Understanding arguments, reasoning and hypotheses

Maria RosalaCalum McNamara

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Proposition

Paradox

WHAT THEY AREHOW TO CONSTRUCT THEMHOW TO CONSTRUCT THE NULL HYPOTHESISCONFIRMING THE HYPOTHESIS

Part 1:

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)

WHICH OF THESE ARE ‘BAD’ ARGUMENTS?

Part 2:

(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.

WHICH OF THESE ARE PROPOSITIONS?

Sentences like the below aren’t propositions.

‘What’s the time?’

‘pass me the salt’

ØWhy?

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!

PART 3:

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?

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’.

Part 4

§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.

§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.

IDENTIFY THE CONSISTENT SETS OF PROPOSITIONS

Part 5

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 mortalSocrates is a manTherefore, Socrates is Mortal



All men are mortal Premise (universal)Socrates is a manTherefore, Socrates is Mortal



All men are mortal Premise (universal)Socrates is a man PremiseTherefore, Socrates is Mortal



All men are mortal Premise (universal)Socrates is a man PremiseTherefore, Socrates is Mortal Conclusion

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.)

IDENTIFY THE ARGUMENTS

Part 6

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 airv I am a human

v Therefore, I breathe air

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.

CONSTRUCT A VALID ARGUMENT

IDENTIFY THE VALID ARGUMENTS

Part 7

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!

CONSTRUCT A SOUND ARGUMENT

§ 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.

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.

Part 1

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 conclusion is true.

v If it’s a raven, then it will be blackv It is a ravenv 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.

CONSTRUCT A DEDUCTIVE ARGUMENT OF YOUR OWN

Part 2

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 Bacon1561-1626Philosopher, scientist

WHERE MIGHT THIS BE A PROBLEM?

In deduction, the truth of the conclusion follows from the truth of the premise. But, in induction, the truth of the conclusion is notguaranteed by the truth of the premises.

The philosopher Ian Hacking calls them ‘risky arguments’ for just this reason. As a result, inductive arguments can be very good arguments—but they can never be valid!

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

PART 3:

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.

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.

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.

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!

§

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

PART 1:

Affirming the consequent (i.e. the conclusion) is a formal fallacy which takes the form:

v If P --> Q

v Q

v Therefore, P


The quantifier-shift fallacy is 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!)

PART 2:

This fallacy is committed when one forms a conclusion from a sample that is either too small or too unique to be representative.


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.

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

The fallacy of equivocation is 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.”


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 certainproperty to members of a set, after observing that the set itselfexhibits that property.


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

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.

SPOT THE FALLACY

§ 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.

PART 1:

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.

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?

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.

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 (PandQ)

T T T

T F F

F T F

F F F

P Q (PorQ)

T T

T F

F T

F F

P Q (PorQ)

T T T

T F T

F T T

F F F

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

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 (PimpliesQ)

T T T

T F F

F T T

F F T

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.

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.

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 (PifandonlyifQ)

T T T

T F F

F T F

F F T

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.

PART 2:

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.

∀ (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.

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!

PART 3:

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.

(SISTER OF MODUS PONENS)

v If P --> Qv -Qv 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.

(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).

Remember propositions?

A hypothesis is a proposition about a state of the world…

E.g. Plants require light for photosynthesisor more advanced: the level of light plants require for photosynthesis is proportionate to the rate of photosynthesis

We have done lots of observation and we have a hunch about a causal mechanism

X causes Y or X is a cause and affects Y to some unknown extent Z

We want to test whether our hunch is true!

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

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

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?

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?

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!

CONSTRUCT A HYPOTHESIS, A NULL HYPOTHESIS AND AN EXPERIMENT TO TEST

YOUR NULL HYPOTHESIS

TRUE OR FALSE?

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Understanding arguments, reasoning and hypotheses

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