3D  Syllogisms


 

p

~p

p Ú ~p

T

F

T

F

T

T

If you say “It is raining or it is not raining” then you speak the truth. The same can be said of any other statement having the symbolic form “p Ú ~p”. Indeed, the truth table at the right confirms that this statement is true whether p itself is true or false. Such a compound statement, necessarily true just because of its logical form, is called a tautology. In a truth table, the column under a tautology consists of all T's.

p

q

p Ù q

(p Ù q) → p

T

T

T

T

T

F

F

T

F

T

F

T

F

F

F

T

 

The symbolic statement “(p Ù q) → p” likewise is a tautology, as demonstrated by the truth table at the left. For instance, if you say

“If it is rainy and windy, then it is rainy” ,

again you have spoken the truth, regardless of the weather outside. The truth table demonstrates that, whether it is raining or not raining, and windy or not windy, your statement is still a true one.

 

p

~p

p Ù ~p

T

F

F

F

T

F

At the other extreme, there are certain compound statements that are guaranteed to be false only because of their logical structure; a statement of this type is said to be a contradiction. The simplest example is the symbolic statement “p Ù ~p”. A column of a truth table under a contradiction will consist of all F's, as we observe in the table at the right. If you were to say

“It is raining and it is not raining” ,

then certainly no one in his right mind would believe you.

p

q

~p

p Ù q

q → ~p

(p Ù q) Ù (q → ~p)

T

T

F

T

F

F

T

F

F

F

T

F

F

T

T

F

T

F

F

F

T

F

T

F

 

A less transparent example of a contradiction is the symbolic statement

(p Ù q) Ù (q → ~p) ,

whose truth table appears at left. An example of this construction is the statement “It is rainy and windy, and if it is windy then it is not rainy.” As the truth table shows, no matter what the weather is doing outside this statement is false, and hence a contradiction. Indeed, any compound statement of this general form is automatically false.

A symbolic statement that is neither a tautology nor a contradiction is a contingency; such a statement can be either true of false, depending on the truth values of the simpler statements from which it is constructed. In a truth table, the column under a contingency will contain at least one T and at least one F. As we have seen, the conjunction p Ù q, the disjunction p Ú q, and the implication p → q all are contingencies, as their truth values depend on the truth values of p and q. In fact, most of the statements we make in everyday life are contingencies. It would make no sense to speak in tautologies, as you would only be pointing out the obvious, and if you spoke in contradictions you would quickly lose all credibility.

 

Joan

Joan at home, sleeping

Let us now examine the following sequence of statements, or argument :


If Joan is at home, then she is sleeping.

Joan is at home.

Therefore, Joan is sleeping.


The person making this argument expects the listener to accept the first two statements as true; these statements are called the hypotheses, or premises, of the argument. The word “therefore” suggests that the third statement follows logically from the first two - that is, if you accept the first two statements and use your God-given common sense, then you have no choice but to accept the third. This third statement forms the conclusion of the argument. But is the speaker correct in the implied assertion that the third statement follows necessarily from the first two? To investigate this question we introduce simpler statements,

p : Joan is at home

,

q : Joan is sleeping .

 p → q 

 p 

--------

 \

 

Then the argument can be briefly summarized as in the table at the left. (The dotted line separates the premises of the argument from the conclusion, while the three dots forming a triangle preceding q represent “therefore”.) As the speaker expects us to accept both premises, it is assumed that the conjunction (p → q) Ù p is true. Moreover, as this conjunction is supposed to imply the conclusion q, we can write the entire argument in symbolic form as the implication

[(p → q) Ù p] → q .

 

p

q

p → q

(p → q) Ù p

[(p → q) Ù p] → q

T

T

T

T

T

T

F

F

F

T

F

T

T

F

T

F

F

T

F

T

The accompanying truth table for this argument, exhibited at the right, has only T's in the final column; therefore, the whole line of reasoning is a tautology - that is, it is a compound statement true under any circumstances. We say that such an argument is valid. If we denote the hypotheses of the argument, (p → q) Ù p, as H, and the conclusion q as C, then the above argument becomes the implication H → C. As this implication, a tautology, is always true, it is impossible that H be true and C be false - thus if you accept the hypotheses as true you must accept also the conclusion.

It should be emphasized that a valid argument does not guarantee that the conclusion of the argument is true; it guarantees only that the conclusion follows in a logical way from the hypotheses. But if you do not accept the hypotheses of an argument, then you are in no way obligated to accept its logical conclusion. For instance, if in our example you know for a fact that Joan is not at home but is out shopping, then obviously you will not accept the hypotheses of the argument, and consequently you are not bound to the conclusion. Likewise, if you know that Joan has much to do today and would not be sleeping even if she were at home, then again you will not accept the hypothesis.

In an implication p → q, the statement p is called the antecedent, and q the consequent. The symbolic line of reasoning just illustrated is referred to as affirming the antecedent, while the fancy Latin name is modus ponens. Affirming the antecedent is quite common in everyday conversation, and is easily recognized.


example 1

Here are some valid arguments, each an example of affirming the antecedent :


the boss

 

a)

If it is raining, Mel took his raincoat. It is raining. Therefore Mel took his raincoat.

b)

Whenever Amy does not wear glasses, she cannot read the blackboard. Amy is not wearing glasses. Consequently, she cannot read the blackboard.

c)

Tim is driving, and Tim drives only if he is sober. I must conclude that Tim is sober.

d)

If the moon is made of green cheese, then the earth is flat. The moon is made of green cheese; therefore the earth is flat.

e)

If the boss did not eat breakfast, he is in a bad mood. The boss missed breakfast - so stay out of his way!


Once again, as demonstrated in d), a false hypothesis and/or conclusion does not necessarily negate the validity of an argument.




 

lady with umbrella

A syllogism is an argument consisting of two statements, called the hypotheses or premises, and a third statement, the conclusion, necessarily following from the premises. Affirming the antecedent is but one example of a form that a syllogism may assume. For a second example, consider the argument


If it rained on the way, the umbrella is wet.

The umbrella is not wet.

Therefore, it did not rain on the way.


 p → q 

 ~q 

--------

 \ ~p 

 

If we introduce the simpler statements

p : it rained on the way

,

q : the umbrella is wet ,

then this argument takes the symbolic form

[(p → q) Ù ~q] → ~p ,

as outlined in the table at the left. This line of argument is called denying the consequent, while its Latin description is modus tollens. In the exercises you are asked to confirm the validity of “denying the consequent” by using a truth table to confirm that the argument is a tautology.


example 2

Here are some valid arguments, each an example of denying the consequent :


night heron

 

a)

If the Wahine played well, they won the game. The Wahine lost the game. Therefore, they did not play well.

b)

If she loves me, she is true to me. She is not true to me. Thus, sadly, she does not love me.

c)

When the night heron is young, its feathers are not yet gray. This night heron has gray feathers; therefore, it is a mature bird.

d)

If Mary is not yet 16, then she has no driving license. However, as Mary has a driving license, she must be already 16.




Thus far our syllogisms have been constructed from only two simple statements p and q. For a syllogism built from more than two statements, consider the following argument:


 

catching a pass

If the pass is good, the receiver scores.

If the receiver scores, we win the game.

Therefore, if the pass is good we win the game.


This argument can be written in terms of the three statements


p : the pass is good ,

q : the receiver scores ,

r : we win the game .


 p → q 

 q → r 

-----------

 \ p → r 

 

Two implications form the hypotheses of the argument, and likewise the conclusion is another implication. The line of reasoning, outlined in the table at the left, has the symbolic representation

[(p → q) Ù (q → r)] → (p → r) .

This general line of argument is called transitive reasoning.

In order to verify that transitive reasoning is valid, we must construct a truth table involving three statements p, q, and r. The left three columns of the table list all possible combinations for truth values of p, q, and r. In the remaining columns we build up the entire argument a step at a time.


p

q

r

p → q

q → r

(p → q) Ù (q → r)

p → r

[(p → q) Ù (q → r)] → (p → r)

T

T

T

T

T

T

T

T

T

T

F

T

F

F

F

T

T

F

T

F

T

F

T

T

T

F

F

F

T

F

F

T

F

T

T

T

T

T

T

T

F

T

F

T

F

F

T

T

F

F

T

T

T

T

T

T

F

F

F

T

T

T

T

T


Since the last column consists of all T's, the corresponding symbolic statement heading that column is a tautology; thus transitive reasoning is valid.


example 3

Each argument below is an example of transitive reasoning, and hence valid :


a)

If the spider has a red spot, it is a black widow. If the spider is a black widow, it is poisonous. Therefore, if the spider has a red spot, it is poisonous.

 

black widow spider

b)

If 2x+1 = 7, then 2x = 6. If 2x = 6, then x = 3. Therefore, if 2x+1 = 7, then x = 3.

c)

If you smoke marijuana, you will use cocaine. If you use cocaine, you will become a drug addict. Therefore, if you smoke marijuana you will become a drug addict.

d)

All ants are insects. All insects will bite you. Therefore, all ants will bite you.


(In argument d), we can rephrase all statements as implications; for example, the first statement of d) is equivalent to “If it is an ant then it is an insect.”)




Another valid reasoning pattern, although it does not quite fit the pattern of a syllogism, is the symbolic argument

(p → q) → (~q → ~p)  .

 p → q 

--------------

 \ ~q → ~p 

 

This argument, as outlined at the left, asserts that if an implication is true, then necessarily also its contrapositive is true. The assertion comes as no surprise, as we are aware that an implication and its contrapositive have always the same truth value. This line of reasoning is really just a variant of denying the consequent; in the exercises you are asked to verify with a truth table that it is valid. Here are examples of this reasoning pattern:


nurse  

a)

If you are a nurse, then you wear a uniform. Therefore, if you do not wear a uniform you are not a nurse.

b)

If a mosquito bites you then it is female. Therefore, if a mosquito is male it will not bite you.

c)

If you drive legally then you have insurance. Thus, if you have no insurance you drive illegally.

d)

If Tom is not here then he is sick. Hence, if Tom is well, he is here.

e)

If the radio is not plugged in, it does not play. Therefore, if the radio plays, it is plugged in.

f)

If Mary is busy she does not answer her phone. Consequently, if she answers her phone she is not busy.


  Sylvia

Certainly not all arguments are valid ones. We next look at a few invalid reasoning patterns. Consider for instance the following argument:

Sylvia exercises daily.

Therefore, Sylvia exercises daily and eats well.

If we designate statements p and q as

p : Sylvia exercises daily     ,       q : Sylvia eats well ,

then the argument takes the symbolic form

p → (p Ù q)  ,

 p 

-----------

 \ p Ù

 

 

p

q

p Ù q

p → (p Ù q)

T

T

T

T

T

F

F

F

F

T

F

T

F

F

F

T

as outlined at the left. But of course the logic here is faulty, as it cannot be assumed that a person who exercises daily must also eat well. We can expose the fallacy in the argument by constructing its truth table, as displayed at the right. The last column shows that the symbolic statement of the argument is not a tautology, but rather that it is false in the event statement p is true and q is false. In this situation, Sylvia exercises daily but does not eat well, making the hypothesis of the argument true but its conclusion false.

Another simple example of an invalid reasoning pattern is the argument

 p Ú

--------

 \

 

(p Ú q) → p  ,

as outlined at left and illustrated with the statements

Sylvia exercises daily or eats well.

Therefore, Sylvia exercises daily.

You should be able to construct a truth table to demonstrate that this symbolic argument is not a tautology. What habits of Sylvia would make the hypothesis of the argument true but its conclusion false?

 p → q 

 q 

--------

 \

 

One invalid reasoning pattern is so often wrongly used that it has a name - the fallacy of affirming the consequent. This pattern, seen at left, arises from the mistake of confusing an implication with its converse. The symbolic representation of this fallacious argument is

[(p → q) Ù q] → p  .

The first hypothesis asserts that p implies q, but not that q implies p; thus the second hypothesis, q, cannot be used to infer anything about p.

For an illustration of this bogus reasoning pattern, consider the argument

  damaged car

Out of Gas ?

If your car is out of gas, then it does not run.

Your car does not run.

Therefore, your car is out of gas.

The statements p and q here are

p : your car is out of gas,

q : your car does not run.


 

p

q

p → q

(p → q) Ù q

[(p → q) Ù q] → p

T

T

T

T

T

T

F

F

F

T

F

T

T

T

F

F

F

T

F

T

As the last column in the truth table at the right confirms, this argument fails when p is false and q is true - or, in this particular example, when the car is not out of gas but still it does not run.

A slight variation on the fallacy of affirming the consequent is the equally invalid reasoning pattern

(p → q) → (q → p) ,

 p → q 

------------

 \ q → p 

 

as outlined at left. You can check with a truth table that this argument fails in the same circumstance as the previous one - when p is false and q is true. An illustration of this ill-advised pattern is the argument


If your car is out of gas, then it does not run.

Therefore, if your car does not run it is out of gas.


 p → q 

 ~p 

--------

 \ ~q 

 

The fallacy of denying the antecedent is another invalid reasoning pattern, arising from confusing an implication with its inverse. This pattern has the symbolic formulation

[(p → q) Ù ~p] → ~q  .

The first hypothesis asserts that p implies q, but it does not suggest that ~p implies anything whatsoever. Thus the second hypothesis, ~p, cannot be used in conjunction with the first to draw any conclusion. A truth table confirms that the argument is not a tautology, and thus invalid. A variation on this reasoning pattern, equally invalid, is the argument

 p → q 

--------------

 \ ~p → ~q 

 

(p → q) → (~p → ~q) .

Both these arguments, outlined at left, make the unwarranted assumption that an implication and its inverse mean the same thing - which as we know is not the case. Below are examples of these invalid reasoning patterns:


ballerina  

a)

If Jan is a ballerina, she is graceful. Jan is not a ballerina. Therefore, Jan is not graceful.

b)

If the child is not crying, it is asleep. The child is crying. Consequently, the child is awake.

c)

If the dog is barking, the neighbor is not happy. The dog is not barking; hence the neighbor is happy.

d)

If the bear is angry it is dangerous. Therefore, if the bear is not angry it is not dangerous.

e)

If the team does not play hard it will lose. Thus, if the team plays hard it will win.

f)

If the phone is off the hook, you will not get a call. So if the phone is on the hook, you will get a call.


We have discussed only a few of the common reasoning patterns that people use - some valid and some invalid. There are quite a few other patterns we could mention, but the goal here is only to offer a brief glimpse of the study of logical reasoning. Perhaps the most striking thing about these patterns is that most of the time we are not aware of them - our minds are wired to think according to these rules even though we hardly ever stop to think about it. It is somewhat like walking down the street - there are certain things you must do to keep your balance and walk straight and not fall down, but you don't think about them, and probably don't even know what most of them are!

For convenience the main reasoning patterns we have discussed are listed in the following table. The ones in the first row, labelled with V and a number, are valid, while those in the second row, labelled with I and a number, are invalid.


V1.  

 p → q 

 p 

--------

 \

 

V2.  

 p → q 

 ~q 

--------

 \ ~p 

 

V3.  

 p → q 

 q → r 

-----------

 \ p → r 

 

V4.  

 p → q 

---------------

 \ ~q → ~p 

 

I1.  

 p → q 

 q 

--------

 \

 

I2.  

 p → q 

------------

 \ q → p 

 

I3.  

 p → q 

 ~p 

--------

 \ ~q 

 

I4.  

 p → q 

---------------

 \ ~p → ~q 

   


EXERCISES 3D


  1. Use a truth table to determine whether the symbolic statement is a tautology, a contradiction, or a contingency.
    1. p → p
    2. p → ~p
    3. p → (p Ù q)
    4. (p Ù q) Ú ~q
    5. [(p Ú q) Ù ~p] Ù ~q

  2. Confirm with a truth table that the following are valid reasoning patterns, by verifying that each statement is a tautology.
    1. [(p → q) Ù ~q] → ~p  (denying the consequent)
    2. (p → q) → (~q → ~p)
    3. [(p Ú q) Ù ~p] → q
    4. [(p → q) Ù (q → r) Ù p] → r

  3. Use a truth table to show that the argument is not a tautology, and hence invalid.
    1. (p Ú q) → p
    2. (p → q) → (q → p)
    3. [(p → q) Ù ~p] → ~q  (denying the antecedent)
    4. (p → q) → (~p → ~q)

In the remaining problems, you are to determine whether the argument is valid or invalid. The last statement in each argument is to be considered the conclusion, and the preceding statements the premises. Remember, whether you agree or not with the premises or conclusion of an argument is irrelevant; you are to accept the premises, and test whether the conclusion then must follow logically.


      ship
  1. If the sails are full, then the ship is moving.
    The sails are full.
    Thus the ship is moving.

  2. If today is Tuesday, the ship is in port.
    The ship is in port.
    Thus it must be Tuesday.

  3. You cannot sail if you have no sailor hat.
    You can sail.
    Consequently, you have a sailor hat.

  4. If the captain is on board, the ship is leaving.
    The captain is not on board.
    Hence the ship is not leaving.

  5. The boat is stopped only if the wind is still.
    Thus, if the wind is still then the boat is stopped.

  6. All fishing boats have a motor.
    Therefore, if the boat has no motor it is not a fishing boat.

  hanging clothes
  1. If it does not rain, the clothes will dry.
    Thus, if it rains the clothes will not dry.

  2. If the pants were dirty they were washed.
    The pants were washed.
    You must agree that the pants were dirty.

  3. The pants are dry if it did not rain today.
    Consequently, it did not rain today if the pants are dry.


  1. It did not rain today if the sun was out.
    Therefore, if the sun was not out it rained today.

  2. If Jeb cleaned the barn his pants are dirty.
    If Jeb's pants are dirty, Gladys will wash them.
    Therefore, if Jeb cleaned the barn Gladys will wash his pants.

      girl and dog
  3. If Jenny went to the beach, she took Rags.
    Jenny did not take Rags.
    Thus Jenny did not go to the beach.

  4. If Rags is wet, he is cold.
    Rags is cold.
    Sadly, then, Rags is wet.

  5. If Rags is cold he is wet.
    It must be that Rags is dry if he is warm.

  6. If the dog is lost, he is afraid.
    The dog is not lost; thus he is not afraid.

  7. Rags does not swim if the weather is not warm.
    Therefore, if Rags swims then the weather is warm.

  8. If Rags is cold, he will be sick.
    If Rags is wet, he is cold.
    It follows that Rags will be sick if he is wet.

  mongoose
  1. If you are quiet you will see the mongoose.
    You are not being quiet.
    Too bad, you will not see the mongoose.

  2. If a mongoose is friendly it will eat from your hand.
    This mongoose will not eat from your hand.
    I guess this mongoose is not friendly.

  3. A frightened mongoose runs away.
    Therefore, a mongoose that runs away is frightened.


  old horse
  1. The horse sleeps whenever it is tired.
    The horse is tired.
    Doubtless, then, the horse is sleeping.

  2. A black horse will not sleep on duty.
    Therefore, a horse sleeping on duty is not black.

  3. If the mare is old and gray, it ain't what it used to be.
    This mare ain't what it used to be.
    You must concur with me that the mare is old and gray.

  4. If the horse is young, it is full of energy.
    But the horse is old.
    That is why the horse is not full of energy.

  mallard duck
  1. If a mallard is brown, it is female.
    Consequently, a male mallard is not brown.

  2. Only male mallards have a green head.
    This mallard has a green head.
    By golly, this mallard has to be a male!

  3. Every mallard with a yellow bill is male.
    All male mallards have a green head.
    Thus, a mallard has a green head if it has a yellow bill

  4. A female mallard has an orange bill.
    Therefore, a male mallard does not have an orange bill.


  moon and clouds
  1. The wolves howl when the moon is full.
    The moon is full tonight; so the wolves will howl!

  2. The wolves howl only if the moon is full.
    The wolves are howling mournfully.
    You can bet that the moon is full.

  3. If the moon is full then the frogs croak.
    Accordingly, when the frogs croak the moon is full.

  4. Persuant to the frogs croaking, the wolves howl at the moon.
    The wolves are quiet tonight.
    Needless to say, the frogs are not croaking.

  baseball players
  1. If the foot was on the bag, the runner is out.
    Thus, the foot was on the bag if the runner is out.

  2. If the runner beat the throw, the runner is safe.
    The runner did not beat the throw.
    We have to conclude that the runner is out.

  3. If the runner was fast, he beat the throw.
    If he beat the throw, he is safe.
    Thus, if the runner was fast he is safe.


  1. If the ball was caught, the runner is out.
    Therefore, if the ball was not caught the runner is safe.


In the following problems, write the argument symbolically. Then use a truth table to determine whether the argument is valid or invalid.


      Siamese cats
  1. We are Siamese, if you please.
    We are Siamese, if you don't please.
    Therefore, we are Siamese.

  2. We are Siamese, or we are tigers.
    We are not Siamese.
    So watch out - we are tigers!

  3. If we are Siamese, then we please.
    If we are Siamese, then we don't please.
    So you see, we are not Siamese!