E. M. Segal:
Primer on Logic
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Psy 416 Syllabus
II. Categorical Logic:
Categorical propositions: Propositions of a subject-predicate
type connected by a copula (is, are).
The subject is an individual (e.g.
Socrates, John, this book) or a category
(eg. men, elephants, green things), and the predicate is a category. Propositions
define a relationship between the subject and predicate.
Types of categorical propositions (Aristotelian):
a) Universal affirmative, (e.g. All men are mortal;
All water is wet)
b) Universal negative, (e.g. No men are mortal; No apple
is sweet)
c) Particular affirmative, (e.g. Some men are mortal;
Some paperclips are plastic), and
d) Particular negative, (e.g. Some men are not mortal;
Some talk show hosts are racists)
In studying Categorical Logic, one can substitute variables
(e.g. A, B, S, P) for the categories.
In a syllogism the two premises each contain a common
category (the middle term) and a unique one. The conclusion contains the
unique categories from the premises. (e.g., Some A are B, No B are C, Therefore
No A are C). Some syllogisms are valid and some (such as this example)
are invalid.
III. Relational terms and linear reasoning.
Relations: There are relations in addition to
a copula. Aristotle did not consider these in his (categorical)
logic. One can specify various relations between the subject and predicate
terms (e.g. greater than, equal to, not equal to, hit). The relationship
has content; different relations imply different logical properties.
If a relation is between
two objects, it is called a binary
relation, and the objects are referred to as arguments of the relation.
In the proposition A is greater than B, (Formally this can be written
aGb) `A' and `B' are the `arguments' of the binary relation greater than.
There are higher order relations (e.g., give, sell; A gave B to C)
which have more than two arguments.
Linear or ordered relations. These are relations
that can be applied to syllogism-like arguments. These syllogisms contain
relations that are binary, transitive, asymmetrical, and nonreflexive (e.g.
greater than, larger than, faster than).
Transitive (def.):
if aRb, and bRc, then aRc.
Asymmetrical (def.):
if aRb, it is not the case that bRa.
Nonreflexive (def.):
it is not the case that aRa.
Intransitive (e.g.
next to, sees): aRb and bRc does not imply aRc.
Symmetrical (e.g.
similar to, near, equal to, not equal to), (def.): If aRb, then bRa.
Reflexive (e.g. identical
to, equal to) (def.): aRa.
IV. Conditional Reasoning and Symbolic Logic.
Symbolic logic refers to formal reasoning systems.
Reasoning principles are rules of inference based on
the form of the representations.
Well-formed formulas:
A notational system will define symbol strings that represent propositions.
Other strings cannot formally represent propositions.
In Elementary propositional or sentential logic,
Simple propositions are not formally analyzed. They are
often represented by a single letter, (e.g., A, B). Compound propositions
are formed by combining simple propositions using "logical terms" to connect
propositions with one another.
Logical terms include
not, which relates to a single proposition (e.g., ~A. If A is The
sky is blue, ~A (not A) is It is not the case that the sky is blue,
or The sky is not blue). Some other logical connectives are: and,
or, if then, and if and only if. These relate two
propositions (e.g., A&B, AÚ B, AÉ
B, Aº B, read A and B, A or B,
If A then B, and A if and only if B, respectively.
In many logics these connectives are truth
functional; that is, they assign truth values to the compound
propositions only as a function of the truth values of the simpler propositions
they contain.
Law of the excluded middle--All
propositions are either true or false. This means we must assign a truth
value to any sentence if we are to analyze an argument using this logic.
Most (if not all) logics of this type obey this law.
Truth functional logic--Truth
of composite propositions is determined by the truth of its component propositions
Rules of inference: Principles by which one can verify
the validity of conclusions from premises. Some rules of inference in propositional
logic include Modus Ponens (If AÉ
B, and A, conclude B), Modus Tollens
(If AÉ B, and ~B, conclude ~A).
Hypothetical Syllogism
(If AÉ B, and BÉ
C, conclude AÉ C)
Tautology: a logically
true sentence--A sentence which has to be true by form alone, i.e. whatever
the truth of its component propositions, it is true. E.g. The Yankees
will either win the World Series this year or they will not win the World
Series this year.
Empirical sentence--A
sentence representing a proposition that may be true or false. E.g.
It is raining in Beijing.
Logically false sentence--A
sentence that must be false as determined by its form alone. E.g. It
is raining and it is not raining in Beijing. (Joe gave 50%
of the pie he baked to each of three friends. is a false sentence and
is necessarily false, but its falseness depends on the meanings of the
terms 50% and three, not on the sentence form, therefore it is not false
because of its form.)
Truth table— The
truth of the composite propositions in certain logics are determined entirely
by the truth of its component propositions. A way to represent truth of
composite propositions as a function of the truth of its component propositions.
These may represent whether a composite proposition is a tautology, an
empirical sentence, or a logically false one. Truth tables may also be
used to evaluate whether an argument is valid or not.
In a truth table the top row contains the elementary propositions
and the set of composite propositions that comprise the proposition under
analysis. For each of the remaining rows of the table the truth of the
elementary propositions are assigned systematically such that each row
has a different combination of truth values. The truth of the composite
propositions are then determined by the rules of assignment, that is what
their truth value would be if the elementary propositions had the truth
of their assignment for that row. One might envision a given row as a possible
world where the truths af the elementary propositions are as assigned.
In that world, in a truth functional logic, each composite proposition
has the truth of its rule-governed assignment. Some elementary examples:
P, Q, and R stand for elementary propositions; ~ means 'not'; & means
'and,' Ú means 'or,' É
means 'if…then…', and º means 'if and only
if'. In the tables T is the semantic assignment of 'true' and F the assignment
of 'false.'
Table 1
Table 1 asserts if P is true, not P is false (row 1);
and if P is false, not P is true (row 2).
Table 2
| P |
Q |
P&Q |
PÚ
Q |
P É
Q |
Pº
Q |
| T |
T |
T |
T |
T |
T |
| T |
F |
F |
T |
F |
F |
| F |
T |
F |
T |
T |
F |
| F |
F |
F |
F |
T |
T |
Table 2 represents several of the logical connectives.
In it you can see that according to this simple logic
a. P&Q is true if P is true and Q is true,
and is false otherwise (From columns 1, 2, and 3).
b. PÚ Q is true
if P is true, if Q is true, or if both P and Q are true. It is false only
when both arguments, P and Q are false.
c. P É Q is false
only when P is true and Q is false. It is true whenever P is false. These
latter cases are called "counterfactual conditionals and are the basis
for much discussion and controversy.
d. Pº Q is true
if the truth values of P and Q are the same and false if they are different.
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