DEDUCTIVE ARGUMENTS

The aim of this tutorial is to help you learn to recognize, analyze, and evaluate deductive arguments.

“When you have eliminated the impossible, whatever remains, however improbable, must be the truth.”

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Deductive arguments

A deductive argument claims that its conclusion necessarily follows from the premises.

Certain words and phrases are commonly used in deductive arguments; these include certainly, absolutely, definitely, conclusively, must be, and it necessarily follows that. However, not all deductive arguments contain indicator words.

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Hot or Not?

Are deductive arguments better than inductive arguments?

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Deductive arguments and syllogisms

Deductive arguments are often presented in the form of syllogisms, with two supporting premises and a conclusion.

A deductive argument is valid if the form of the argument is such that the conclusion must be true if the premises are true. The form of an argument is determined by its layout or pattern of reasoning. An argument is sound if both (1) it is valid, and (2) the premises are true.

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No fish are dogs.All dogs are mammals.

Therefore, some mammals are fish.

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Types of deductive arguments

There are several types of deductive arguments. Three types are used in everyday reasoning:

Arguments by elimination rule out different possibilities until only one possibility remains.

Arguments based on mathematics depend on mathematical or geometric equations to generate conclusions.

In an argument from definition, the conclusion is true because it is based on a key term or essential attribute in a definition.

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A mouse locates the prize at the end of a maze through the deductive process of elimination.

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Bo Dietl

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The smooth landing of the Mars rovers was a result of the deductive reasoning skills of NASA scientists.

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“Marilyn and Jessica cannot be married, since a marriage is a union between a man and a woman.” This argument may no longer be sound because the legal definition of marriage is undergoing change.

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Hypothetical syllogisms

Hypothetical reasoning involves “If…then…” reasoning.

A hypothetical syllogism is a form of deductive argument that contains two premises, at least one of which is a hypothetical or conditional if…then statement.

There are three basic patterns of hypothetical syllogisms: Modus ponens (affirming the antecedent) Modus tollens (denying the consequent) Chain arguments

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Modus ponens arguments

In a modus ponens argument, the following structure is used:

If A, then B.

A.

Therefore, B.

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Modus tollens arguments

In a modus tollens argument, the following structure is used:

If A, then B.

Not B.

Therefore, not A.

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Chain arguments

In chain arguments, the following structure is used:

If A, then B.

If B, then C.

Therefore, If A, then C.

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Using hypothetical syllogisms

Not all valid arguments are sound. Rewording arguments in ordinary language in the form of hypothetical syllogisms can help you expose faulty premises. They are also useful as clarification tools, and as decision-making aids.

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Categorical syllogisms

A categorical syllogism is a type of deductive argument that sorts things into specific classes or groups. It is composed of a conclusion, two premises, and three terms, each of which occurs exactly twice in two of the three propositions.

Categorical syllogisms can be written in any of 256 standard forms or combinations.

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Standard-form categorical syllogisms

A standard form for categorical syllogisms is shown here:

All P are/are not M. (P=predicate, M=middle term) Some S are/are not M. (S=minor term, M=middle term) Some S are/are not P. (S=minor term, P=major term)

As with hypothetical syllogisms, if the form of a categorical syllogism is valid, then the argument will be valid regardless of term substitutions.

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Quality and qualifier

Each proposition in a standard-form categorical syllogism is written in one of four forms based on quality (universal or particular) and qualifier (affirmative or negative).

Three forms predominate. Universal affirmative: All S are P. Universal negative: No S are P. Particular affirmative: Some S are P.

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Venn diagrams

Venn diagrams are useful instruments for diagramming and evaluating categorical syllogisms. They directly engage our spatial reasoning ability and help us to visualize group relationships effectively.

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P MNo (dogs) are (cats).

S MSome (mammals) are (cats).

S PTherefore, some (mammals) are not (dogs.)

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Translating arguments into standard categorical form

Rewrite each proposition in standard form, starting with the conclusion.

Use the context and grammar of the original argument to decide on which qualifier to use.

Identify the three terms in the argument. Where necessary, rewrite each term as a noun or noun

phrase. Each proposition uses a form of the to be verb. Assemble in standard form: major premise, minor

premise, conclusion.

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Conclusions

Knowledge of deductive arguments—including arguments from definition, mathematical arguments, arguments by elimination, and hypothetical and categorical syllogisms—is essential for us to effectively function in the world. As good critical thinkers, we must constantly identify and evaluate these types of arguments, both our own and those presented to us by others.

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How can learning about deductive logic help us make better-informed decisions?

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Perspectives on the death penalty

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