Propositional
logic:: Rules of
inference
Logical Connectives
In propositional logic, we
will study the common inferences we make in everyday thought and speech as well
as the common inferences we make in more complex arguments in philosophy and
science. We will use of symbols to represent the logical form of these
arguments and then we will use of a small number of inference rules to generate
and test arguments of any complexity.
Thus we will discuss compound
statements -- statements that are made up of other, simpler propositions. We
will learn the rules to make or introduce compound statements out of simple
statements and to break up compound statements.
Please learn the rules of inference
as they apply to the logical connectives of compound statements: dot,
implication, conjunction, negation. Learn the few rules of inference (most
common argument patterns) as logical connective introductions and
eliminations. For example, simplification is dot elimination and
conjunction is dot introduction.
(See Peter
Suber's following webpages on propositional logic)
I. Propositional Logic Terms and Symbols
II. Translation Tips
III. Paradoxes of Material Implication
IV. Introduction to Derivations
V. Justifying the Rule of Rigor
See Layman's The Power of logic
Categorical Propositions are propositions that affirm or deny that some class S is included or is excluded from some other class P in whole or in part.
Class is defined as “the collection of all objects that have some specified characteristic in common.”
There are four ways that two classes can be related to each other:
1. Universal affirmative: Every member of one class (called the subject
class) is included as a member of the second class (called the predicate
class).
2. Universal Negative: The two classes have no members in common. The
entire logical subject class is excluded from the entire logical predicate
class.
3. Particular affirmative: Some but not all members of the first class
are included in the second class.
4. Particular negative: Some but not all members of the first
class are excluded from the entire
second class.
These relationships of class inclusion and exclusion are affirmed and
denied by categorical propositions. Thus there are four categorical
propositions.
FORM Quantity
and Quality
All S is
P
Universal Affirmative
No S is P Universal Negative
Some S is P Particular Affirmative
Some S is not P Particular Negative
For example,
1. All <Politicians> are <moral>.
2. No <politicians> are <moral>.
3. Some <politicians> are <moral>.
4. Some <politicians> are not <moral>.
The first is a universal affirmative proposition. A Universal
affirmative proposition is about two classes and the entire first class (in
this case, the class of all politicians) is affirmed to be included in the
second class (the class of all men): every member of the first class is also a
member of the second class. All universal affirmative propositions have the
form “All S (logical subject) is P (logical predicate).”
The second is a universal negative proposition. The entire logical
subject is asserted to be excluded from the entire logical predicate. The
proposition denies that the relationship of class inclusion holds between the
two classes. It asserts that no member of the first class is a member of the
second class.
The third type of categorical proposition is particular affirmative. It
affirms that some members of the first class are members of the second. A
particular affirmative proposition does not affirm or deny that all members of
the first class are included in the second class; also, it does not imply that
“some S are not P”. A particular affirmative proposition says
that the two classes have one member in common: Some S is P.
The last type is particular negative. It does not refer to the logical subject class universally but only to some members of the logical subject class. It denies that the particular members of the first class referred to by the logical subject term are included in the second class.
Quantity
The quantity of a categorical proposition depends on whether the statement
makes a claim about the entire logical subject or only a claim about some
members of the logical subject class. If the claim made by the categorical
proposition is about every member of the logical subject class, then the
categorical proposition is universal. The categorical proposition is particular
if the proposition makes a claim about only some member of the logical subject
class. All categorical propositions are universal or particular.
Quality
Every standard form categorical proposition has a quality and its quality is
either affirmative or negative. If the proposition affirms that part or the
entire subject class is included in the predicate class, then it is
affirmative. If the proposition denies inclusion, then it is negative.
The E and O propositions are negative. A and I propositions are affirmative.
Distribution
Propositions may refer to all members of a class or only to some of them.
Universal propositions are about all members of the subject class. Particular
propositions are about some members of the subject class. Affirmative
propositions do not refer to every member of the predicate class: no assertion
is made about the entire predicate class of affirmative propositions. Thus the
subject of a universal affirmative proposition is distributed and the predicate
of a universal affirmative (A) proposition is not distributed. The entire
subject class and the entire predicate class of an E proposition are referred
to in an E proposition. Both terms are distributed.
Conversion, Obversion, and Contrapositive
An inference is the implication of a conclusion from one or more premises. The
inference is mediate if there is more than one premise. If the conclusion is
drawn from only one premise, then the inference is called immediate infernce.
Conversion and Obversion are immediate inferences. Other useful immediate
inferences are displayed by the square of opposition.
The principle of conversion: you can switch the subject and predicate
of an E and and I proposition and the new proposition is logically equivalent
to the original proposition. Conversion is not valid for the A and the O
proposition. Conversion is valid for the E and the I. Why is conversion not
valid for the A and the O? The reason is: the distrubution of
the logical subject and logical predicate terms in the premise and the
conclusion. If you only know something about part of a class in the premise and
then claim to know something about every member of the same class in the
conclusion, you then have an illict process of conversion. For example, "Some
animals are not cats." (premines) is true. But the converse (conclusion)
"Some cats are not animals" is false. The logical subject term in the
premise is undistributed but when the terms are switched by the process of
conversion, the term "animals" becomes the prediate of a particular
negative propositon (Distributed or the term that designed the class of animals
now refers to each and every animal (dogs, monkeys, ....).
The principle of obversion. This principle states that you can change
a proposition from affirmative to negative, or negative to affirmative, if you
change the predicate term to its complement or the complement of the predicate
term to the predicate term:
All S are
P = No S are (non-P)
No S
are P = All S are (non-P)
Some S
are P = Some S are not (non-P)
Some S
are not P = Some S are (non-P)
Contraposition is not a new process of inference: if you obvert,
convert, and the obvert the given categorical proposition, the new proposition
is the contrapositive of the original proposition: Contraposition is valid for
the A and the O:
'All S are P' is equivalent to 'All (non-P) are (non-S)'
'Some S are not P' is equivalent to 'Some (non-P) are not (non-S)'
Mood of syllogism:
The mood of a standard form syllogism is determined by the type ( A,E, I, O) of
propositions it contains. The mood is represented by three letters and three
letters is a specific order. The first letter stands indicates the major
premise is a certain type of proposition, the second letter the minor premise,
and the third letter the conclusion. Thus a syllogism who mood is EAO has a
univeral negative major premise, a universal affirmative minor premise, and a
particular affirmative conclusion.
Figures of the syllogism:
1. MP 2. PM 3. MP 4.
PM
SM
SM
MS MS
Rules
and Fallacies:
Rule 1. Any syllogism must have the middle term distributed in at
least one of its premises. The violation of this rule is called the
fallacy of undistributed middle.
Rule 2. No term can be distributed in the conclusion that is not distributed in the premises. Thus if a term is distributed in the conclusion it must be distributed in the premises. The violation of this rule is called the fallacy of (2a) illicit minor or (2b) illict major. When a syllogism has its major term undistributed in the major premise but distributed (as the predicate) in the conclusion, the syllogism is said to commit the fallacy of illicit major. And when a syllogism has its minor term undistributed in the minor premise but distributed (as the subject) in the conclusion, the syllogism is said to commit the fallacy of illicit minor.
Rule 3. No syllogism can have two negative premises. The violation of this rule is called two exclusive premises.
Rule 4. If (4a) either premise is negative, then the conclusion has to be negative and (4b) if the conclusion is negative, then one of the premises has to be negative. Thus, 4a is called the fallacy of Drawing an affirmative conclusion from a Negative Premise. and 4b is called the fallacy of having Affirmative Premises and a Negative conclusion.
Rule 5. No syllogism can have two universal premises and a particular conclusion. To violate this rule is to draw a conclusion which does have existential import based upon two universal premises that does not have existential import (remember we are assuming the Boolean interpretation of categorical propositions). Any syllogism that violates rule 5 is said to commit the Existential Fallacy.