Logic
A short primer on Deduction and Inference
We will look at Symbolic Logic in order to examine how we employ
deduction in cognition.
Revised 4/13/2015
Slide # 1
Logic
What is Logic?
• Logic
• For CS/Math Types: from Factasia
• A formal language, given a precisely defined syntax and
semantics, becomes a logic when rules for correct
reasoning in the language are formally described
• For Philosophers: from Factasia
• Logic is the study of necessary truths and of formal
systems for deriving them.
• The study by which arguments are classified into
good ones and bad ones. (Irving Copi)
Revised 4/13/2015
Slide # 2
Logical Systems
• There are actually many logical systems
• The one we will examine in class is called RS1
(I think…it is a simple propositional calculus, in any
event.)
• It is comprised of
• Statements
• "Roses are red“
• "Republicans are Conservatives“
• “P”
• Operators
• And
• Or
• not

Revised 4/13/2015
Some Rules of Inference
Slide # 3
Logic
Compound Statements
• Conjunctions (Conjunction Junction)
• Two simple statements may be connected with a
conjunction
• The conjunction “and”
• The disjunction “or”
Revised 4/13/2015
Slide # 4
The conjunction operator


“and”
Symbolized by “•”



Revised 4/13/2015
"Roses are Red and Violets are blue.“
"Republicans are conservative and Democrats
are liberal.“
P • Q (P and Q)
Slide # 5
The disjunction operator


“or”
Symbolized by “v”


Revised 4/13/2015
"Republicans are conservative or
Republicans are moderate
PvQ
Slide # 6
Negation


Not
Symbolized by ~



Revised 4/13/2015
That is not a rose
Bob is not a Republican
~A
Slide # 7
Operators


These may be used to symbolize
complex statements
The other symbol of value is


Revised 4/13/2015
Equivalence ()
This is not quite the same as “equal to”.
Slide # 8
Truth Tables


Statements have “truth value”
For example, take the statement P•Q:

P
T
T
F
F
Revised 4/13/2015
This statement is true only if P and Q are both
true.
Q P•Q
T
T
F
F
T
F
F
F
Slide # 9
Truth Tables (cont)


Hence “Republicans are conservative and
Democrats are liberal.” is true only if both
parts are true.
On the other hand, take the statement PvQ:

P
T
T
F
F
Revised 4/13/2015
This statement is true only if either P or Q are
true, but not both. (Called the “exclusive or”)
Q PvQ
T
F
F
T
T
T
F
F
Slide # 10
The Inclusive ‘or’


Note that ‘or’ can be interpreted differently.
Both parts of the disjunction may be true in
the “inclusive or”. This statement is true if
either or both P or Q are true.
P
T
T
F
F
Revised 4/13/2015
Q
T
F
T
F
PvQ
T
T
T
F
Slide # 11
The Exclusive ‘or’


With the exclusive or, of p is true, than q
cannot be.
Only one part of the disjunction may be true
in the “exclusive or”. This statement is true if
either P is true or Q is true, but not both.
P
T
T
F
F
Revised 4/13/2015
Q
T
F
T
F
PvQ
F
T
T
F
Slide # 12
The Conditional

The Conditional




(antecedent)
(consequent)
It is also called the hypothetical, or
implication.
This translates to:




if a
then b
A implies B
If A then B
A causes B
Symbolized by A  B
Revised 4/13/2015
Slide # 13
The Implication


We use the conditional or implication a
great deal.
It is the core statement of the scientific
law, and hence the hypothesis.
Revised 4/13/2015
Slide # 14
Equivalency of the Implication

Note that the Implication is actually
equivalent to a compound statement of
the simpler operators.


~p v q
Please note that the implication has a
broader interpretation than common
English would suggest
Revised 4/13/2015
Slide # 15
Rules of Inference


In order to use these logical
components, we have constructed
“rules of Inference”
These rules are essentially “how we
think.”
Revised 4/13/2015
Slide # 16
Modus Ponens
AB
A


B
This is the classic rule of inference for
scientific explanation.
Revised 4/13/2015
Slide # 17
Modus Tollens
AB
~B
 ~A

This reflects the idea of rejecting the
theory when the consequent is not
observed as expected.
Revised 4/13/2015
Slide # 18
Disjunctive Syllogism
Av B
~A
 B
Revised 4/13/2015
Slide # 19
Hypothetical Syllogism
AB
B C
A  C

Classic reasoning



Revised 4/13/2015
All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.
Slide # 20
Complex propositions

Take for instance - US




If Weapons  USAttack
If Weapons  ~Insp
 If ~Insp  Weapons
 If ~Insp  USAttack
Revised 4/13/2015

Take for instance - Iraq



If Insp  ~Weapons
If Weapons  ~USAttack
If Insp  USAttack
Slide # 21
Logical Systems


Logic gives us power in our reasoning
when we build complex sets of
interrelated statements.
When we can apply the rules of
inference to these statements to derive
new propositions, we have a more
powerful theory.
Revised 4/13/2015
Slide # 22
Tautologies





Note that p v ~p must be true
“Roses are red or roses are not red.” must be
true.
A statement which must be true is called a
tautology.
A set of statements which, if taken together,
must be true is also called a tautology (or
tautologous).
Note that this is not a criticism.
Revised 4/13/2015
Slide # 23
Tautologous systems

Systems in which all propositions are by
definition true, are tautologous.





Balance of Power
Why do wars occur? Because there is a change in
the balance of power.
How do you know that power is out of balance? A
war will occur.
Note that this is what we typically call circular
reasoning.
The problem isn’t the circularity, it is the lack
of utility.
Revised 4/13/2015
Slide # 24
Useful Tautologies

Can a logical system in which all
propositions formulated within be true
have any utility?




Try Geometry
Calculus
Classical Mechanics
But not arithmetic

Revised 4/13/2015
Kurt Gödel & his Incompleteness Theorem
Slide # 25

The Liars Paradox


The Paper Paradox (a variant of the Liar’s paradox)




Epimenedes the Cretan says that all Cretans are liars.“
< The next statement is true.
< The previous statement is false.
For further info
Russell’s Paradox

The paradox arises within naive set theory by considering
the set of all sets that are not members of themselves.
Such a set appears to be a member of itself if and only if it
is not a member of itself.
Hence the paradox
Revised 4/13/2015
Slide # 26


Grelling’s Paradox

Homological – a word which describes itself



Heterological – a word which does not
describe itself



Short is a short word
English is an English word
German is not a German words
Long is not a long word
Is heterological heterological?
Revised 4/13/2015
Slide # 27
Paradox of voting







It is possible for voting preferences to result in elections in
which a less preferred candidate wins over a preferred one.
See Paradox of Voting
Suppose you have 3 individuals and candidates A, B and C
 Individual 1: A > B > C
 Individual 2: C > A > B
 Individual 3: B > C > A
Now if these individuals were asked to make a group choice
(majority vote) between A and B, they would chose A;
If asked to make a group choice between B and C, they would
chose B.
If asked to make a group choice between C and A, they would
chose C.
So for the group A is preferred to B, B is preferred to C, but C is
preferred to A! This is not transitive which certainly goes against
what we would logically expect.
Revised 4/13/2015
Slide # 28