# logic by stariya

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Logic & Argumentation

I. Preliminaries
A. Rational Arguments
 Arguments are commonly taken to be disagreements or disputes often involving an
emotional component. In general, emotional appeals and the strength of one’s
convictions have nothing to do with rational argumentation. Rational or logical
argumentation is a form of discourse that attempts to show how certain statements
called conclusions are supported by or follow from other statements called premises.
Arguments consist of a set of statements, divided into premises and conclusion, for
which it is claimed the conclusion follows from the premises.

P1 All men are mortal.
P2 Socrates is a man.                              (1)

C Socrates is mortal.

The premises are written above the line; the conclusion below. The conclusion C is
suppose to follow from the premises P1 and P2. The above argument has exactly two
premises; all such arguments are called syllogisms.

B. Conditionals (If-then statements)
 A conditional is a statement whose standard form is “If P is the case then Q is the case.”
If P then Q                                   (2)

where P and Q are particular statements. For example, if we take P to be “I am in
College Park.” and Q to be “I am in Maryland.” then we have:

If I am in College Park then I am in Maryland.                 (3)

 P or the if-part of the if-then statement is called the antecedent. Q or the then-part is
called the consequent.

C. Necessary & Sufficient Conditions
1. Necessary Conditions
 A condition N is called necessary for the occurrence of an event E if the event in
question could not have happened with out the presence of N. Necessary conditions
of event E must be present for E to occur.
 The consequent of a conditional is a necessary condition. If the antecedent is true
then the consequent must be true. A necessary condition for my being in College
Park is that I be in Maryland. I cannot be in College Park without being in
Maryland.
 Necessary conditions N for an event E do not guarantee E will occur though.
Ex: Being in College Park
As already noted, a necessary condition for being in College Park is being in
Maryland. But being in Maryland does not guarantee that I am in College Park.
I may for instance be in Baltimore or Annapolis.
Being in Boston
A necessary condition for being in Boston is being on this planet. Another
necessary condition for being in Boston is being somewhere in North America.
Further necessary conditions are being in the United States, being in
Massachusetts, and being in the Eastern half of Massachusetts. None of these
conditions alone or together though guarantee I am in Boston.

2. Sufficient Conditions
 A sufficient condition S for the event E is one that guarantees E will occur.
 The antecedent of a conditional is a sufficient condition for the occurrence of the
consequent. If the antecedent is true then the consequent must be true.

Ex: Being in Maryland
Being in College Park is a sufficient condition for being in Maryland. One
cannot be in College Park without also being in Maryland because College
Park is in Maryland.
Go to the Store!
Your significant other sends you to the store to buy a gallon of milk and a loaf
of bread. You are broke; so, they fork over two dollars. You tell them this is
not enough to purchase the milk and bread. The two dollars is certainly
necessary if you are going to buy the bread, but it is not sufficient. In a huff
they throw you a five. You now have seven dollars. This is sufficient for
making your purchase since the milk and bread will cost no more than three
fifty. (Seven dollars is not necessary to make your purchase though.)

II. Validity
 A valid argument is one such that if the premises were true then the conclusion would have
to be true.
 Validity concerns the form or structure of an argument, and analyzing the validity of an
argument does not require knowing anything of the truth or falsity of the premises or the
conclusion.

If flames burn then Dr. Levinson is Michael Jordan.
Flames burn.                                                (3)

 Dr. Levinson is Michael Jordan.

All arguments having this form are valid. They possess the proper logical structure such
that if the premises are true then the conclusion must be true.
 But wait! Something seems wrong here. The argument is not true. How can (3) be a valid
argument when the first premise and the conclusion are false? Also, what does the
potential for flames to burn have to do with Dr. Levinson being Michael Jordan?
 The issue of the truth or falsity of the premises and/or the conclusion of an argument is
irrelevant when assessing the validity of an argument. Also, what physical connection
exists or does not exist between the antecedent and the consequent of a conditional is of no
concern to the logician, qua logician. Such concerns as these fall out side the domain of
logic. Logic is only concerned with analyzing the structure of arguments to find out if they
possess the proper form. The logical form or structure of argument (3) is:
If B then J.
B                                         (4)

J

Any argument possessing this form is a valid argument, even if the premises and
conclusion are all false, even if the antecedent B in no way appears to be connect with the
consequent J.

III. Soundness
 A sound argument is a valid argument whose premises are all true.
 We have so far only presented two arguments, (1) and (3). Argument (1) is a sound
argument but (3) is not.
 Is the following argument sound?

P1 All forms of prejudice are morally wrong.
P2 Racism is a prejudice.                             (5)

C Racism is morally wrong.

This is not a sound argument. Why? We certainly want to agree that racism is wrong. It
is also pretty clear that racism is a prejudice. But (5), though valid, is not sound because
P1 is not necessarily true. There are a number of definitions of prejudice, most having
some sort of negative connotation to them. A neutral definition of prejudice is of a
preconceived notion or idea, a bias. If we take this to be our definition of a prejudice then
P1 is not true. A preconceived notion may or may not be morally wrong. If someone has
a bias against buying foreign cars is this morally wrong? There maybe something
“wrong” with it, but it certainly is not morally wrong. Thus, even though (6) is valid, it is
not sound argument one of its premises is false.

IV. A Sampling of Valid Argument Types
A. And-Elimination
 Arguments of this type have the following form:

P and Q                                  P and Q
                  (6a)                                    (6b)
P                                       Q
B. Modus Ponens
 Arguments of this type involve conditional statements and possess the following
structure:
If P then Q.
P                                           (7)

Q

This is the same form as (4), and argument (3) possesses this very structure as we
have already seen. Another argument possessing this structure is:

If I am in College Park then I am in Maryland.
I am in College Park.                                      (8)

I am in Maryland.

C. Modus Tollens
 Arguments of this type have the form:

If P then Q.
~Q                                          (9)

~P

Ex: College Park Revisited

If I am in College Park then I am in Maryland.
I am not in Maryland.                                       (10)

I am not in College Park.

V. Logical Fallacies
A. Fallacy of Affirming the Consequent
 Fallacious arguments of this sort possess the following form:

If P then Q.
Q                                            (11)

P
Ex:
P1 If I am in College Park then I am in Maryland.
P2 I am in Maryland.                                        (12)

C I am in College Park
Take careful note of the following: It may in fact be the case that I am in College
Park. Being in College Park means I must be in Maryland. Thus C and P2 are true.
But P1 is also true. Both premises are true as is the conclusion; so, why is (12)
considered a fallacious argument? Why is (12) not sound? (12) is not sound
because it is not valid. The conclusion does not follow from the premises.
 Arguments possessing the form of (11) are not valid because it is possible for the
premises to be true and the conclusion false. We know P1 of (12) is true. Now, let us
assume P2 of (12) is true, viz., that I am in Maryland. It is possible for these conditions
to hold if I am in Baltimore. If I am in fact in Baltimore then (12) has true premises
and a false conclusion; but a valid argument cannot have all true premises and a false
conclusion therefore (12) is not valid.

B. Fallacy of Denying the Antecedent
 Fallacies of this sort have the form:

If P then Q.
~P                                            (13)

~Q
Ex:
If I am in College Park then I am in Maryland.
I am not it College Park.                                    (14)

I am not in Maryland.

C. Equivocation
 Equivocation involves drawing an unwarranted conclusion by making a word or phrase
used in two different senses appear to have the same meaning.

Ex: Determined or Determined
Holbach claims all our actions are determined. I know people who drift through
life, having no real plans, and really not caring much what they do. Certainly the
actions of these people are not determined. Therefore Holbach’s claim that all our
actions are determined is false.

P1 All actions are determined.
P2 The actions of some persons are not determined.
P3 Some actions are not determined. (P2 restated)              (15)

C1 Not all actions are determined.

There is an equivocation on the word determined. In P1 determined is used to
mean necessarily brought about by antecedent conditions. The second occurrence,
in P2 and P3, means carried out with strong convictions.
D. Questioning Begging Arguments (QBA)
 QBA come in a variety of types (and we will here only be concerned with a few of the
more common ones). In one way or another they all manage to smuggle a
controversial point about the issue at hand into the premises. Many times the point
being smuggled into the premises is just a restatement of the conclusion. In such
arguments the premises do not provide sufficient reason for the truth of the conclusion.
Because the evidence is insufficient to establish the conclusion the arguer is in essence
begging you to accept the conclusion.

1. Circular Reasoning (CR)
 CR either explicitly or implicitly makes use of the conclusion in the premises of the
argument, or they make use of premises that are questionable and would only be
accepted if the conclusion were already demonstrated.

Ex: Determinism
Holbach in defending his Hard Determinism is confronted with an event E that
has no apparent cause. Holbach ask, “What could have produced it if it was
not a cause that brought it about?”

Is this a circular argument? We need to look at it in greater detail before we
can decide.

P1Event E has no apparent cause.
P2(If an event occurs then something must produce it.)
P3All events must be assumed to be produced.                     ()
P4(Produced and caused are synonymous.)

C Every event must be caused

The circularity is immediately obvious when the implicit premises P2 and P4
are made explicit. P2 as interpreted by P4 is just the conclusion stated in a
different form.

2. Question-Begging Definition
 Fallacies of this sort attempt to firmly establish a particular position by employing
questionable definitions. A key claim in a dispute is rendered impervious to attack
by definition.

Ex: Free Will and Indeterminism
Hard determinism claims that determinism and free will are incompatible and
snce determinism does in fact hold there can be no free will. But the hard
determinist (as well as the libertarian) has defined free will to mean
indeterminism. If one does not agree to define free will as such then there is
the possibility that determinism and free will are compatible. (Stace takes this
very position and builds his soft determinism on it.)

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