Download Logic 5: Material Implication and Argument Forms

Review The Conditional Logical symbols Argument forms
Logic 5: Material Implication and Argument
Forms
Jan. 28, 2014
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Overview I
Review
The Conditional
Conditional statements
Material implication
Logical symbols
Argument forms
Disjunctive syllogism
Disjunctive syllogism
Disjunctive syllogism
Disjunctive syllogism
Modus ponens
Modus tollens
Hypothetical syllogism
Constructive dilemma
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Review
3 basic operators we considered:
p q p·q
T T
T
1 Conjunction:
T F
F
F
F T
F F
F
2
p
Negation: T
F
3
p
T
Disjunction: T
F
F
∼p
F
T
q
T
F
T
F
Logic 5: Material Implication and Argument Forms
p∨q
T
T
T
F
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Conditional statements Material implication
Conditional Statements
Most arguments contain a sentence of “If A, then B” form
— this is called a hypothetical, or an implication.
A: antecedent
B: consequent
Example
If Mr. Jones is my neighbor, then Mr. Jones has three children.
antecedent: “Mr. Jones is my neighbor”
consequent: “Mr. Jones has three children.”
A few things to notice:
A conditional statement tells us that in any case in which the
antecedent is true, the consequent is also true.
It does NOT tell us that the antecedent is actually true.
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Conditional statements Material implication
Material Implication I
There are various kinds of conditional statements.
Example
1
If all humans are mortal, and Socrates is a human, then
Socrates is mortal. — Logical implication
2
If John is a bachelor, then John is unmarried. — Follows from
the definition
3
If a piece of wood is placed in fire, then it will burn. —
Causal implication
4
If we lose this game, then I’ll eat my hat. — A decision
Is there any common meaning in these?
One way to find a common meaning is to check when the
above sentences are false.
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Conditional statements Material implication
Material Implication II
All the above sentences are false when their antecedents are
true and their consequent are false.
Material Implication
The logical symbol for the conditional is ⊃ (called the
‘horseshoe’). A conditional p ⊃ q is false if and only if p is true
and q is false (it is true otherwise).
p q p⊃q
T T
T
T F
F
F T
T
F F
T
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Conditional statements Material implication
Material Implication III
Things to keep in mind:
Material implication includes conditionals where there is no
real connection between the antecedent and the consequent.
Example
“If Hitler was a military genius, then I’m a monkey’s uncle.” —
This only wants to suggest that the antecedent is false.
Material implication is true when the antecedent is false —
this is often counter-intuitive.
Example
“If the world is flat, then the moon is made of green cheese.”
It can be shown that p ⊃ q is true just when ∼ q ⊃∼ p is
true — Contrapositive
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Logical Symbols — Summary
We have considered four basic logical concepts.
1
Conjunction: a · b is true if and only if a is true and b is true.
2
Negation: ∼ a is true if and only if a is false.
3
Disjunction: a ∨ b is false if and only if a is false and b is
false.
4
Conditional: a ⊃ b is false if and only if a is true and b is
false.
There are various relations between these; e.g., it can be shown
very easily that a ⊃ b is equivalent to ∼ a ∨ b.
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma
Argument Forms I
Reminder: an argument is valid if whenever the the premises
are true, the conclusion is also true.
Thus, whenever you accept the premises of a valid argument,
you must also accept the conclusion!
We can test argument-validity by the truth-table method.
Of course, different arguments can have the same form:
Example
The blind prisoner has a red hat
or a white hat.
The blind prisoner does not have
a red hat.
Therefore, the blind prisoner has
a white hat.
Logic 5: Material Implication and Argument Forms
Either Shakespeare wrote the
Hamlet or he was not a great
writer.
Shakespeare was a great writer.
Therefore, Shakespeare wrote
Hamlet.
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma
Argument Forms II
Thus, it seems that if the first argument is valid, then the second
one also has to be valid (they have the same form).
Argument Form
An argument form is a string of symbols, containing letters
(variables) for statements. When these letters are filled in with
sentences, we get an argument. If an argument form is valid, then
every argument that falls under that form is also valid.
Example
The argument form exemplified by the arguments above:
p∨q
It is called disjunctive
∼q
syllogism. (We justified its
∴p
validity last time.)
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma
Argument Forms III
The most important argument forms:
1
Disjunctive syllogism
2
Modus ponens
3
Modus tollens
4
Hypothetical syllogism
5
Constructive dilemma
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma
Modus Ponens
Example
If Shakespeare wrote Hamlet, then Shakespeare was a good writer.
Shakespeare wrote Hamlet.
Therefore, Shakespeare was a good writer.
p = Shakespeare wrote Hamlet.
q = Shakespeare was a good writer.
1
Formalize:
p⊃q
p
∴q
2
Truth-table:
p q
T T
T F
F T
F F
p⊃q
T
F
T
T
p
T
T
F
F
q
T
F
T
F
Thus, the argument form Modus Ponens is valid.
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma
Modus Tollens
Example
If Mozart was not a composer, then he did not write the Requiem.
Mozart did write the Requiem.
Therefore, Mozart was a composer.
p = Mozart was not a composer.
q = Mozart did not write the Requiem.
1
Formalize:
p⊃q
∼q
∴∼ p
2
Truth-table:
p q p⊃q
T T
T
T F
F
F T
T
F F
T
∼q
F
T
F
T
∼p
F
F
T
T
Thus, the argument form Modus Tollens is valid.
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma
Hypothetical Syllogism I
Example
If Walter Scott was British, he was not Italian.
If Walter Scott was not Italian, he did not live in Italy.
Therefore, if Walter Scott was British, he did not live in Italy.
p = Walter Scott was British.
q = Walter Scott was not Italian.
r = Walter Scott did not live in Italy.
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma
Hypothetical Syllogism II
1
Formalize:
p⊃q
q⊃r
∴p⊃r
2
Truth-table:
p q
T T
T T
T F
T F
F T
F T
F F
F F
r
T
F
T
F
T
F
T
F
p⊃q
T
T
F
F
T
T
T
T
q⊃r
T
F
T
T
T
F
T
T
p⊃r
T
F
T
F
T
T
T
T
Thus, the argument form Hypothetical Syllogism is valid.
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014
Review The Conditional Logical symbols Argument forms
Modus ponens Modus tollens Hypothetical syllogism Constructive dilemma
Constructive Dilemma
Example
If Descartes was a philosopher, he wrote the Meditations, but if
Beethoven was a composer, he wrote a symphony.
Descartes was either a philosopher or Beethoven was a composer.
Therefore, either Descartes wrote the Meditations or Beethoven
wrote a symphony.
p = Descartes was a philosopher.
q = Descartes wrote the Meditations.
r = Beethoven was a composer.
s = Beethoven wrote a symphony.
1 Formalization:
(p ⊃ q) · (r ⊃ s)
p∨r
∴q∨s
Logic 5: Material Implication and Argument Forms
Jan. 28, 2014