So far, our study in logic has dealt primarily with *deductive* arguments. When you intentionally craft a valid argument, you have made a *deductive argument.* An examination of a *deductive* argument will find that the premises conclusively support the conclusion. Deductive arguments claim validity and are subject to scrutiny. Here we will look at the various forms of deductive arguments. After that, we will consider inductive arguments. Buy the book to help you learn and continue learning with us.

Another type of statement, the *explanation*, can look like an argument and contain premise and conclusion indicators. Such a statement does not attempt to convince the reader or listener to agree to a conclusion. Instead, an *explanation* seeks to expound on or provide a reason for an accepted fact.

## Forms of Deductive Arguments

An inductive argument, which we will consider in the next chapter, tries to illustrate the probability of its conclusion. Contrariwise, a deductive argument conclusive either proves or fails to prove a conclusion based on stated or implied premises. In fact, the premises of a *valid* deductive argument necessitate its conclusion.

### Syllogisms

A syllogism is either a standard-form argument or an argument that can be reduced to standard form without changing its meaning. A valid syllogism has two premises and one conclusion. Though syllogisms don’t often naturally occur, they help us illustrate logical principles.

Valid deductive arguments, in the form of syllogisms, appear as one of the following seven patterns.

The first 2 patterns are called *mixed hypothetical syllogisms* because they contain one conditional premise and one absolute premise.

### Pattern 1: Modus Ponens: The affirming mode.

A modus ponens argument *affirms the antecedent*. For example:

If today is Monday, then this is Math class.

Today is Monday.

**∴** This is Math class.

The “if” part is *antecedent*. The “then” part is the *consequent*.

The first premise of a *modus ponens* argument contains a conditional statement as its *antecedent.* The second premise of the syllogism *affirms* the antecedent. The conclusion of a modus ponens argument *affirms the consequent. *

The generic format of *modus ponens* follows:

If P is true, then Q is true. – OR – Q, if P

P

**∴** Q

#### Fallacy alert

Sometimes, a fallacious argument looks like modus ponens:

**Invalid:**

If P then Q

Q

**∴** P

If George Washington was beheaded, he is dead.

George Washington is dead.

**∴** he was beheaded.

The above is a fallacy called “Affirming the consequent.”

### Pattern 2: Modus Tollens

The next *mixed hypothetical syllogism, *modus tollens, has a conditional premise followed by a categorical premise that denies the consequent of the first. For example, consider:

If Johnson is seven feet tall, then he is at least six feet tall.

Johnson is not at least six feet tall.

∴Johnson is not seven feet tall.

Modus tollens arguments follow the following generic format:

If P then Q

Not Q

∴Not P

Here’s another example of modus tollens:

If it is healthy to ingest mercury, Fish from polluted streams are good to eat.

Fish from polluted streams are not good to eat.

∴It’s not healthy to ingest mercury.

#### Fallacy alert:

You must be careful to avoid mistaking modus tollens for a fallacy. Is the following valid?

If P then Q

Not P

∴Q

No. The above argument is called denying the antecedent. Compare the following two fallacious (and invalid) arguments:

If eating Fritos is immoral, it should be illegal.

Eating Fritos is not immoral.

∴It should not be illegal.

and

If George Washington was beheaded, he’s dead.

George Washington wasn’t beheaded.

∴George Washington is not dead.

The above arguments *deny the antecedent* and are, therefore, invalid.

### Pattern 3: Hypothetical Syllogism

Pure hypothetical syllogisms contain *only* conditional premises. In other words, hypothetical syllogisms consist of two conditionals linked by consequent of one of them is antecedent of the next:

If P then Q.

If Q then B.

∴if P then B

Both premises are conditional and the conclusion is conditional.

If Hillary wins New Hampshire, then Bernie Sanders will drop out.

If Bernie Sanders drops out, Hillary will win the nomination.

∴if Hillary wins New Hampshire, then Hillary will win the nomination.

Here’s another example of a hypothetical syllogism:

If today is Wednesday, then the rape trial is being held in Courtroom Annex G

The rape trial is being held in Courtroom Annex G only if the larceny trial is being held in Courtroom Q

So, if today is Wednesday, then the larceny trial is being held in Courtroom Annex Q

### Pattern 4: Disjunctive Syllogism Either P or Q

Syllogisms are two-premise arguments. Either P or Q is known as a disjunction.

Either A or B is true.

A is not true.

∴B

Here’s an example of a deductive argument in the form of a disjunctive syllogism:

Either I’ll go to the party or go to the library.

I am not going to the party.

∴I’m going to the library.

### Pattern 5: Barbara

All cats are mammals.

All American Bobtails are cats.

∴All American Bobtails are mammals.

and

All A are B.

All C are A.

∴All C are B.

### Pattern 6: Predicate Instantiation

Given an instance of A, you conclude the following logical arguments:

All A are B.

X is an A.

∴X is a B.

Every A is a B.

X is an A.

∴X is a B

Each A is a B.

Any A is a B.

∴ Any A is a B

Here’s an example of predicate instantiation:

All Olympic athletes are ambassadors of some country.

Bo is an Olympic athlete.

∴Bo is an ambassador of some country

### Pattern 7: Predicate Negation No A are B.

Predicate negation syllogisms have the following format:

m is an A.

\m is not a B.

Note: What’s the meaning of “only if” <== a conditional that points toward a consequent.

X only if Y.

if X [antecedent] then Y [consequent]

Here’s an example of predicate negation:

No 20th century artists are baroque painters.

Picasso is a 20th century artist.

∴Picasso is not a baroque painter.

## Deductive Arguments: What does all this mean?

Deductive arguments are intended to be valid and conclusive. You, therefore, cannot improve upon it. Inductive arguments make a case for a conclusion but are not absolute.

Syllogisms are various forms of standard-form arguments. Although naturally occurring syllogisms are hard to find, you can re-write deductive arguments in syllogistic form as long as you don’t change the meaning of any of the premises or the conclusion.

You will find syllogisms useful while analyzing arguments because they can help you uncover and expose fallacies.

Share your opinions of **deductive arguments** by sending an email to info@truthintheword.org.

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