From the Greek logos, “reason,” logic is the study of our methods for evaluating whether the premises of an argument adequately support its conclusion.

In formal logic, both premises and conclusions are called statements. A statement is a sentence that is either true or false.

• E.g., the statement, “All men are mortal,” is either true or false.

A premise is a statement on the basis of which a conclusion is affirmed. That is, premises lead to conclusions.

• E.g., “All men are mortal. Socrates is a man. Therefore, Socrates is mortal.”

A conclusion is a statement affirmed on the basis of some premises. That is, conclusions follow from premises.

• E.g., “All men are mortal. Socrates is a man. Therefore, Socrates is mortal.”

In formal logic, an argument is a set of statements, one of which, the conclusion, is affirmed on the basis of the others, the premises.

All arguments have two basic properties: validity and soundness. Validity concerns the truth of the conclusion in relation to the premises. Soundness does too, but it also concerns the truth of the premises in relation to the world.

An argument is valid when its premises support its conclusion: if the premises are true, then the conclusion is true. As such, an argument is invalid when the conclusion does not necessarily follow from the premises: the premises may in fact be true, and the conclusion may in fact be true, but the conclusion is not the necessary consequence of the premises.

• Valid: All biologists are scientists. John is not a scientist. So, John is not a biologist.
• Invalid: All biologists are scientists. John is not a biologist. So, John is not a scientist.

An argument is sound when its premises support its conclusions and those premises are true: that is, evidence affirms the truth of the premises, and the premises affirm the truth of the conclusion. As such, an argument is unsound when the premises are false. Note that an argument can be valid but not sound: it can be logical but not true; however an argument cannot be sound but not valid: validity implies soundness.

• Sound: All collies are dogs. All dogs are animals. So, all collies are animals.
• Unsound: All collies are cats. All cats are animals. So, all collies are animals.

Deductive logic is the part of logic that concerns validity. Deductive logic uses general statements to arrive at a specific statement.

• E.g., “All men are mortal. Socrates is a man. Therefore, Socrates is mortal.”

Inductive logic is that part of logic that concerns soundness. Inductive logic uses specific statements to construct or evaluate general statements.

• E.g., “Socrates has a beard. Socrates has a penis. Socrates stands when he pees. Socrates likes beer and football. Therefore, Socrates is a man.”

Conditional statements describe cause and effect; they usually take the form of “if-then” statements. In a conditional, the if-clause is called the antecedent, and the then-clause is called the consequent.

If x is a necessary cause of y, then the presence of y necessarily implies the presence of x. The presence of x, however, does not imply that y will occur.

If x is a sufficient cause of y, then the presence of x necessarily implies the presence of y. However, another cause z may alternatively cause y. Thus the presence of y does not imply the presence of x.

A syllogism is an argument in which the conclusion is inferred from two or more of the premises, usually through conditional statements . If A, then B. If B, then C. So, if A then C.

When presenting information in a paper, it would be inelegant to present a simple logical proof of your idea. An enthymeme is an informally stated syllogism: that is, an argument with an unstated premise, and unstated sub-conclusion, and/or an unstated conclusion. If a syllogism is designed to demonstrate an argument, an enthymeme is designed to persuade someone to accept the argument. While your thinking should be rigidly structured by syllogism, your writing should be closer to enthymeme.

You should be able to draw or chart the logical progression of your ideas, and to chart your logic in a flow chart.

¹[Campaign reform is needed] because ²[many contributions to political campaigns are morally equivalent to bribes].

2

↓

1

¹[Charles is unpleasant to work with] since ²[he interrupts people constantly]. Therefore, ³[I do not want to serve on a committee with Charles].

2

↓

1

↓

3

Although ¹[Americans like to think they have interfered with other countries only to defend the downtrodden and helpless], ²[there are undeniably aggressive episodes in American history]. For example, ³[the United States took Texas from Mexico by force]. ⁴[The United States seized Hawaii, Puerto Rico, and Guam.] And ⁵[in the first third of the 20^th century, the United States intervened militarily in all of the following countries without being invited to do so: Cuba, Nicaragua, Guatemala, the Dominican Republic, Haiti, and Honduras.]

3     4     5

↘ ↓  ↙

2

 ¹[No physical object can travel faster than light.] ²[A hydrogen atom is a physical object.] Hence ³[no hydrogen atom can travel faster than light].

1  +  2

↓

3

¹[The defendant is guilty.] After all, ²[he confessed to stealing the jewels] and ³[he was undoubtedly present at the scene of the crime] since ⁴[his fingerprints are on the safe].

       4

       ↓

2     3

↘  ↙

1

¹[If China attacks Taiwan, Taiwan will fight], for ²[the Taiwanese are ready to defend themselves]. ³[Their air force is formidable.] ⁴[And their navy is well trained and well equipted.]

3     4

↘ ↙

2

↓

1

Although ¹[some have argued that nuclear weapons introduce nothing genuinely new into the disputes about the morality of war], I believe that ²[nuclear weapons raise novel moral issues]. First ³[nuclear weapons have new and undreamed-of long-term effects] since ⁴[the radioactive fallout pollutes the environment and alters human genes]. Second, ⁵[a nuclear war could destroy human civilization in its entirety]. Third, ⁶[in case of nuclear war, the dust caused by the explosions would prevent the sun’s rays from reaching the earth’s surface]. So, ⁷[a nuclear war would result in a drastic lowering of the earth’s temperature]. In other words, ⁸[a nuclear war would result in a “nuclear winter”]. And ⁹[no human or human group has a right to gamble with the very climate upon which life itself is based].

4            6

↓            ↓

3     5     7  +  9

↘ ↓  ↙

2