 1 / 1

Given this, it is possible to find a combination of AND, OR, NOT gates which will do the job. Well, of course we start with our “sum-of-products” approach producing the miniterms for each row outputting a “1” and OR-ing these terms. So, in the notation of digital logic, we have

Now we must simplify this using Boolean algebra. Don’t worry, I’ll give you this for free. Here goes:

This derivation is not too hard to understand. The second line uses the distributive theorem of Boolean Algebra. The third line uses the inverse theorem. The fourth line is tricky, This uses the distributive theorem to expand the sum. The fifth line uses the identity theorem once more. But the result is important and simple

Let’s test this out. We can easily build ¬P Q>, Wire it up as shown below.

Now get its truth table and verify this is the same as the Philonian Conditional, as given in the table above.

P / Q / P  Q
0 / 0
0 / 1
1 / 0
1 / 1

Yep! The truth tables are identical. The equality does sound reasonable, the sentences "If I am a boy, then I am male" is the same as "I am not a boy or else I am male"

3.CONSISTENCY

A set of declarative sentences is said to be “consistent” if one or more combinations can be found which is true. Think about this set of two sentences;

a) Holmes and Watson are on the case

b) Either Holmes was not on the case, or else Watson was.

They could be consistent if a combination can be found which is true. Can you think it out? You must say who was on the case, if possible.

3.1 The method of Consistency Checks.

Let's set out this example as a template for the method we shall use for all consistency checks. Take;

P = Holmes was on the case

Q = Watson was on the case

These sentences are derived from the set (a) and (b). Your first job is always to extract these simplest sentences from the complex set of sentences given in any problem.

Secondly, you must transcribe the sentences a) and b) in terms of your found atomic sentences P andQ.;

a)

b) <¬P Q>

Thirdly, to test for consistency, we AND these sentences, to look for any case where both a) and b) are true. So we form

¬P Q>

Fourthly, we make the electronic wire-up. The circuit is given below, but it is built up as follows;

1.build

build < ¬ P Q>
AND the outputs of 1 and 2

Now what to do? Make up the truth table by trying the various P’s and Q’s (0 and 1) and look for any

true combination of P and Q.

P / Q
0 / 0
0 / 1
1 / 0
1 / 1

If your circuit was correct, there should be a single true combination, P = 1, Q = 1. What does this mean? 'That the only consistent combination of P and Q is

a)Holmes is on the case

b)Watson is on the case

So we have proved that the original set of sentences (a) and (b) is consistent.

3.2 Watson and Holmes in Confusion

Test the consistency of thisset of sentences using the method above.

a) Holmes and Watson were on the case.

b) If Holmes was on the case, then Watson was not.

First find the atomic sentences P and Q;

Q =

Now transcribe the set of sentences a) and b) in terms of P and Q;

Now AND these sentences,

Now build the circuit and get the truth table. Draw your truth table and the circuit.

Did you expect this result? Look at the starting sentences and think out the answer.

3.3 Smiley's People.

Test the consistency of this set of sentences;

"Smiley is an English spy. Smiley is not both a Russian and an English spy. If Smiley is a Cad, then he is a Russian spy".

Transcription;

Truth Table:

P / Q / R
0 / 0 / 0
0 / 0 / 1
0 / 1 / 0
0 / 1 / 1
1 / 0 / 0
1 / 0 / 1
1 / 1 / 0
1 / 1 / 1

And build a circuit to verify your solution. If all has gone well, you should find only one consistent solution. Write this out in English

3.4Too many Detectives.

Test the consistency of these sentences;

"If Holmes. solved the crime, then Lestrade took the criminal. Of course, if the criminal escaped, then Lestrade did not take him. On this particular day, Holmes solved the crime, but the criminal escaped"

Transcription ;

Circuit

Write out in English any consistent solutions;

3.5Who does what and Where whom lives..

Here's a test where there are four sentences in the starting set. Your task, should you choose to accept it, is to find out who does what and who lives where.

“Either David or Warren live in Bromsgrove. Either Warren or Fozzy is a Muppet. Warren is not a Muppet and doesn't live in Bromsgrove!

Transcribe, build the Circuit and complete the truth table ,and write out in English any consistent solutions.

3.6Gunmen and Hostages.

Again, we have a set of four sentences, and we wish to find if there is/are any consistent solution(s) to this set.

"If the gunmen are tired, then they are on edge. If they are armed and on edge, then the hostages are in danger. The gunmen are armed but the hostages are not in danger."

Transcription;

Circuit;

and write out in English any consistent solutions;

4.TESTING THE VALIDITY OF ARGUMENTS

Here is an example of an ARGUMENT:

"Worcester Uni. students who are take Computing may use Room 113"

"I am in a Worc. Uni. Student and I take Computing"

"Therefore I may use the Room 113"

The first two lines are called the PREMISES of the argument, and the final line is the CONCLUSION.

How to proceed? Well, via a somewhat vicious route. We take the premises as given, but then we take the conclusion and negate (invert) it. Then we do a consistency check (as above). If we find any single consistent solution (with true premises and false conclusion) then clearly the argument is incorrect! Only is there is no consistent solution to this set of sentences (true premises and negated conclusion) do we conclude that the argument is correct. The method's called REDUCTIO AD ABSURDUM.

4.1 A Vicarious Example

Look at this argument;

"Either the vicar or the butler shot the "Earl. If the butler shot the Earl, the butler was not drunk. Unless the vicar is a liar, the butler was drunk. Therefore, either the vicar is a liar, or he shot the Earl.”

a) The first thing to do is as usual find the atomic sentences

P= <The vicar shot the Earl>

Q=<The butler shot the Earl>

R= <The butler was drunk>

S=<The vicar is a liar>

b)Now transcribe the Premises :

1st sentence

2nd sentence ¬ Q ¬ R >

3rd sentence < S R >

c)Now transcribe the NEGATION of the conclusion;

Concln.= <S P>

NEGATED concln. ¬ <S P>

d)Now AND all the sentences and wire up the circuit and get the truth table.

¬Q ¬R> <S R> ¬ <S P>

e)If all has gone well, you should have found the final column to be all O’s. This means there is no consistent solution. There is no case where the argument could be false, so it MUST be a VALID argument.

4.2 A Question of Dialect.

Test the following argument for validity;

"If Higgins born in Bristol, then Higgins is not a Cockney. Higgins is either a Cockney or an impersonator. Higgins is not an impersonator. Therefore Higgins was born in Bristol”.

P =

Q =

R =

Now transcribe, not forgetting to negate the conclusion and build up the circuit. You should find here that there is One Consistent solution.Write this out in English;This is a counterexample to the argument. Since it exists, then the argument was false.

4.3Problem for a Medic.

Check the Validity of this argument;

" If the boy has spots in his mouth, then he has measles. If the boy has a rash on his back, then he has heatspots. The boy has a rash on his back. Therefore he hasn't got measles."

Transcription (plus inverted conclusion), CircuitIf you find any counter-examples,then write these down in English:

4.4Three girlfriends too many!

Test the validity of this argument;

"If I love Tatiana but not Diana, then I love Maria. If I love Diana, then I love Tatiana. I love either Diana and Maria or else I love neither of these two. Hence I do not love any."

Transcribe (and negate conclusion). Make the circuit. Comment on the validity of the argument.

4.5 to 4.8 Crime and Punishment.

For these following sections, use the set of sentences

P = Fred has hit Joe with a stick

Q = Joe is a Slave

R = <Fred is liable for a 100 Rouble fine

S = Fred is liable for a 1000 Rouble fine>

You do not need to know that 100 Roubles will buy you a train ticket for a Worcester-London (like) journey, a good bottle of vodka. That’s semantics.

4.5 Fred Hit Joe.

“If Fred hit Joe with a stick then he is liable to a 1000 frank fine. Fred has not hit Joe with a stick. therefore, Fred is not liable to a 1000BF Fine".

Transcription (plus negation). Circuit. Comments on the Validity;

4.6 Can Fred hit a Slave?

"If Fred has hit Joe with a stick, then he is liable to a100 frank fine, unless Joe is a Slave. But Joe is not a Slave. therefore, either Fred is liable to a 100 frank fine, or else Fred has not hit Joe with a stick".

Transcription. Circuit. Comments on the Validity.

4.7Fred Hits a Slave.

"Only if, Joe is not a Slave is Fred, who hit him a stick, liable to a 100 frank fine. Joeis a Slave. therefore Joe is a Slave and Fred is not liable to a 1000 frank fine eventhough he has hit Joe with a stick"

5. ZAP THEM WITH LOGIC!

Here followeth, dear student, some classic thorny bushes of logic. They are in the form of arguments, which you should test using the validity method used in the previous section. Actually the first two bushes are not thorny at all.

5.1 Modus Ponens

Prove the validity of this argument: "If there is a fault, it will blow up. There is a fault to therefore it will blow up.

5.2 Modus Tollens

Prove the validity of this argument; "If there is a fault it will "blow up. It will not blow up. therefore there is no fault".

5.3Fallacy of Denying the Antecedent.

Think about (and investigate) the validity of the following argument; "If there is a fault, it will blow up. There is no fault. therefore it will not blow up”.

5.4 Fallacy of Affirming the Consequent.

Discuss the validity of this argument; "If there is a fault, it will blow up. It will blow up. therefore there is a fault"

5.5Fallacy of Conversion.

Check out the truth of the following argument; "If there is a fault then it will blow up. therefore if it blows up, then there is a fault!

5.6Hyperbole

Examine this argument. "I'll have a second cup. therefore I'll die if I don't have a second cup"

5.7 True Grit.

And, finally this one… "I'll play football tomorrow. therefore I'll play football tomorrow if I break my leg today".

6. EX CAMERA

6.1 Paper Tigers

Exercise 1. Express the following sentence sets as variables (P,Q,R,…) linked by truth functors ( ,¬ ,, and )