As researchers working in government, influencing service design, we need to know that our research is methodologically sound, our research findings are grounded in empirical data and our recommendations are logically derived. 'Understanding arguments, reasoning and hypotheses' is the first in a series of 5 short courses, covering introduction courses to various aspects of methodology in research, from the use of grounded theory in discovery research, to hypothesis testing and sampling in more experimental research. In this course, you'll learn: About arguments - what we mean by an argument - how to identify a valid/invalid argument - what we mean by premises - what validity and soundness of arguments mean About reasoning - what is deductive reasoning and where do we use it - what is inductive reasoning and where do we use it - what is abductive reasoning and where do we use it About hypotheses - what is a hypotheses and a null hypothesis - how do we test them

1. Maria Rosala
Calum McNamara

2. § Refreshments and Breaks
Coffee break 11:30 – 11:45
Lunch break 13:00 - 14:00
§ Fire exits
§ Toilets

3. § Name & Role
§ What you know already about arguments,
reasoning and hypotheses
§ What you’d like to learn today

5. Proposition
Paradox

7. WHAT THEY ARE
HOW TO CONSTRUCT THEM
HOW TO CONSTRUCT THE NULL HYPOTHESIS
CONFIRMING THE HYPOTHESIS

9. Part 1:

10. Logic is the study of arguments,particularly
of valid arguments.
A logician is a person who worries about
arguments and they can be from anywhere
science,technology,politics…etc.
How can we tell good arguments from bad
arguments,from assessing the sentences
themselves.
Fathers of classical
logic:
Frege (right)
Leibniz (bottom right)

16. WHICH OF THESE ARE ‘BAD’ ARGUMENTS?

20. Part 2:

22. (SOMETIMES REFERRED TO
AS BELIEFS)
Propositions can be thought of as simple declarative sentences
ü It is raining
ü Elephants like peanuts
ü Paris is the capital of Germany
A proposition is something which can be either true or false.
In logical jargon, we’d say that a proposition must have a truth-
value.

24. WHICH OF THESE ARE PROPOSITIONS?

25. Sentences like the below aren’t propositions.
‘What’s the time?’
‘pass me the salt’
ØWhy?

26. Sentences like the below aren’t propositions.
‘What’s the time?’
‘pass me the salt’
ØWhy?
A useful tool for ‘testing out’a proposition is to stick the
phrase‘It’s true that…’at the start of the sentence,and see
whether it makes sense.
‘It’s true that “it’s raining outside’’makes perfect sense!

27. PART 3:

29. A counter-example is a statement which disproves the assertion
made by a given proposition.Take the following proposition:
‘All birds can fly’
Can you think of a counter-example?

30. The beauty of arguing by counter-examples is that,if you can
find just one, then you can be sure that the form of the argument
is invalid.
Note:The argument itself might be intuitively ‘good’,but if you
discover a counter-example,it can no longer be classed as
‘logical’.

31. Part 4

33. § Consistency applies to sets of propositions.
§ For a set of propositions to be consistent,then,there must be at
least one situation in which they could all be true together.
§ If that hypothetical situation is impossible—that is, if the
propositions in our set couldn’t all be true at the same time—
then we say the set is inconsistent.

34. § A set of propositions is consistent if, and only if, it’s possible that
all those beliefs could be true at the same time.
§ A set of propositions is inconsistent if, and only if, it’s impossible
that the beliefs could all be true at the same time.

36. IDENTIFY THE CONSISTENT SETS OF
PROPOSITIONS

37. Part 5

42. Philosophers (mathematicians,scientists,and so on) use the
term‘argument’in a precise and narrow sense.
An argument is made up of‘propositions’ which either act as the
premise(s) or the conclusion.
Here’s an example of an argument:
All men are mortal
Socrates is a man
Therefore, Socrates is Mortal

43. Philosophers (mathematicians,scientists,and so on) use the
term‘argument’in a precise and narrow sense.
An argument is made up of‘propositions’ which either act as the
premise(s) or the conclusion.
Here’s an example of an argument:
All men are mortal Premise (universal)
Socrates is a man
Therefore, Socrates is Mortal

44. Philosophers (mathematicians,scientists,and so on) use the
term‘argument’in a precise and narrow sense.
An argument is made up of‘propositions’ which either act as the
premise(s) or the conclusion.
Here’s an example of an argument:
All men are mortal Premise (universal)
Socrates is a man Premise
Therefore, Socrates is Mortal

45. Philosophers (mathematicians,scientists,and so on) use the
term‘argument’in a precise and narrow sense.
An argument is made up of‘propositions’ which either act as the
premise(s) or the conclusion.
Here’s an example of an argument:
All men are mortal Premise (universal)
Socrates is a man Premise
Therefore, Socrates is Mortal Conclusion

46. Arguments attempt to expand our knowledge.If you have
good reason to believe an argument’s premises,then a well-
structured argument will give you good reason to believe the
conclusion too.
In logic,the argument is the smallest individual piece of
reasoning.(If arguments are the ‘atoms of reasoning’,then
propositions are the sub-atomic particles.)

48. IDENTIFY THE ARGUMENTS

49. Part 6

51. Validity refers to arguments:arguments are either valid or
invalid
A valid argument is one where it is impossible that the premises
all be true and the conclusion false.
v All humans breathe air
v I am a human
v Therefore,I breathe air

52. Another way to think of this is as follows: if the argument’s
premises are all true, then the conclusion must be true also.
If there is even one situation in which the argument’s premises
are all true, but its conclusion is false, then we say the argument
is invalid.

54. CONSTRUCT A VALID ARGUMENT

56. IDENTIFY THE VALID ARGUMENTS

57. Part 7

59. A valid argument whose premises are all actually true is called a
sound argument.
§ Note:Every sound argument is a valid argument;it is not
possible for an argument to be invalid and sound!

61. CONSTRUCT A SOUND ARGUMENT

62. § Arguments can only be valid or invalid. Propositions can only be
true or false.
§ There is no such thing as a ‘true argument’ or ‘false argument.’
Likewise, there is no such thing as a ‘valid belief’ or an ‘invalid
belief.’
§ A valid argument whose premises are all true is called a sound
argument.

64. So far,we’ve said that an argument is valid if
(and only if) the conclusion follows necessarily
from the premises.
However, we can hone this idea a little more by
introducing the notions of deduction and
induction.

67. Part 1

69. We have already covered deduction,when we covered
arguments;a deductive argument is where the premises supply
all the information we need to see that the conclusionis true.
v If it’s a raven,then it will be black
v It is a raven
v So,it will be black
Here,the premises of the argument supply all the information
we need to say whether the conclusion is true or false.

71. CONSTRUCT A DEDUCTIVE ARGUMENT OF
YOUR OWN

74. Part 2

76. An inductive argument is one which moves from
observations to a universal statement.
It takes the following form:
Raven no.1 is black
Raven no.2 is black
Raven no.3 is black
…
Raven no.n is black
So, all ravens are black!
Francis Bacon
1561-1626
Philosopher, scientist

78. WHERE MIGHT THIS BE A PROBLEM?

79. In deduction,the truth of the conclusion
follows from the truth of the premise.But, in
induction,the truth of the conclusion is not
guaranteed by the truth of the premises.
The philosopher Ian Hacking calls them
‘risky arguments’for just this reason.As a
result,inductivearguments can be very
good arguments—but they can never be
valid!

81. WHICH OF THESE ARGUMENTS ARE
INDUCTIVE, AND WHICH ARE DEDUCTIVE?

84. PART 3:

85. Fans of Sherlock Holmes might recall that
character’s frequent references to the
‘science of deduction’.
ØIs this use of the word‘deduction’
correct?
Pixabay.com. Creative Commons license CC0.

86. Actually, what Sherlock is usually doing could be described as
abductive reasoning.
Abductive reasoning, then, can be thought of as inference to
the best explanation.
A lot of what you’ll do as researchers will involve some form of
abductive reasoning.

87. Suppose I come home and find that all the milk that I had in the
fridge has disappeared.How did this happen?
Any number of hypothetical situations is possible: perhaps a
thirsty burglar broke into my house! Perhaps,for some
inexplicable reason,the fridge became very hot and all the milk
was evaporated.However,much more likely than these is that I’d
drank all the milk, and had simply forgotten.

88. We should note,however,that,like inductive arguments,
arguments based on abductive reasoning carry an element of
risk. But, just because an argument isn’t valid in the narrow sense
we’ve described doesn’t mean that it’s a bad argument!

90. §
Image sourced and adapted from
https://bookofbadarguments.com. Ali Almossawi. Creative
Commons BY-NC license

91. PART 1:

92. Affirming the consequent (i.e. the
conclusion) is a formal fallacy
which takes the form:
v If P --> Q
v Q
v Therefore,P
Image sourced and adapted from
https://bookofbadarguments.com. Ali Almossawi. Creative
Commons BY-NC license

93. The quantifier-shift fallacyis a logical fallacy in which
the different quantifiers used in a statement get mixed
up.
‘Every event has a cause.So,there must be one cause
for every event.’
(We’ll return to this one!)

94. PART 2:

95. This fallacy is committed
when one forms a
conclusion from a sample
that is either too small or
too unique to be
representative.
Image sourced and adapted from
https://bookofbadarguments.com. Ali Almossawi. Creative
Commons BY-NC license

96. When one event is believed to have caused by
another because of their co-occurrence or where
one event is seen to have preceded another.

97. A false dilemma occurs
when only limited options
are presented,despite the
fact that at least one other
option is possible.
“Your either with us,or with
the fanatics”
Image sourced and adapted from https://bookofbadarguments.com. Ali Almossawi.
Creative Commons BY-NC license

98. The fallacy of equivocationis an
informal,semantic fallacy where
the a term is used which has
more than one meaning (but the
meaning which is intended is
not made clear).
It makes for a lot of our British
jokes…
“The sign said "fine for parking
here",and since it was fine,I
parked there.”
Image sourced and adapted from https://bookofbadarguments.com. Ali Almossawi.
Creative Commons BY-NC license

99. The fallacy of composition
involves attributing a certain
property to a set of things,after
observing that each individual
member of the set has that
property.
And,the fallacy of division,
involves attributing a certain
property to members of a set,
after observing that the set itself
exhibits that property.Image sourced and adapted from https://bookofbadarguments.com. Ali Almossawi.
Creative Commons BY-NC license

100. A slippery slope
argument attempts to
discredit a proposition
by arguing that its
acceptance will
undoubtedly lead to a
sequence of events,one
or more of which are
undesirable
Note the probability of that
eventuality can sometimes be
infinitesimally small!
P(0.2) x Q(0.1) x R(0.6)…
Image sourced and adapted from https://bookofbadarguments.com. Ali Almossawi.
Creative Commons BY-NC license

101. A.K.A. the existence of
[God, aliens, you fill in the
blank] argument
This kind of argument
assumes a proposition to be
true simply because there is
no evidence proving that it
false.
Hence,absence of evidence
is taken to be evidence of
absence.

102. REDO

103. SPOT THE FALLACY

104. § Propositions are entities that can either be true or false
§ Arguments are sets of propositions containing some premises and a
conclusion
§ Propositions are said to be consistent when they can all be true
together
§ Arguments are valid when it is impossible that their premises be true
and their conclusions false.

105. PART 1:

106. Propositional logic (the logic of propositions) has five different
‘connectives’ for linking sentences up.
You’ll probably be familiar with all of them already: they
correspond (roughly) to the English words ‘and’, ‘or’, ‘implies’, ‘if
and only if’, and ‘not’.
Let’s take a look at these more closely.

107. P and Q be two arbitrary English sentences.
P might be‘it’s raining’,
while Q might be‘it’s wet outside’.
Now suppose we join these sentences up with the connective ‘and’
to form the new sentence ‘P and Q’.
What’s different about this new sentence?

108. The difference is that the truth of the new sentence depends on
the truth of each of the parts represented by the sentence-letters
P and Q, respectively.
If P is ‘It’s raining’ and Q is ‘It’s wet outside’…
then for the new proposition ‘It’s raining and it’s wet outside’
to be true, both of the ‘smaller’ sentences must be true in turn.

109. We can represent this idea
in something called a
Truth-Table.
Note,then,that the
proposition ‘P and Q’ (‘It’s
raining and it’s wet
outside’) is true only when
P is true and Q is true.
P Q (P and Q)
T T T
T F F
F T F
F F F

110. P Q (P or Q)
T T
T F
F T
F F

111. P Q (P or Q)
T T T
T F T
F T T
F F F

112. This one is the easiest of the
logical connectives. For any
sentence,P, adding the word‘not’
simply reverses the truth-value of
P.
To assert that ‘It is not raining’ is
just to say that the proposition ‘It
is raining’ is false.
P not-P
T F
F F

113. Suppose we postulate that P
implies Q.What does this
actually mean?
Going back to our natural
language equivalents,we might
say that ‘It is raining implies that
it is wet’.
Another way to say this is ‘If it is
raining, then it must be wet’.
P Q (P implies Q)
T T T
T F F
F T T
F F T

114. This has unintuitive implications. Suppose we have an implication
like ‘If pigs can fly, then I will be king.’
In this case, both parts of the implication are false.
But, consult the truth-table:if both parts of the implication are
individually false, then the implication as a whole is true.

115. If you find this one hard to stomach,then don’t worry: you’re not
alone!
Just remember that we’re just interested in saying when a certain
state of affairs must entail another,and not in whether or not
either of those states of affairs is actually possible.

116. In essence,it’s just another way of
saying‘P implies Q and Q implies
P.’
Another way to write this is P
implies Q if and only if, Q implies
P.
P Q (P if and only if Q)
T T T
T F F
F T F
F F T

117. Knowing the truth-tables provides you with a useful method for
recognising when propositions,hypotheses,etc., are legitimate and
when they are not.
For instance,knowing that the presence of fire implies the presence
of smoke does not necessarily give you any reason to believe that
the reverse holds too.

118. PART 2:

119. Another key insight of modern formal logic is into how words like
‘all’, ‘some’, or ‘none’affect the truth of a sentence,or the validity
of an argument.
Words like these are known as quantifiers. There are 2 types of
quantifier.

120. ∀(an upside down ‘A’).You can read this symbol as‘for all’ or
‘everything’, etc.You might like to think about statements
involving this quantifier as being always true or always false.
∃(backwards‘E’), and it can be read as ‘there exists’,‘there is at
least one’,‘some’,‘many’, etc.Basically, it’s anything other than
‘all’. Conversely,you might like to think about statements
involving this quantifier as sometimes true, and sometimes false.

121. People mix them up all the time! (Quantifier shift fallacies)
ØLooks plausible?
‘Every event has a cause. So, there must be one cause for each
event.’
ØWhat about this one?
‘Everyone has a mother.Therefore,someone must be the mother
of everyone’—this is an obviously invalid assertion!

122. PART 3:

123. v If P --> Q
v P
v Therefore,Q
§ EXAMPLE: I know that if it rains, it will be wet outside. I also
known that it is raining now.
Thus, by modus ponens, I am logically justified in inferring that
it’s wet outside as well.

124. (SISTER OF MODUS PONENS)
v If P --> Q
v -Q
v Therefore,-P
§ EXAMPLE: Suppose I know that if it’s raining,it will be wet
outside.However,upon looking outside,I find that it is not
wet outside.From this, I am justified in inferring that it’s not
raining either.

125. (PROOF BY CONTRADICTION)
this argument form allows us to infer one
proposition,P,by showing that its negation,not-P,
leads to contradiction.
§ EXAMPLE: Galileo’s proof of the law of falling
bodies
(which says that the distance travelled by a falling
body is proportional to the square of the time).

128. Remember propositions?
A hypothesis is a proposition about a state of the world…
E.g. Plants require light for photosynthesis
or more advanced:the level of light plants require for
photosynthesis is proportionate to the rate of photosynthesis

129. We have done lots of observation and we have a hunch about a
causal mechanism
X causesY or X is a cause and affectsY to some unknown
extent Z
We want to test whether our hunch is true!

130. To test the hypothesis, we construct the null hypothesis (which is
the exact negation of the hypothesis).
E.g.
Hypothesis: Plants require light for photosynthesis
Null Hypothesis: Plants do-not require light for photosynthesis

132. WHY DO WE TEST THE NULL HYPOTHESIS
AND NOT THE HYPOTHESIS?

133. Let’s suppose we test the hypothesis.
vIf the theory is correct, it implies that we could observe
Phenomenon X or Data X.
vX is observed.
vHence,the theory is correct.
What’s wrong with this?

134. Let’s look at a similar example..
vIf Jefferson was assassinated,then Jefferson is dead.
vJefferson is dead.
vTherefore Jefferson was assassinated.
It’s invalid and a fallacy. Remember affirmation of the
consequent?

135. If we want to validly confirm the hypothesis,we therefore test the
null, in the attempt to reject it.
Remember our double negative?
Not (Not P) = P
If we can validly conclude P, this acts as confirmation for the
hypothesis.
N.B.We never say we’ve proved the hypothesis!

137. CONSTRUCT A HYPOTHESIS, A NULL
HYPOTHESIS AND AN EXPERIMENT TO TEST
YOUR NULL HYPOTHESIS

139. TRUE OR FALSE?

141. HTTP://WWW.SMARTSURVEY.CO.UK/S/BY103/