We have discussed many logical fallacies in this book. Now we will learn how to use and recognize good reasoning by introducing five traditional valid argument forms. For centuries, logicians and scientists have used these forms as templates for structuring valid arguments. These forms are: modus ponens, modus tollens, disjunctive syllogism, hypothetical syllogism, and constructive dilemma. Familiarity with these arguments is essential in the study of logic. They should be learned and remembered as prototypes of valid argumentation.
Modus ponens is made up of two premises and a conclusion. It begins with a conditional statement as its first premise. A conditional statement is an “if-then” statement. For example: “If I drop the cup of coffee then it will spill.” The “if” component is called the antecedent (“If I drop the cup of coffee . . .”), and the “then” component is called the consequent (“. . . then it will spill”). The second premise of modus ponens always affirms the antecedent of the conditional statement. For example:
If the printer prints in color, then it is out of black ink. (First premise)
The printer is printing in color. (Second premise)
Therefore, the printer is out of black ink. (Conclusion)
In this example, we see clearly that the antecedent of the first premise is affirmed in the second premise. In the conclusion, we see another characteristic always found in modus ponens: the consequent of the conditional premise is affirmed. In fact, modus ponens is translated from the Latin as the “mode of affirming.” In symbolic form, modus ponens looks like this:
If A then B.
A.
Therefore, B.
Modus ponens always follows this pattern. Here are some more examples of modus ponens.
1. If Bertrand Russell was a philosopher then he was an intellectual.
Bertrand Russell was a philosopher.
Therefore, Bertrand Russell was an intellectual.
2. If Mr. Jones murdered the bank teller in the botched robbery, then Mr. Jones is guilty and should be punished.
According to the video evidence, Mr. Jones did murder the bank teller in the botched robbery.
So, Mr. Jones is guilty and should be punished.
3. If I see the chef sneeze into the spaghetti, then I will not eat it.
I see the chef sneezing into the spaghetti.
I will not eat the spaghetti.
In all of these examples, one can clearly see the form of modus ponens. We should memorize this valid argument form and use it as a guide for structuring our own arguments.
This valid argument form is similar to modus ponens. However, where modus ponens affirms the antecedent of the conditional statement, modus tollens denies the consequent of the conditional statement. This is why it is translated from the Latin as the “mode of removing.” For example:
If the candle is lit, then there is a flame.
There is no flame.
Therefore, the candle is not lit.
Here, the consequent of the conditional statement is denied (or negated) in the second premise. And from this, it logically follows that the conclusion is true. Modus tollens takes the following symbolic form:
If A then B.
Not B.
Therefore, not A.
Here are some more examples of modus tollens:
1. If Steven was born in the United States, then he is a citizen of the United States.
Steven is not a citizen of the United States.
Therefore, Steven was not born in the United States.
2. If Sally has a cat, then she is a pet owner.
Sally is not a pet owner.
Therefore, Sally does not have a cat.
3. If the magician’s assistant had been decapitated, then she would be dead.
The magician’s assistant is not dead.
Therefore, the magician’s assistant was not decapitated.
In these examples we see clearly the modus tollens pattern. Modus tollens, like modus ponens, is a valid and coherent logical form that should be remembered and employed.
The disjunctive syllogism is perhaps the easiest valid form to remember. This form always has a disjunctive statement—an “either/or” statement—as one of its premises. For example:
Either Bach wrote Mass in B Minor or Wagner wrote it.
Wagner did not write Mass in B Minor.
Therefore, Bach wrote Mass in B Minor.
In the above, we are presented with two options in the first premise: Bach or Wagner. If these really are the only two choices, and if Wagner did not write Mass in B Minor, then the conclusion follows logically from the premises. If there were other choices that were intentionally left out, then this would be the fallacy of bifurcation.
Here we are reminded that all valid argument forms rest on the assumption that the premises are true. That is, the forms provide the structure for valid arguments, but in reality, the content of the forms may or may not be true. For the sake of learning the traditional argument forms and how to create good arguments, we will take the examples herein at face value. And, on this assumption, the above argument is valid.
The disjunctive syllogism has the following symbolic form:
Either A or B.
Not B (or, not A).
Therefore, A (or, B).
From this symbolic form we can clearly see that it does not matter which side of the disjunctive statement is denied. If one option is negated, then the other must be true. Consider these examples.
1. Either Holmes solved the crime or Dr. Watson solved the crime.
Dr. Watson did not solve the crime.
Therefore, Holmes solved the crime.
2. Either you go to college or you get a job.
You did not get a job.
So, you will go to college.
3. Truth is either a construct of institutions or it is the correspondence of language to reality.
Truth is not a construct of institutions.
Therefore, truth is the correspondence of language to reality.
The disjunctive syllogism is a valid form of reasoning when there are only two options to choose from. If possible, when creating good arguments, we should try to use forms like the disjunctive syllogism.
The hypothetical syllogism is made up of three conditional statements: one in each of the two premises and one in the conclusion. This is also known as the chain argument because it links together all components of the premises and conclusion. For example:
If I move to the Bay Area, then I will need an income.
If I need an income, then I will need a job.
So, if I move to the Bay Area, then I will need a job.
The symbolic form of the hypothetical syllogism is as follows:
If A then B.
If B then C.
So, if A then C.
This argument form is quite easy to recognize because it consists entirely of conditional statements. Like the other valid argument forms, the hypothetical syllogism always provides an argument with good structure, but not necessarily true content. For the purpose of illustrating the hypothetical syllogism, we will assume the premises are true in the following three final examples of this form.
1. If community colleges continue to cut classes, then fewer Americans will have access to education.
If fewer Americans have access to education, then poverty will increase.
Therefore, if community colleges continue to cut classes, poverty will increase.
2. If I rob the bank, then I may kill someone.
If I kill someone, then there is a possibility that I will spend my life in prison.
So, if I rob the bank, then there is a possibility that I will spend my life in prison.
3. If the atheists are correct, then there is no God.
If there is no God, then intelligent design is false.
Therefore, if the atheists are correct, then intelligent design is false.
This is the final of the five valid argument forms we will discuss. The constructive dilemma uses both disjunctive and conditional statements. It also is the only valid argument form that uses three premises instead of two. Here is an example:
We will either rent or buy a home.
If we rent, then we will need to move.
If we buy, then we will stay.
So, we will either move or we will stay.
The symbolic form of this constructive dilemma is as follows:
Either A or B.
If A then C.
If B then D.
So, either C or D.
Here we see that the constructive dilemma begins with a disjunctive statement as its first premise. In the second and third premises, it contains conditional statements. Finally, it concludes with a disjunctive statement that affirms the consequents of the second and third premises. Here are some more examples of the constructive dilemma:
1. Either we celebrate Hanukkah or we celebrate Christmas.
If we celebrate Hanukkah, then we will drive to Los Angeles.
If we celebrate Christmas, then we will drive to San Francisco.
Therefore, we will either drive to Los Angeles or to San Francisco.
2. Humans either have instrumental value or intrinsic value.
If humans have instrumental value, then their worth is measured by their efficiency.
If humans have intrinsic value, then they have worth regardless of how efficient they are.
So, either human worth is measured by efficiency or humans have worth regardless of their efficiency.
3. Either Sigmund Freud was correct or C.S. Lewis was correct.
If Sigmund Freud was correct, then the idea of God is simply a projection of unconscious desires.
If C.S. Lewis was correct, then the theism of the Bible is true, and God exists independently of our unconscious desires.
So, either God is a projection of our unconscious desires, or God exists independently of our unconscious desires.
In these examples, we clearly see the constructive dilemma employed. It is an especially helpful argument form to use when making plans in everyday life, or when debating abstract theories.
Like the other four valid argument forms we have discussed, the constructive dilemma is simply a template to be used when creating good arguments. Of course, it is not always possible to use one of these five forms. However, they are vital to the study of logic, and useful when one is seeking solid argument structure.