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# The Basic Concepts of Logical Theory

© 2011 By Paul Herrick

### A BRIEF OVERVIEW OF LOGICAL THEORY

### 1. Some Basic Concepts of Logic

### Argument

Logical theory begins with the concept of an argument. An “argument,” as the word is used in logic and in intellectual contexts generally, is *reasoning that has been put into words*. When you put your reasoning into words, you produce what logicians call an “argument.” Simple enough, but for the purposes of logical theory, a more precise definition is needed. Most logic textbooks include a more detailed definition, usually one that sounds much like this:

An argument is one or more statements, called premises, offered as a reason to believe that a further statement, called the conclusion, is true, that is, corresponds to reality.

It is true that in some contexts we use the word “argument” differently, to refer to people angrily yelling at each other, or to people having a heated emotional dispute. But in logic, and in academic and intellectual contexts generally, the word just means “one or more premises offered as reasons or as evidence for the truth of a conclusion.”

When we listen to an argument, it is sometimes difficult to tell which statements are premises and which statement is the conclusion. This is why the English language contains what logicians call “argument indicator words.” To tell your audience that you are drawing your conclusion, introduce your statement using a word or phrase such as “therefore,” “in conclusion,” “thus,” “consequently,” and so on. To indicate a premise, introduce a statement using words such as “because,” “since,” “for the reason that,” and so on. Premise and conclusion indicator words help your audience follow the “flow” of your reasoning.

### Deductive vs Inductive Arguments

Following Aristotle, logicians divide all arguments into two broad types: deductive arguments and inductive arguments.

Adeductiveargument is any argument that aims to show that its conclusionmustbe true. In other words, a deductive argument aims to conclusively establish its conclusion. The implicit or explicit claim, in a deductive argument, is therefore that if the premises are true then the conclusion is completely certain to be true as well.

Aninductiveargument is any argument that aims to show that its conclusion is probably true although not certainly true. An inductive argument, in other words, claims (in so many words) only that if the premises are true, then the conclusion is likely or reasonable (but not certain).

You can explicitly tell your audience that your argument is deductive by introducing your conclusion with wording such as “therefore it must be that,” or “it necessarily follows that,” or “therefore it is certain that,” or “it is conclusively proven that,” and so on. These phrases are called “deductive indicators.”

You tell your audience that your argument is inductive by introducing your conclusion with wording such as “therefore it is *probably* the case that,” or “it is *likely* that,” or “therefore it is reasonable to conclude that,” and so on. These phrases are called “inductive indicators.”

### Valid, Invalid, and Sound Deductive Arguments

A deductive argument that succeeds in showing that its conclusion *must* be true if its premises all are true is called a **valid** deductive argument. A deductive argument that fails to show that its conclusion *must* be true if its premises are true is called an **invalid** deductive argument.

Thus a valid argument may be defined as a deductive argument in which it *is* the case that if the premises are true then the conclusion *must* be true. An invalid argument may be defined as a deductive argument in which it is *not* the case that if the premises are true the conclusion must be true.

Both of the following arguments are deductive, because each obviously *aims* to show that its conclusion *must* be true if its premises all are true. However, only the first is valid, the second is *in*valid:

- All human beings are mammals.
- All mammals are warm-blooded.
- Therefore it must be that all human beings are warm-blooded.

- All human beings are mammals.
- All dogs are mammals.
- Therefore it must be that all human beings are dogs.

Do you see the difference between these two arguments? Again: Not all reasoning is equal. Some reasoning is better than other reasoning.

When logic students first learn the concept of validity, they almost always find one thing extremely puzzling. An argument can be valid even though it has *false* premises and a *false* conclusion. Consider the following deductive argument:

- All students are millionaires.
- All millionaires are Buddhists.
- Therefore, all students must be Buddhists.

Although the premises are false, and although the conclusion is false, the argument is valid. It is valid simply because *if* the premises were to be true then the conclusion would have to be true as well. The argument fits the definition of a valid argument. Does this seem puzzling to you? The premises are false, and yet the argument is perfectly valid! This shows that true premises are not required for validity. In logic, “valid” does not mean true. An argument is valid as long as it is the case that *if* the premises are true then the conclusion must be true. True premises are not required.

However, validity is not all we want in a deductive argument. We normally also want truth. If an argument is valid, and in addition its premises are all true, then the argument is called a **sound** argument. Thus, a sound argument has two characteristics:

- All of its premises are true.
- It is valid.

Since truth—correspondence with reality—is the ultimate goal of reasoning, *soundness* is the ultimate goal of deductive argumentation, not mere validity. You know you’ve made it if your deductive argument is sound as well as valid. The following argument is both valid and sound:

- All whales are mammals.
- All mammals are animals.
- Therefore, it must be that all whales are animals.

The previous argument, about students and Buddhists, was valid, but *un*sound.

### Strong, Weak, and Cogent Inductive Arguments

An inductive argument that succeeds in showing that its conclusion is probably (but not certainly) true if its premises are true is called a **strong **inductive argument. An inductive argument that fails to show that its conclusion is probably (but not certainly) true if its premises are true is called a **weak **inductive argument.

Thus a strong argument may be defined as an inductive argument in which it *is* the case that if the premises are true then the conclusion is *probably* true. A weak argument may be defined as an inductive argument in which it is *not* the case that if the premises are true then the conclusion is probably true.

Both of the following arguments are inductive, because each *aims* to show that its conclusion is probably (but not certainly) true. However, the first is strong while the second is weak:

- In all of recorded history, it has never snowed 6 inches in Dallas in August.
- Therefore, it probably will not snow 6 inches in Dallas next August.

- Joe is a member of the Democratic Party.
- Some known Communists have been members of the Democratic Party.
- Therefore, Joe is probably a Communist.

Again, not all reasoning is equal. Some acts of reasoning are better than others. Do you agree?

Many logic students find this aspect of strength puzzling at first: An inductive argument can be strong even though it has *false* premises and a *false* conclusion. Consider the following inductive argument:

- For the past six months it has been snowing every day in Dallas, it is below 30 degrees in Dallas, and the sky in Dallas is full of snow clouds.
- Therefore, it will probably snow in Dallas today.

Although the premise is false, and although the conclusion is false, the argument is strong. It is strong because if the premise were to be true then the conclusion would probably be true as well: If the premise is true, then the conclusion is likely to be true although not certain. The argument fits the definition of a strong argument!

But strength is not all we want in an inductive argument. We normally also want truth. If an argument is strong, and in addition its premises are all true, then the argument is called a **cogent** argument. Thus, a cogent inductive argument has two properties:

- All of its premises are true.
- It is strong.

Since truth is the ultimate goal of reasoning, cogency is the ultimate goal of inductive argumentation. The following argument is both strong and cogent:

- Most cars burn gasoline.
- The Presidential Limousine is a car.
- Therefore the Presidential Limousine probably burns gasoline.

The earlier argument, about Dallas and snow, was strong but not cogent.

### Consistency, Implication, and Equivalence

We have been using our faculty of reason to judge deductive arguments as valid or invalid and to assess the strength of inductive arguments, but we also use reason when we decide whether or not two of our beliefs stand in logical conflict and when we look for certain logical relations among our beliefs. For this reason, logical theory also studies the logical relationships that exist between declarative statements and the logical properties of statements. Four terms are especially important: Consistency, inconsistency, implication, and equivalence. Here are the first two definitions:

- Two statements are
**consistent**if and only if it is possible both are true - Two statements are
**inconsistent**if and only if it is not possible both are true.

For example, the following statements, given their standard meanings, are consistent:

- Sue is 33 years old.
- Sue is an accountant.

And the following statements, given their standard meanings, are inconsistent:

- Sue is 33 years old.
- Sue is a teenager.

Next:

- One statement
**implies**a second statement if and only if it is not possible that the first statement is true and the second statement is false.

In other words, a statement P implies a statement Q when and only when it is the case that if P is true then Q is true. For example, in the following case, the first sentence implies the second:

- Sam is 33 years old.
- Sam is older than 21.

However, in the next case, the first sentence does not imply the second:

- Sam is a Republican.
- Sam is a millionaire.

Next:

- Two statements P and Q are
**equivalent**if and only if P implies Q and Q implies P.

In other words, two statements are equivalent when and only when it is not possible that they differ as to truth and falsity: if one is true then the other is true and if one is false then the other is false. In the following case, the two statements are logically equivalent.

- Ann is taller than Bob.
- Bob is shorter than Ann.

But these two statements are not equivalent:

- Ann is older than Bob.
- Bob is 33 years old.

### Necessity and Contingency

We also use our faculty of reason when we decide whether a statement is necessary or contingent, and logic is concerned to define the relevant terminology so that our thinking can be as clear as possible on this matter as well. Four additional terms are important: Necessary truth, necessary falsehood, contingent truth, contingent falsehood. Here are the definitions:

- A statement is
**necessarily****true**if it is true and it cannot possibly be false.

In other words, it is true in all possible circumstances, there are no possible circumstances in which it would be false. For the purposes of logical theory, a “possible circumstance” is defined as any circumstance whose description is not self-contradictory. This is the broadest concept of possibility humanly and consistently conceivable.

- A statement is
**necessarily****false**if it is false and it cannot possibly be true.

In other words, it is false in all possible circumstances, there are no possible circumstances in which it would be true.

For example, the following statements, given their standard meanings of course, are all necessarily true:

- All triangles have three sides.
- The derivative of a constant is zero.
- The number 3 is greater than the number 2.

Given their standard meanings, the following statements are necessarily false:

- All triangles have 9 sides.
- 1 + 1 = 5.
- The number 12 is less than the number 3.

Next:

- A statement is
**contingently****true**if it is true but there are possible circumstances in which it would be false.

In other words, it is true but it might have been false if circumstances had been sufficiently different.

- A statement is
**contingently****false**if it is false but there are possible circumstances in which it would be true.

In other words, it is false but it might have been true if circumstances had been sufficiently different.

Examples of contingent truth would include:

- Crosby, Stills, and Nash performed at Woodstock in 1969.
- It is sunny in Seattle on September 9, 2011.

And examples of contingent falsehoods would include:

- Bob Dylan performed at Woodstock in 1969.
- Richard Nixon was elected President in 1960.

### Conclusion

There you have it: A “cook’s tour” of the main concepts of logical theory. I hope you found it helpful.