Introduction to Formal Logic

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Introduction to Formal Logic
Richard.Pettigrew_at_bris.ac.uk
Lecture 2
The form of arguments
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Recap
Logic is the study of arguments.
An argument is a set of premises along with the
conclusion.
The premises and conclusions of an argument are
propositions.
Propositions are the sort of thing that may be
true or false. Many sentences may express the
same proposition. Two sentences express the same
proposition if they have the same meaning.
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Recap on propositions
Three different sentences all express the same
single proposition.
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What properties of arguments will concern us?
An argument is valid if the truth of its premises
guarantees the truth of its conclusion that is,
if there is no possible circumstance in which the
premises are true and the conclusion is false.
An argument is invalid if the truth of its
premises does not guarantee the truth of its
conclusion that is, if there is a possible
circumstance in which the premises are true and
the conclusion is false.
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Examples of valid arguments
If Cameron wins the election, he will abandon his
environmental zeal. He will win the election.
Therefore, he will abandon his environmental zeal.
If grass is red, then Jade Goody is from East
Anglia. The grass is red. Therefore, Jade Goody
is from East Anglia.
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The moral of the story
An argument may be valid even if all its premises
and its conclusion are false. Whether an
argument is valid or not concerns only the
relationship between the premises and the
conclusion. It does not concern the relation
of truth between the premises and the world, or
the conclusion and the world.
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The sun has risen on the Earth every morning
since at least 5000BC. Therefore, it will rise
tomorrow morning.
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The moral of the story
An argument may be invalid even if its premises
make its conclusion very likely.
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How can we show that an argument is valid?
Strategy
1) Identify a general form of which the argument
in question is a particular instance.
2) Show that all arguments of this form are
valid. (We do this by one of two techniques
truth-tables or tree proofs.)
3) Conclude that, in particular, the argument in
question is valid.
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What is the form of an argument?
Melodic form in music
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Rhyme form in poetry
Chorus Hey, Mr. Tambourine Man, play a song for
me, I'm not sleepy and there ain't no place I'm
going to. Hey, Mr. Tambourine Man, play a song
for me, In the jingle jangle morning I'll come
following you. Verse Though I know that
evenings empire has returned into
sand, Vanished from my hand, Left me blindly here
to stand but still not sleeping. My weariness
amazes me, I'm branded on my feet, I have no one
to meet, And the ancient empty street's too dead
for dreaming.
A B A B
C C D E E D
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Argument form in logic
If Cameron wins the election, then he will
abandon his environmental zeal. He will win the
election. Therefore, he will abandon his
environmental zeal.
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Argument form in logic
If Cameron wins the election, then he will
abandon his environmental zeal. He will win the
election. Therefore, he will abandon his
environmental zeal.
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Argument form in logic
If Cameron wins the election, then he will
abandon his environmental zeal. He will win the
election. Therefore, he will abandon his
environmental zeal.
p Cameron wins the election. q Cameron will
abandon his environmental zeal.
If p, then q. p. Therefore, q.
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Argument form in logic
If porridge oats are good for you, then he should
eat them. Porridge oats are good for you.
Therefore, he should eat them.
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Argument form in logic
If porridge oats are good for you, then he should
eat them. Porridge oats are good for you.
Therefore, he should eat them.
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Argument form in logic
If porridge oats are good for you, then he should
eat them. Porridge oats are good for you.
Therefore, he should eat them.
p Porridge oats are good for you. q He should
eat them.
If p, then q. p. Therefore, q.
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Argument form in logic
Spiders have eight legs. So, it is not true that
spiders are insects, for if they were insects,
then it would not be true that they have eight
legs.
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Argument form in logic
Spiders have eight legs. So, it is not true that
spiders are insects, for if they were insects,
then it would not be true that they have eight
legs.
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Argument form in logic
Spiders have eight legs. So, it is not true that
spiders are insects, for if they were insects,
then it would not be true that they have eight
legs.
p Spiders have eight legs. q Spiders are
insects.
If q, then p is not true. p. Therefore, q is not
true.
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Argument form in logic
If the grass is green, then it is not true that
Jade Goody lives in East Anglia. Jade Goody does
live in East Anglia. Therefore, it is not true
that the grass is green.
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Argument form in logic
If the grass is green, then it is not true that
Jade Goody lives in East Anglia. Jade Goody does
live in East Anglia. Therefore, it is not true
that the grass is green.
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Argument form in logic
If the grass is green, then it is not true that
Jade Goody lives in East Anglia. Jade Goody does
live in East Anglia. Therefore, it is not true
that the grass is green.
p The grass is green. q Jade Goody lives in
East Anglia.
If p, then q is not true. q. Therefore, p is not
true.
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Argument form in logic
Mrs. White must have done it, because either she
did it or Revd. Green did, and if hed done it,
it would have been in the Kitchen, and it was not.
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Argument form in logic
Mrs. White must have done it, because either she
did it or Revd. Green did, and if hed done it,
it would have been in the Kitchen, and it was not.
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Argument form in logic
Mrs. White must have done it, because either she
did it or Revd. Green did, and if hed done it,
it would have been in the Kitchen, and it was not.
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Argument form in logic
Mrs. White must have done it, because either she
did it or Revd. Green did, and if hed done it,
it would have been in the Kitchen, and it was not.
p Mrs. White committed the murder. q Revd.
Green committed the murder. r The murder was
committed in the Kitchen.
p or q. If q, then r. r is not true. Therefore, p.
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Argument form in logic
Matt or Tarisai won. Matt didnt win.
Therefore, Tarisai won.
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Argument form in logic
Matt or Tarisai won. Matt didnt win.
Therefore, Tarisai won.
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Argument form in logic
Matt or Tarisai won. Matt didnt win.
Therefore, Tarisai won.
p Matt won. q Tarisai won.
p or q. p is not true. Therefore, q.
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Summary
Logic is the study of arguments.
An argument is a set of propositions, called
premises, along with a single proposition, called
the conclusion.
The logical form of an argument is obtained by
letting letters stand for the various
propositions that occur in the argument we use
the same letter when a proposition occurs a
further time in an argument.