PROPOSITIONAL LOGIC

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PROPOSITIONAL LOGIC
Propositions and arguments
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Propositions

• It is two oclock.
• The mind is the brain.
• The mind is not the brain.
• He believes in God.

Where have you been? Oh no, not logic
again. Thank you. Help!
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Propositions

• It is two oclock.
• The mind is the brain.
• The mind is not the brain.
• He believes in God.

• Declarative sentences
• Can be believed (or not believed)
• Can be true or false
• They express beliefs they express propositions

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Propositions

• It is two oclock.
• The mind is the brain.
• The mind is not the brain.
• He believes in God.

• Propositions
• Things we can believe
• Things which can be true or false
• Things expressed by declarative sentences

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Arguments

• If everything is determined, then people are not
  free.
• People are free.
• Therefore, not everything is determined.

Its raining or its snowing. Its not
raining. Therefore, its snowing.
If the battery is flat, then the starter will be
dead. You wont get the car started unless we
push it. Therefore, you will stay here.
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Arguments

• If everything is determined, then people are not
  free.
• People are free.
• Therefore, not everything is determined.

Argument List of propositions (premises)
followed by another proposition (conclusion)
Valid argument Argument such that whenever all
its premises are true, its conclusion is also
true.
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Arguments

• If people are mammals, then people are not cold
  blooded.
• People are cold blooded.
• Therefore, people are not mammals.

• Is this a (valid) argument?
• Whats wrong with it?

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Arguments

• If people are cold blooded, then people are not
  mammals.
• People are mammals.
• Therefore, people are not cold blooded.

Sound argument Valid argument all its premises
are true.
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Arguments

• It will rain tomorrow.
• Although Sean drives a fast car, John drives a
  slow car.
• Therefore, logic is a boring subject.

The mind is the brain. The mind is not the brain.
Therefore, both Dennett and Searle are right.
Its Wednesday. The Earth is flat. Therefore, 11
or 1?1.
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Arguments

• Propositions
• Things we can believe
• Things which can be true or false
• Things expressed by declarative sentences

Argument List of propositions (premises)
followed by another proposition (conclusion)
Valid argument Argument such that whenever all
its premises are true, its conclusion is also
true.
Its Wednesday. The Earth is flat. Therefore, 11
or 1?1.
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Propositional forms

• Its not raining.
• If everything is determined, then people are not
  free.
• Its raining or its snowing.
• The battery is flat and the starter is dead.

Its not true that its raining. If everything is
determined, then its not true that people are
free. Its raining or its snowing. The battery
is flat and the starter is dead.
Sub-proposition Part of proposition which is
itself a proposition
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Propositional forms
Its not true that its raining. If everything is
determined, then its not true that people are
free. Its raining or its snowing. The battery
is flat and the starter is dead.
Its not true that p. If q, then its not true
that r. p or s a and ß
p Its raining. q Everything is determined. r
People are free. s Its snowing.
Propositional form is created by replacing
sub-propositions by letters.
A sentence is an instance of a propositional form
if we get it (or a sentence with the same
meaning) by replacing the single letters by
sentences.
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Argument forms

• If everything is determined, then its not true
  that people are free.
• People are free.
• Therefore, its not true that everything is
  determined.

If p, then its not true that q. q Therefore,
its not true that p.
Argument form List of propositional forms
(premises) followed by another propositional form
(conclusion)
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Argument forms
An argument is an instance of an argument form if
we get it by replacing the single letters by
sentences.
When every instance of an argument form is valid,
we call it a valid argument form.
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Argument forms
If p, then its not true that q. q Therefore,
its not true that p.
If p, then q. p Therefore, q.
If p, then q. q Therefore, p.
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Argument forms
If p, then q q Therefore, p
If its 3am, then Im inside. Im
inside Therefore, its 3am.
If 224, then 4-22. 4-22 Therefore, 224.
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Argument forms
Modus Tollens
Modus Ponens
If p, then q p Therefore, q
If p, then q not p Therefore, not p
Hypothetical Syllogism
Affirming the Consequent
If p, then q If q, then r Therefore, if p, then r
If p, then q q Therefore, p
Disjunctive Syllogism
Disjunction Introduction
Either p or q not p Therefore, q
If p, then r If q, then r Therefore, if p or q,
then r
NOT ALL ARE VALID FORMS!
to be continued