Hidden Logic
An Introduction to the Analysis of Informal Argument
©1999 Edward G. Rozycki

edited 3/30/12

Reconstructing Thought

The point of this article is to help the reader to state explicitly the train of thought that underlies a decision reached or a conclusion drawn. We make the reasoning explicit not only to examine it critically but to enable us, if we are so inclined, to reconstruct it as a stronger argument.

Suppose someone, let us call him John Smith, were to say,

(A) Since this is only her first year on the job, Mary can't be put in charge.
John, who is Mary's supervisor -- let us suppose -- has decided not to put Mary in charge. What argument could be made against this? What assumptions are being made by John? Can you make these assumptions explicit?

Clearly John must believe something like

First year people can't be put in charge.

This belief of John's may itself be a conclusion deriving from further considerations such as

No first year employees have sufficient experience.

Inexperienced persons cannot be put in charge.

We can understand John's reasoning even if we disagree with some of his assumptions. Perhaps some first year people do have sufficient experience. Mary may be one of them. Thus, we might disagree with John's decision to exclude her from leadership by pointing out that he is making some dubitable, if not clearly false, assumptions.

By way of contrast, suppose John says

(A) Since this is only her first year on the job, Mary can't be put in charge.
When we ask John if he is assuming that first year people haven't enough experience to be put in charge, he might respond that it isn't so much experience that is lacking, since even new hirees have a lot of skill, but rather that new employees lack the personal relationships with their coworkers necessary to leadership and that only develop with time. For this reason, they cannot be put in charge.

Again, we can understand John's reasoning, even if we might disagree with some of the premises of his argument, e.g.

New employees lack the personal relationships with their coworkers necessary to leadership.

These relationships only develop with time.

We might protest that Mary has a kind of charisma that can overcome such lack of familiarity, so that she should be put in charge none the less. Note that we are not questioning John's rationality, his ability to draw reasonable conclusions from what beliefs he uses as premises. Rather we are questioning the necessity or factuality of those premises.

Consider now this very different situation. John tells us, as he did before,

(A) Since this is only her first year on the job, Mary can't be put in charge.
When we ask him to explain his reasoning he replies,
Mushrooms can be poisonous, so you have to avoid jumping.
We might conclude that John was either joking, or refusing to answer our question, or mentally disturbed. We cannot intuit any relationship between mushrooms, jumping, and Mary's being a first year person.

What the exercises in this article are intended to do is to make explicit as an argument what we intuit to be the structure of reasoning behind the conclusions people come to. This "intution" rests on a capacity for informal reasoning that is shared by most healthy animals, even those not possessing language. We who possess language can reason more subtlely, and also, commit profounder errors in thought. The exercises herein will provide a framework for articulating the forms of valid reason and the types of reasoning errors we might encounter. Also, it will help the reader understand the difference between disputes over the form of reason and disputes over the content of the reasoning process. Just because we find a person's reasons to be wrong-headed or repugnant, doesn't mean that person has committed a logical fallacy, or is "irrational."

Articulating Our Intuitions

Most people, by the time they reach adulthood, have developed an informal sense of "logic." That is, they can more or less consciously -- if haphazardly -- "dope out" what went into the thinking of a person who presses for a particular point or proposes a specific undertaking. The point of this essay is to provide you with a means for consciously and methodically casting your "intuitions" into a common language -- an ancient, honored tradition -- which can be used to communicate your "intuitions" with others. In this way, you will discover how very much of a common species we are: human beings are a particular kind of reasoner. Our shared traits of intelligence are often obscured by the informal varieties of reasoning we normally encounter. This obscurity is often intensified by lack of training in a more formal manner of presentation.

The set of examples and exercises in this booklet will provide you with the means for analyzing informal arguments and evaluating them explicitly. The structure of the presentation is as follows:

a. The first section assumes you have some acquaintance with Venn diagrams. It demonstrates how to interpret them as statements involving implication, necessity and sufficiency.

b. The next two sections use Venn diagrams to define and illustrate the two basic argument forms, modus ponens and modus tollens.

c. Next we discuss the distinction between a valid argument and a sound argument and show how the fallacious reasoning is a matter of overlooking logical possibilities.

d. Finally, we deal with enthymemes, the elliptical, compressed and apparently incomplete forms of argument that make up much of our day-to-day attempts to persuade each other to action.

We will use our basic argument forms to help us reconstruct the reasoning underlying the enthymeme. This will make obvious -- and vulnerable -- the hidden premises of our enthymemic conclusion.

Interpreting Venn Diagrams as Argument Forms

Most of us have had some encounter with formal arguments, if only in our high school geometry course. Geometric proofs are forms of argument. The structure of such proofs is generally a list of premises -- called axioms, postulates and lemmas in Geometry --from which follows, hopefully, something called a conclusion -- Q.E.D. The problem is that hardly anyone attempts to persuade others by using such formal arguments. Actual argumentation in a natural setting appears to be more varied and complex than the forms we may have encountered in school.

We can deal with a great many informally stated arguments by working with only two argument forms. If you are familiar with Venn diagrams, the relationships that constitute valid, sound argument are easily represented by such diagrams. In the charts below, each Venn diagram is given interpretations that enable them to illustrate argument forms.

Venn Diagram
Necessity & Sufficiency
1a. All A is B

Some B is A.

e.g. All cats are animals.

A implies B

B does not imply A.

A is sufficient for B

B is necessary for A.

1b.  No A is B.

No B is A.

e.g. No dog is a chair.

A implies not-B

B implies not-A.

A is neither necessary nor sufficient for B.
1c.  Some A is B.

Some B is A.

e.g. Some chairs are wood.

(no implications) A is neither necessary nor sufficient for B.

It is important to note that implications are only possible for relationships 1a. and 1b. Such implications depend entirely upon terminological consensus. Reasoning depends upon implication, and implication only works to the extent to which consensus exists on how different terms relate to one another.

Modus Ponens

The first argument form is called modus ponens and is illustrated in the chart below.

Venn Diagram
Premisses and Conclusion
Alternative Form
2a. A implies B, and

x is in A;

therefore x is in B.

e.g. Every dog is a mammal, and Fido is a dog, therefore Fido is a mammal.

Let A = p.

Let B = q,

Restate the argument as

p implies q


therefore q.

Tell what is wrong with this:

(B) Some animals are cold-blooded and Fido is an animal, so Fido is cold-blooded.

Can you illustrate the fallacy using Venn diagrams? This brings up an important point. A logical fallacy is made when a possible option (a conceivable situation) is dismissed out-of-hand. What possible option has been dismissed by (B)?

There are two options in locating Fido in the Venns that relate being cold-blooded to being an animal: Fido is an animal is given. He may or may not be cold-blooded. The fallacy in the argument comes from ignoring the first option (Fido is an animal and not-cold-blooded) and concluding that because he is an animal, he is necessarily cold-blooded.

Venn Diagram
Option 1
Option 2

x = Fido, A = cold-blooded;
B = animal

Fido is an animal that is cold-blooded.

Fido is an animal but not cold-blooded.

The more terms we introduce into our reasoning, the greater the number of possible relations. For example, suppose we know X is an animal. Does this information allow us to draw any conclusions as to whether X is cold-blooded, or aquatic?


A = cold-blooded; B = animal;
C = aquatic

Given that X is an animal there are still several possibilities:
1. X is a cold-blooded, aquatic animal.

2. X is a warm-blooded aquatic animal.

3. X is cold-blooded, but not aquatic animal.

4. X is warm-blooded but not aquatic animal.

Since there are both cold-blooded and warm-blooded animals and also aquatic animals, non-aquatic animals and aquatic non-animals (e.g. water-lilies) then several possibilities present themselves. These are given above next to the Venn diagram.

Given the relationships indicated in the above diagram, which of the following are fallacious arguments?

a. Since X is warm-blooded, X is an animal.

b. Since X is non-aquatic, X is cold-blooded.

c. Since X is not aquatic, X is warm-blooded.

d. Since X is not-aquatic, X is not an animal.?

e. Since X is not an animal, X is not aquatic.

Exercise: Consider the relationships among leadership, charisma and experience. Draw a Venn diagram illustrating this relationship. Decide on the basis of that diagram which of the following statements about Mary are true:
a. Since Mary lacks charisma, she exhibits no leadership.

b. Since Mary has experience, she exhibits leadership.

c. Since Mary is charismatic, she exhibits leadership.

d. Since Mary is inexperienced, she exhibits no leadership.

e. Since Mary has experience and charisma, she exhibits leadership.

f. Since Mary exhibits leadership but no charisma, she lacks experience.

Modus Tollens

The only other argument form we need consider is called modus tollens. The chart below illustrates it.

Venn Diagram
Premisses and Conclusion
Alternative Form
2b.  A implies B, and

x is not in B, 

So, x is not in A.

e.g. Dogs are animals, and 

New York is not an animal, so

New York is not a dog.

Let A = p

Let B = q, 


p implies q

not q

therefore not-p.

With just these two forms, modus ponens and modus tollens, we can reconstruct the arguments behind a good deal of informal reasoning.

Soundness vs. Validity

Now we must observe a highly important distinction: that between a valid argument and a sound argument.

A valid argument, commonly called a "logical" argument is one which has the form of either modus ponens or modus tollens (or is a permissible combination or reiteration of any of the two -- see the next section.)
The reason we should be concerned that our arguments be valid is that invalid arguments prematurely dismiss possible options. Now, if our valid argument has true premisses -- and we may decide that they are true in a variety of ways this paper will not discuss -- the argument is not only valid, but sound. That is,
A sound argument is a valid argument with true premisses. The conclusion of a sound argument must be true.
We have a second important reason for constructing valid arguments: as long as our argument is valid, then, if our premisses can be demonstrated to be true -- by whatever means -- we can rest assured our conclusions are true. (Clearly, establishing premisses as true may be the bigger part of our task.)

It follows that even if the premisses of an argument are in doubt -- or even false -- the argument can be valid. This is extremely important. It means that we do not have to determine in advance the truth of our premisses to be assured we have not overlooked a conceivable possibility. Valid argument assures us that no necessary assumptions will remain hidden.

The following interrelationships between an argument's validity and soundness exist:

an argument may be both valid and sound

an argument may be valid but not sound

an argument may be neither valid and nor sound

In the chart that follows examples of each are given showing an argument form (valid and invalid).with examples below.
1) Valid, sound

(true premisses)

2) Valid, unsound (false premisses)
3) Invalid

(despite true premisses and conclusion)

4) Invalid,

(false premisses, true conclusion.

A implies B


therefore B

A implies B


therefore B

A implies B


therefore A

A implies B


therefore A

Any cat is a mammal.

A Manx is a cat;


a Manx is a mammal.

Any cat is a boat.

A Manx is a cat; therefore,

a Manx is a boat.

Any cat is a mammal.

A Manx is a mammal; therefore,

a Manx is a cat

Any cat is a canine.

A Manx is a canine; therefore,

a Manx is a cat.

We mentioned above that the trouble with invalid argument is that it hides conceivable possibility. Note that in the chart, 3) obscures the possibility that there are mammals that are not Manxes. This is why the truth of premisses alone, even when the truth of the conclusion has been ascertained, does not insure the validity of the argument. Truth without valid argument is at best good luck.

Combining for Longer Arguments

Above we defined a valid argument to be one with either a modus ponens or modus tollens form, or which combined or reiterated these forms. The simplest method is through chaining, where the conclusion of one argument becomes a premise for the next. The order of the premisses of a valid argument are not significant. Let's look at some examples.

modus ponens chain (compressed)
modus ponens-modus tollens chain (compressed)
A implies B

B implies C


therefore C.

A implies B

C implies not-B


therefore not-C

Any wolf is a canine.

Any canine is a mammal

therefore any wolf is a mammal.

Any dog is a canine.

A feline is not a canine.

A beagle is a dog.

therefore a beagle is not a feline.

Can you decompress the examples given above? (Hint: write each one as two arguments in sequence.)


Finally, we get to the point of the previous exercises and discussion: to examine forms of argument where premisses have been hidden. Consider again,

(A) Since this is only her first year on the job, Mary can't be put in charge.

The way we will proceed is to

1) Treat this claim as the conclusion of an argument.

2) Assume that a valid argument form underlies this claim. (Use your intuition!)

3) Reconstruct plausible premisses which together with the conclusion form a valid argument.

Since modus ponens is the easiest form to deal with, let's try this:
a. First year people can't be put in charge. (What kind of Venn diagram would this make?)

b. Mary is a first year person.

c. Therefore, Mary can't be put in charge.

This is simple enough, perhaps too simple, since if you challenge the conclusion, the person advocating it will probably show more ingenuity than our reconstruction indicates.

Let's try something a little more complex, yet more realistic.

a. First year people have insufficient experience for leadership positions. (Venn diagram?)

b. Putting someone in charge, even temporarily, is a leadership position.

c. Mary is a first year employee.

d. Therefore, Mary cannot be put in charge.

Basically what is being said here is that experience is necessary to leadership, that is leadership implies experience
Leadership implies Experience.
Also, first year people do not have experience , that is
First Year implies not Experience.
We see obviously that Mary is a first year person and being Put in Charge is a case of Leadership,

so if we look at the formulas we recognize a combination of modus ponens and modus tollens underlying the enthymeme.

Leadership implies Experience.

First Year implies not Experience.

Mary implies First Year

Put in Charge implies Leadership,

If we rearrange these and group them we see what's up:

Part A:

Mary implies First Year

First Year implies not Experience.

Clearly, Mary, being a first year person, has no experience. Let's continue:

Part B:

Put in Charge implies Leadership,

Leadership implies Experience.

But by part A, not-Experience, that is Mary has no experience. Therefore by modus tollens,
not-Put in Charge.
This may seem a roundabout way to explain what possibly went on in someone's head when they announced A). But unless someone could justify it in such a manner, the decision it announces would be considered whimsical, arbritrary, unreasoned. On the other hand, our analysis enables us to identify the assumptions underlying the judgement so as to bring them up for questioning.

Try to work out a similar analysis for

(C) If you want to get ahead, go to college!

Where do we go from here? It depends upon what we want to do. Notice, for example. the some of the premises are slogans. We might want to criticize them as obscuring important distinctions or imputing false causes. Or, rather than criticize, we might want to reconstruct the argument using different premises so as to make the conclusion stronger. The analysis we have just completed is a preparatory step to other undertakings.

Other examples for analysis (work these out first intuitively)

(D) Sam can't write curriculum for Black Studies; he's Puerto Rican.

(E) Federal money has to be spent to serve everyone, not just a religious sect.

(F) Your students won't improve unless you raise your expectations.

(G) Gun control will reduce violent crime.

(H) How can we be surprised about violence in the schools considering how much violence there is on TV?

(I) Show me a school with problems and I'll show you one lacking leadership!

(J) Only when humankind finally gets serious about peace will wars stop.

(K) Since real education involves our deepest values, we might as well close down all non-religious schools.

(L) Because so many kids today lack direction, our schools must incorporate Values Education into the curriculum.

(M) As international mathematics tests have shown, Japanese education is better than American.

(N) SAT declines indicate a general degradation of IQ probably the result of TV.

(O) Money is not the issue, since many schools have shown no improvement despite higher educational expenditures.

(P) SAT decreases over the past twenty-five years point to the failure of our schools.

(Q) Want to know why there's violence in the schools? Testosterone!