|
I. Logic, Thinking, and Language
Logic is the study of good thinking: you determine and evaluate the standards of good thinking (i.e., rational
thinking). One way to study thought and thinking would be through introspection, but this sort of approach
is problematic for two reasons: (a) thought and thinking are ethereal and elusive when accessed that way,
and so it is difficult to have much confidence in the results of one's inquiry, and (b) the results of the inquiry
are ineliminably subjective, which will not be of much use to us if we wish to develop a general account of
good thinking. Logicians opt for another method, a method that is grounded in the assumption that thoughts
and thinking are expressible (in principle) in language. According to this method, you study the structure
of language with a view to determining the structure of thought, and in particular, with a view to determining
what separates a good argument from a bad one.
II. Arguments and Types of Arguments
Think of an argument as a sequence of claims, the last of which--call this the conclusion--is supposed to
follow from the claims that precede it. Arguments understood in this way are what we construct as we solve
problems, plan actions, make decisions, and reason our way through life. Our inquiry into good thinking
invariably focuses on the arguments that are produced in the course of such thinking, and so arguments serve
as the focus of logic. Logic, as it concerns us, is devoted to identifying the principles that distinguish good
arguments from bad ones. From the perspective of these principles, good arguments are those whose reasons
support their conclusions, whereas bad arguments are those that fail to offer their conclusions such support.
Arguments understood in this way--as rationales--are traditionally divided into two groups. In the first
group, known as deductive arguments, the good ones are those whose reasons, when true, force their
conclusion to be true as a matter of necessity. The second group, known as non-deductive arguments, covers
good arguments whose conclusions are more likely true given the truth of their reasons.
III. Deductive Arguments
- Principles
- An argument is valid if the conclusion is true whenever the sentences that precede
it are true. (This notion of validity is the logician's theoretical analysis of the
intuitive notion of following.)
- An argument is sound if it is valid and the sentences that precede the conclusion are
all true.
- Good Arguments: The following arguments are schematic representations of certain types
of good arguments. (The capital letters are sentence variables, which is to say that to get
actual examples of the arguments below, you would need to replace the variables with
sentences, making sure to replace all instances of a single variable with the same sentence.)
- Modus ponens: If A, then B; A; therefore,
B.
-
Modus tollens: If A, then B; not B; therefore, not A.
Together, modus ponens and modus tollens expose the fact that the conditional (i.e.,
"if ... then" claims) specifies both a sufficient condition--A is sufficient for B--and
a necessary condition--B is necessary for A.
- Proof by cases: (You would use this if you wanted to prove that a disjunctive
sentence--i.e., an "or" sentence--implied the truth of some other sentence.) A or B or C; if A, then D; if B, then D; if C, then D; therefore, D.
-
Proof by contradiction: (You would use this indirect method if you had no direct
way of proving your claim.) If A is true, then we can derive an absurdity; therefore, A must not be true, which implies that the sentence "not A" is true.
-
Existential Instantiation: If you know that someone did something, then you can
refer to this person with a name so long as you use a name that is not currently in
circulation. (This might be called the "Jack the Ripper Rule", for reasons that
should be obvious.)
-
Universal Generalization: Say you wanted to prove that every member of a certain
class has a property P. You could do this if you selected an arbitrary element of the
class and demonstrated that it has P. IMPORTANT: in demonstrating this, you must
only use those properties of your arbitrary element that it has in common with every
other member of the class; that is, the element is to be treated as a representative
of the class in question.
- Bad, or fallacious, arguments: The following arguments are schematic representations of
argument types that are bad; that is, argument types that don't convey you from reasons to
a well-supported conclusion. Work through an example of each so as to convince yourself
that these are fallacious.
- Affirming the consequent: If A, then B; B; therefore, A.
-
Denying the antecedent: If A, then B; not A; therefore, not B.
-
Begging the Question: This fallacy is committed by anyone who responds to a
challenge or a question with an argument that assumes an answer to that challenge.
This is inadequate because the person advancing the challenge will want the
argument to convince them of a certain resolution, and they will not be convinced
if you straightaway assume a resolution without defense. (If you assume what you
wish to prove, i.e., if you produce a circular argument, you will also beg the
question in most contexts.)
-
Equivocation: This is committed by anyone who uses two senses of an ambiguous
term in an argument that requires the term to be used in only one way.
IV. Non-Deductive Arguments
- These are arguments that are not valid by their very nature, but can nevertheless qualify
as good.
- They are good when the truth of their reasons increases the likelihood that their
conclusion is true.
- The nature of the specific principles varies with the type of argument
- Types of Non-Deductive Arguments
- Inductive Arguments: Inductive inferences begin with the observation that
certain events or conditions cause to other events or conditions; armed with this
observation, one infers from the presence of the same type of events or
conditions to the conclusion that the events or conditions they cause will also
obtain.
- These are not valid, since one might just get lucky in one's observations;
that is, one might make a number of observations where event A leads to
event B, but as a matter of fact, B does not depend on A and only
follows accidentally from A in these circumstances. Thus, any general
inference to B based on these observations will be fallacious.
- However, if you increase the number and variety of observations, then
confidence in the inference will increase.
- Examples: the water example above; I've seen hundreds of white swans
and not swans of different colors, so all swans must be white; FDA
testing of drugs.
- Argument By Analogy: When one argues by analogy for a certain conclusion,
one tells a story that is supposed to parallel the issue in the relevant structural
respects---this is what makes the story an analogy. This story has the relevant
structure, but it lacks elements that muddy the water on a straight consideration
of the issue. One points out that in the story a certain conclusion follows, and so
because the story and the issue have parallel structure, one should also be able to
derive that conclusion when considering the issue.
- This is also clearly invalid, since an analogous story must be different
from the issue in question in several, if not many, respects, and so there
is quite a lot of room for dispute. Once again, the conclusion one reaches
is not deductively forced.
- An argument by analogy will be more forceful the more parallel the
story is to the issue, where this can be increased by an increase either in
detail or spread. It also helps if the story used is not too far-fetched.
- Examples: the acorn/oak tree analogy used to argue for the permissibility
of abortion; "My son said something like that to my daughter when he
was trying to trick her into giving him her allowance; are you trying to
swindle me?"
- Inference to the Best Explanation: When one makes an inference to the best
explanation, one is typically confronted with a wide variety of data---clues, you
might call them---that you wish to explain. As it is, there is nothing about this
data that forces a single explanation on you, since a wide variety of explanations
are available (some more far-fetched than others). In this case, you infer the
explanation that makes the most coherent sense out of the data you have.
- Clearly this is invalid, since there are any number of possible
explanations available that are consistent with the data---the truth of a
particular conclusion is not forced on you. If one conclusion is forced,
then you would be better served representing this as a deductive
inference.
- The more data an explanation accommodates, the better it will be;
furthermore, if the explanation can account for apparent relationships
between the data, that will also recommend it as a good one. It may be
the case that no available explanation accounts for all the data, but this
could simply be because the data set contains noise.
- Examples: Most Sherlock Holmes stories and the game Clue gives you
many examples; "There is dirt on the floor by the plants and one of the
kids' stuffed animals is ripped up on the floor---the dog must have gotten
loose again."
- Hypothesis Testing Arguments, or Confirmation Arguments: This type of
inference is employed to test hypotheses that are not themselves observable. One
deductively derives an observable implication O from the hypothesis H, and then
sets up an experiment to determine if O obtains in the predicted circumstances;
if it does, then this provides non-deductive support for H; if not, then that refutes
H by modus tollens.
- This type of inference is obviously invalid; in fact, it corresponds to the
famous fallacy "Affirming the Consequent". (For this reason, it is
particularly important to determine that an argument is non-deductive if
it embodies this inference, since it will be dismissed as fallacious if it is
deemed deductive.)
- Support for this inference is increased if the prediction is complex and
detailed, and if the background factors are strictly controlled. And while
it is true that failure of a prediction is a serious problem, one can often
avoid the force of modus tollens by claiming that the prediction failed
because of failure to properly attend to a background condition or an
auxiliary hypothesis.
- Examples: scientific practice; if you know calculus, you would be able
to solve this integration problem---you can solve it, therefore you must
know calculus; if you are who you say you are, you should remember
that time when we went out to the dam ...---you do remember, so it must
be you!
|