3. In this lecture we will learn:
How to formulate valid arguments/explanations
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How to test whether an argument/explanation is valid
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The core methods of so called ``propositional logic’’ and ‘‘syllogistic logic’’
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How to generalize and specify concepts and statements
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4. Part 1: How to formulate valid
arguments/explanations.
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6. Arguments consist of premises and a conclusion
Logic is the analysis and appraisal of arguments
An argument is valid means: if all premises are
true it is impossible that the conclusion is wrong
If you are reading this, you aren’t illiterate
You are reading this
You aren’t illiterate
What is logic?
Premise.
Conclusion.
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7. This is wonderful. With the help of logic you can find out
whether a statement (conclusion) is true if you know whether
other statements (premises) are true.
!
Thus, if your assumptions are plausible you can make your
hypotheses/predictions plausible too (no matter how counter
intuitive they are)
No matter how counter intuitive they are???
What is logic?
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8. We distinguish valid from sound arguments.
An argument is valid means: If all premises are true
it is impossible that the conclusion is wrong.
Logic provides techniques to test whether a given argument is valid
An argument is sound means: The argument is
valid plus all premises are true.
Only if an argument is sound, we can be 100% certain that
the conclusion is true.
If the argument is valid and at least one premise is false, the
conclusion can be false
You need empirical research (and further arguments) to test
whether all premises are true.
Note: Arguments are not true or false (statements are!)
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9. Which argument is valid and which is sound?
If economic welfare increases, the rate of unemployment decreases
In the 1990s, in the US, the economic welfare increased
In the 1990s, in the US, the rate of unemployment decreased
If the rate of unemployment decreases, the rate of violent crimes
decreases
In the 1990s, in the US, rate of unemployment decreased
In the 1990s, in the US, the rate of violent crimes decreased
1
2
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11. The members of group x are integrated
The citizens of Leipzig protest
People who hold similar opinions tend to form friendships
Propositions are statements, for instance:
Propositions are statements which are either true or false
(not valid or invalid)
Shut up! (commands)
Why did nobody bring cookies? (question)
This is a bad song (normative statements)
Hence, statements which are not true or false are not
considered propositions. For instance,
Basic Propositional Logic
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12. e.g., “I am a sociologist”
Propositions are translated into so called “wff’s” (pronounce as
woof as in wood). Wff stands for ``well formed formula”
s
Propositions are analyzed using truth tables. Truth tables give
a logical diagram for a given wff, listing all possible truth-value
combinations.
Propositional language and truth tables
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S
1
0
Symbol of the proposition
Truth values: s can be true (1) or false (0)
13. Truth-functional operators
Propositions can be combined, forming new propositions. This is done
with so called operators
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Operators define the truth-value of the combined proposition based on the
truth-values of the propositions that it consists of.
Operator 1: Negation
e.g. Assume, s (“I am a sociologist”) is true (1).
Then, the negation of s (~s) is false (“I am not a sociologist”).
Symbol: ~ (squiggle)
Read: “not”
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s ~s
1 0
0 1
If s is true, then the negation is false
If s is false, then the negation is true
14. Operator 2: Disjunction
Symbol: ⋁ (vee) or || or +
Read: “or”
p q p
1 1 1
1 0 1
0 1 1
0 0 0
The disjunction of p and q is
false if both p and q are false
Operator 3: Conjunction
Symbol: ⋅ (dot) or & or ⋀
Read: “and”
p q p
1 1 1
1 0 0
0 1 0
0 0 0
The conjunction of p and q is
true if both p and q are true
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15. Operator 4: Implication
Symbol: ⊃ (horseshoe) or →
Read: “if p then q”
p q p
1 1 1
1 0 0
0 1 1
0 0 1
The implication of p and q is false
only if p is true and q is false
Example: If Popper is a sociologist, then he is a Marxist.
Popper is a sociologist + Popper is a Marxist : wff is valid
Popper is a sociologist + Popper is not a Marxist : wff is invalid
Popper is not a sociologist + Popper is a Marxist : wff is valid
Popper is not a sociologist + Popper is not a Marxist : wff is valid
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16. Operator 5: Equality (biconditional)
Symbol: ≡ (threebar) or = or
Read: “if and only if p then q”
p q p
1 1 1
1 0 0
0 1 0
0 0 1
The equality of p and q is true if either
p and q are both true or both false
Example: If and only if Popper is a sociologist, then he is a Marxist.
Popper is a sociologist + Popper is a Marxist : wff is valid
Popper is a sociologist + Popper is not a Marxist : wff is invalid
Popper is not a sociologist + Popper is a Marxist : wff is invalid
Popper is not a sociologist + Popper is not a Marxist : wff is valid
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17. Other operators
Exclusive disjunction: true if one but not if
both operands are true (XOR, ≠, ⨁)
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Logical NAND: false if bot operands are true
and true if at least one operand is false (↑,|)
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Logical NOR: true if both operands are false
and false if at least one operand is true (↓,⊥)
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19. True and False
~p
p
Area inside the circle: possible states where p is true
!
Area outside: possible states where p is false
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20. Disjunction
White area: states where the disjunction of p and q is true
Pink area: states where the negation of the disjunction of
p and q is true
p q⋁
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21. Conjunction
Pink area: p and not q
Blue area: q and not p
White area: q and p
Gray area: not (q or p)
p qp⋅q
~(p⋁q)
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23. Working with truth tables
Example: Let us demonstrate for which combination of truth values of
p and q is it is correct to state: “p and q are equivalent (p≡q)”. Thus,
we want to show that:
(if p, then q) and (if q, then p)
p q p q (p
1 1 1 1 1 1 1 1 1 1 1
1 0 1 0 0 0 1 1 0 1 0
0 1 0 1 1 1 0 0 1 0 0
0 0 0 0 1 0 0 1 1 1 1
Definition of
an equality
This proves that: (p≡q)≡((p⊃q)∙(q⊃p))
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25. When we formulate an argument, we infer the conclusion from
the premises.
An argument is valid means:
If all premises are true it is impossible that the conclusion is wrong.
Thus, if all premises are true, then the conclusion is true.
This is an implication (⊃)
In order to show that an argument is valid (that the inference is
correct), we need to demonstrate that the conjunction (·) of all
premises implies (⊃) the conclusion.
There are three important forms of argument
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Rules of Inference
26. Rule 1: Hypothetical Syllogism
Example:
If Popper is a sociologist (p), then he is is a Marxist (q)
If Popper is a Marxist (q), then he hates capitalism (r)
If Popper is a sociologist (p), then he hates capitalism (r)
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General form:
p⊃q
q⊃r
----------
p⊃r
27. Demonstrations that the hypothetical
syllogism is a valid argument form
Thus, we want to demonstrate that the conjunction (·) of all premises
implies (⊃) the conclusion.
Therefore, we need to demonstrate:
if the premises are true, then the conclusion is always true.
This is an implication
This means: (p⊃q)∙(q⊃r) This means: (p⊃r)
We need to show that ((p⊃q)∙(q⊃r))⊃(p⊃r) is true independent
of the truth-values of p, q, and r.
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28. p q r p q (p p ((p
1 1 1 1 1 1 1 1 1 1 1 1
1 1 0 1 0 1 0 0 0 0 0 1
1 0 1 0 1 0 1 0 1 0 1 1
1 0 0 0 1 0 1 0 0 0 0 1
0 1 1 1 1 1 1 1 1 1 1 1
0 1 0 1 0 1 0 0 1 0 1 1
0 0 1 1 1 1 1 1 1 1 1 1
0 0 0 1 1 1 1 1 1 1 1 1
The conjunction of the premises logically implies the conclusion.
Thus, the hypothetical syllogism is always valid (independent of the
truth-values of the truth of the premises)
Is ((p⊃q)∙(q⊃r))⊃(p⊃r) always valid?
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29. Rule 2: Modus Ponens
Example:
If Popper is a sociologist (p), then he is is a Marxist (q)
If Popper is a sociologist (p)
Popper is a Marxist (q)
pq
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General form:
p⊃q
p
----------
q
Venn diagram of an implication
30. Rule 3: Modus Tollens
Example:
pq
Venn diagram of an implication
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General form:
p⊃q
~q
----------
~p
If Popper is a sociologist (p), then he is is a Marxist (q)
If Popper is not a Marxist (~q)
Popper is not a sociologist (~p)
32. Propositional logic focuses on propositions which refer to single
objects (i.e., Popper).
In contrast, syllogistic logic is concerned with domains of objects
Propositional logic:
It rains (r)
Popper is cool (c)
Syllogistic logic:
All swans are white (all S is W)
Societies with high anomie suffer from high
crime rates (all A is C)
With syllogistic logic, we study the implications of general statements
(laws). Remember that our theories are general statements
Typical wffs from:
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Syllogistic Logic
Like propositional logic, it is a branch of logic.
33. Formulating wffs in syllogistic logic
To formulate a correct wff, you need only five words:
all
no
some
is
not
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34. Formulating wffs in syllogistic logic
There are only eight (8) forms of wffs:
all A is B All swans are white
no A is B There are no white swans
some A is B Some swans are white
some A is not B Some swans are not white
x is B This swan is white
x is not B This swan is not white
x is y This is the only white swan
x is not y This is not the only white swan
Any sentence can be translated into a wff of one of these forms
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35. Implications in syllogistic logic
General form of an implication: all A is B
Read: For all objects in the domain, if an object is A then it is B
Use capital letters to refer to domains of objects (all)
Use small letters to refer to single objects (me, Popper)
Why is “all A are B” an implication?
(a1⊃b1)∙(a2⊃b2)∙(a3⊃b3)...(an⊃bn)
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37. Rule 1: Hypothetical Syllogism
Example:
All sociologists (S) are Marxists (M)
All Marxists (M) are against capitalism (C)
All sociologists (S) are against capitalism (C)
Venn diagram
SMC
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General form:
all S is M
all M is C
----------
All S is C
38. Rule 2: Modus Ponens
Example:
All sociologists (S) are Marxists (M)
Popper (p) is a sociologist (S) [p is S]
Popper (p) is a Marxist (M) [p is M]
SM ★ Popper
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General form:
all S is M
p is S
----------
p is M
Venn diagram of an implication
39. Rule 3: Modus Tollens
Example:
All sociologists (S) are Marxists (M)
Popper (p) is not a Marxist (M) [p is not M]
Popper (p) is not a sociologist (S) [p is not S]
SM
★
Popper
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General form:
All S is M
p is not M
----------
p is not S
Venn diagram of an implication
41. Testing whether a syllogism is valid:
The star test
The star test consist of three steps:
Step 1: Find the “distributed letters”
A letter is distributed if it occurs just after “all” or
anywhere after “no” or “not”
Underline the distributed letters
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all A is B
no A is B
x is A
x is not y
42. Step 2: Star premises letters which are distributed
and conclusion letters which are not distributed
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Testing whether a syllogism is valid:
The star test
all A* is B
some C is A
-----------------
some C* is B*
43. Step 3: Decide. A syllogism is valid if and only if
every capital letter is starred exactly once
&
if there is exactly one star on the right hand side
Each capital letter is starred exactly once
There is exactly one star at the right hand
side (see the B)
Thus, this syllogism is valid.
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Testing whether a syllogism is valid:
The star test
all A* is B
some C is A
-----------------
some C* is B*
44. Second example:
Is it a valid syllogism?
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no A* is B*
no C* is A*
-----------------
no C is B
45. Second example:
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no A* is B*
no C* is A*
-----------------
no C is B
A is starred twice.
There are two stars on the right hand side (see A and B)
Thus, there are two reasons why this syllogism is not valid.
47. Abstract & Generalize
Specify - Classify
Relation between humans
Social relations
Friendships
Friendships between students
Friendships between first-years
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48. dyads
Relation between
humans
Social relations
Friendships
Friendships between
students
Friendships
between
first-years
Specify:
Generalize:
Include more characteristics in the
definition of the concept
Fewer objects fall under the
concept
Abstract more details
More objects fall under the
concept
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49. All sociologists (S) are good statisticians (G). (S⊃G)
S=df. Everybody with at
least a Doctor’s degree
in Sociology
S=df. Everybody with a
university degree in Sociology
G=df. Everybody who
can interpret a
regression
G=df. Everybody who
can explain what a
regression is
1
2
4
3
Generalizing and specifying implications
Generalize the implication: from 1 to 3, or from 2 to 4
Specify the implication: from 3 to 4, or from 1 to 2
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50. The information content of an implication
Scientists seek to formulate informative statements. Thus, they
should inform us about many things and make precise predictions
Independent variable (if) should
be general (true for many cases)
Dependent variable (then) should
be very specific (true for few cases)
S=df. Only modern
human societies
S=df. All human societies
D=df. Increase in
complexity
D=df. Increase in
stratification
⊃
Societies (S) Differentiate (D)
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52. Suggested assignment
The suggested assignment is to solve the
first five exercises of each section
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This chapter is part of your reading material,
you must read it before the tutorial
Solutions for these exercises will be provided
in Nestor after the tutorial for you to check
The practical activities in the tutorial assume
you are able to solve exercises as those
suggested to do as assignment
