DEDUCTIVE VS. INDUCTIVE ARGUMENTS

If a conclusion follows necessarily from the premises, then the argument is deductive. If the premises only indicate or offer some support for the conclusion, then the argument is inductive. For example, although daily evidence tends to support the conclusion that the sun rises in the east, this conclusion does not necessarily follow. It is possible that a cosmic catastrophe during the night could change the rotation of the earth. Therefore, the inductive truth "The sun rises in the east," is contingent, i.e., dependent upon certain empirical circumstances and conditions.

On the other hand, the conclusions of Euclidean geometry follow necessarily from Euclid's axioms, even though it is possible to describe reality (using Riemannian geometry for example) in non-Euclidean terms. The conclusions of Riemannian geometry also follow necessarily from Riemann's axioms. Mathematical truths are necessary truths, i.e., non-contingent. Please note that science and mathematics do not have the same methods, nor do they produce the same type of truths. Most of the truths of science are contingent, whereas the truths of math (except for statistics and probability) are necessary.

VALID VS. INVALID ARGUMENTS

Validity is the relationship that obtains among premises, not single propositions. A valid argument is one that follows the correct rules of inference. A valid argument need not be sound, nor its conclusion true. (See Example D below.) A sound argument must be both valid and have true premises.

TRUTH VS. FALSITY

Truth is a characteristic of propositions only. An individual proposition is true only if it classifies a thing or event correctly. "Jimmy Carter is an American" is true, while "Jimmy Carter is an atheist" is false. The fact that the conclusion of an argument is true does not make an argument valid or sound. It might be an invalid argument, premises constructed in such a way as to have a true proposition tacked on to the end. (See Example C.)

SYLLOGISMS

A syllogism is a form of deductive argument the conclusion of which follows from two premises, a major and minor premise. The examples below are both valid and invalid syllogisms. A syllogism is formally invalid if its structure or form is faulty (Examples B & C); a syllogism is informally invalid if it uses ambiguous or irrelevant terms (e.g., Socrates was Greek, and metaphysics is Greek to me).

EXAMPLE A

All humans are mortal (major premise).
Nixon is human (minor premise).
Hence, Nixon is mortal (conclusion).

Until we have conclusive evidence that some humans do not suffer a physical death, then we must take the major premise as a true proposition. If Nixon is genetically a human being -- all of us have assumed this all along -- then we must also accept the minor premise as true. The syllogism is formally valid--the conclusion follows directly from the premises--so that means we must accept the argument as sound.

But the strength of argument is weakened by the fact that we cannot secure absolutely the truth of the premises. A hundred years form now medical science may prolong human life indefinitely; and, although it is highly improbable, Nixon may be a non-human alien who has successfully maintained a human disguise all his life. The soundness of a valid syllogism is only as strong as the premises.

EXAMPLE B

All Athenians are Greeks.
All Spartans are Greeks
Hence, All Spartans are Athenians.

This syllogism, although the premises are true, contains a formal fallacy–namely, there is no formal link to secure the identity of Athenians and Spartans. Furthermore, it is logically impossible to provide such a link.

EXAMPLE C

All humans are mortal.
All Greeks are Europeans.
Hence, all Greeks are mortal.

This example differs from B in that not only are the premises true, but we also recognize the conclusion as true. But, as we said above, a true conclusion does not mean that the argument is valid. This syllogism suffers form the formal fallacy usually called "no middle term." Unlike Example B, this formal link can be constructed by making two syllogisms and using the conclusion of the first as the major premise of the second.

All humans are mortal.
All Europeans are humans.
Hence, all Europeans are mortal.
All Greeks are European.
Hence, all Greeks are mortal.

EXAMPLE D

All higgledy-pigs are gobbledy-gooks.
All mubbley-moos are higgledy-pigs.
Hence, all mubbley-moos are gobbledy-gooks.

This syllogism is formally valid, but of course not sound, because the premises are nonsense. Such an example shows clearly the difference between validity and truth.

REDUCTIO AD ABSURDUM

If you can show that any assumption, premise, or conclusion of your opponents argument states or implies an absurdity, then you have reduced that argument to absurdity. Note that Mencius uses this technique against Gaozi several times.

THE INFORMAL FALLACIES

nFallacy of Equivocation.  Using different meanings of the same word. Physical equality vs. equality before the law.
nAd hominem.  An argument “against the person” rather than against her argument. A good example: "Communism is wrong because Karl Marx allowed two of his children to die of neglect on the streets of Soho."
nAppeal to Authority.  “It’s true because the Bible says it’s so.”  An obvious questions presents itself: what makes the Bible true?
nBegging the Question. “Persons are the most valuable beings in the universe.” What is your argument for the value of persons?  Is it possible that human persons have arbitrarily made themselves the centers of value?  Animal rights philosophers call this the fallacy of "specieism," the mistake of making our own species a standard for morality.  This is a view just as unjust as sexism and racism.

Note: An informal fallacy differs from a formal fallacy such as invalidity in an argument, which has to do with a mistake in the formal structure of the argument.

ARGUMENT BY ANALOGY

Another widely used mode of philosophical argument is analogical reasoning. Philosophical analogies approximate the form of mathematical proportions and therefore might appear to be tight deductive systems. For example, A is to B as C is to D has the same form as 1/2=2/4, but the "numerators" and "denominators" of philosophical analogies are never mathematically identical. This ultimately makes mathematical proportions and philosophical analogies quite different. It makes then inductive arguments by the definitions above.

In assessing the value of philosophical analogies, we must ask two questions: Are the things compared similar? and are the things similar in the particular respect in question? If these two questions can be answered in the affirmative, then a convincing argument from analogy probably exists.

In his book Practical Logic Monroe C. Beardsley contends that there is no such thing as an argument from analogy. "Analogies illustrate, and they lead to hypotheses, but thinking in terms of analogy becomes fallacious when the analogy is used as a reason for a principle" (p. 107).

Beardsley does, however, give a good example of an analogy which is "strong" and which can be used to represent one thing as another. This is the analogy of a map: "The dots on the map are not very much like actual cities, and the lines on the map are not all like mountains or wet like rivers...But the structure of the map, if it is a good one, corresponds to the structure of the country it represents. That is, the shapes of the states are like the shapes on the map;...and the relative distances between actual cities are like the relative distances between the dots on the map" (p. 106).