Simple Proofs in Propositional Logic
ƒ We do not need to use truth tables or the
shorter truth table technique in order to asses
the validity of arguments in propositional
form.
ƒ Instead, we can show the validity of an
argument by deriving its conclusion from its
premises using argument forms that are
known to be valid.
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Valid Argument Forms
ƒ Each of the following argument forms can be
shown to be valid (e.g., using truth tables).
Once we have established that these
argument forms are valid (and, possibly,
committed them to memory), we can use
them to derive the conclusion of an argument
from its premises.
If we can do this, we will have shown that the
argument is valid.
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Valid Argument Forms
P ∴ ~(~P)
~(~P) ∴ P
or
Double negation
PvP∴P
Tautology
P; Q; ∴ P . Q
Conjunction
P.Q∴P
P.Q∴Q
Simplification
P > Q; P ∴ Q
Modus Ponens
P > Q; ~ Q ∴ ~P
Modus Tollens
P > Q ∴ ~Q > ~P or
~Q > ~P ∴ P > Q
Transposition
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P > Q; Q > R ∴ P > R
Hypothetical syllogism
P v Q; ~P ∴ Q
P v Q; ~Q ∴ P
Disjunctive Syllogism
P∴PvQ
P∴QvP
or
Addition
or
~(P . Q) ∴ ~P v ~Q
~P v ~Q ∴ ~(P . Q)
P > Q ∴ ~P v Q
~P v Q ∴ P > Q
or
or
De Morgan’s Rule
Implication
P > Q; R > S; P v R ∴Q v S Constructive dilemma
P > Q; R > S; ~Q v ~S ∴
~P v ~R
Destructive dilemma
~(P . ~P)
Noncontradiction
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Simple Proof: An Example
Let’s say we had an argument that, once formalized,
looked like this:
1. (A v B) > ~ C
2. ~C > D
3. A
Therefore,
4. D
Once again: If we can derive the conclusion from the
premises using only known to be valid argument forms,
we will have shown that the argument is valid…
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1.
2.
3.
4.
5.
6.
(A v B) > ~ C
~C > D
A
AvB
~C
D
(premise)
(premise)
(premise)
(from 3 by
(from 1 by
(from 2 by
to be proven: D
addition)
modus ponens)
modus ponens)
Valid (D was the conclusion of the original argument,
and we have derived it from the premises of that
argument using only known-to-be-valid argument
forms)
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Another example
1. A . B
2. A > ~C
3. B > ~D
Therefore,
4. ~C . ~D
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1. A . B
2. A > ~C
3. B > ~D
4.
5.
6.
7.
8.
A
B
~C
~D
~C . ~D
(premise) to be proven: ~C . ~D
(premise)
(premise)
(from
(from
(from
(from
(from
1,
1,
2,
3,
6,
simplification)
simplification)
modus ponens)
modus ponens)
7, conjunction)
Valid
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4
Example from Exam #2
1. C > D
2. D > E
3. C
(premise) to be proven: E
(premise)
(premise)
4. D (from 1 and 3, modus ponens)
5. E (from 2 and 4, modus ponens)
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Something to Notice About Simple Proof
ƒ If can derive the conclusion from an argument
using only valid argument forms, you have
shown that that argument is valid.
ƒ However, if you cannot derive the conclusion
this way, you have not shown that the
argument is invalid.
I.e., the simple proof technique requires a
certain amount of ingenuity and you may
have failed to hit upon a workable proof
strategy.
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ƒ So, if you cannot construct a proof for an
argument, you cannot say for certain that the
argument is invalid: It may be invalid or you
may have failed to find the right proof
strategy.
(By contrast the truth table method, you’ll
recall, always provides an answer.)
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Conditional Proof
ƒ In conditional proof, a statement is assumed
“for the sake of argument.”
In particular, we assume the antecedent of a
conditional and then see if that conditional
follows from the other premises using valid
argument forms.
ƒ Notice that a conditional proof is conditional in
two senses: i) the proof of the conclusion is
conditional upon the assumption and ii) the
conclusion proven is itself a conditional
statement.
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Conditional Proof: Simple Example
1. A > B (premise)
2. B > C (premise)
3.
4.
5.
6.
A
B
C
A>C
to be proven: A > C
(assumption)
(from 1 and 3, modus ponens)
(from 2 and 4, modus ponens)
(from 3-5, conditional proof)
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ƒ Making assumptions in this way may seem
arbitrary. Remember, however, that > asserts
only that if the antecedent is true, then the
consequent cannot be false.
So, if from the stated premises and some
assumption A we can derive X, then we can
derive from the stated premises “if A, then X”
P (assumption) … ∴ P > Q Conditional Proof
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So, another rule (another valid argument
form):
P (assumption) … ∴ P > Q Conditional Proof
Meaning: If Q can be derived from the
assumption P through a series of valid
intermediate steps, represented here by the
ellipsis (…), then we can validly conclude P >
Q.
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ƒ Conditional proof is actually a fairly common
technique in everyday reasoning.
ƒ Imagine, e.g., that we believe that 1) we can
solve the problems of ‘Third World’ nations
and still maintain a reasonable standard of
living by developing alternative forms of
energy and that 2) there will be a greater
chance of lasting peace if we solve problems
of Third World nations…
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This can be formalized as follows:
E = we develop alternative forms of energy
S = we solve problems of Third World nations
R = we enjoy a reasonable standard of living
P = there is greater chance of lasting peace
1.
2.
3.
4.
5.
6.
7.
E > (S . R)
S>P
E
S.R
S
P
E>P
(premise)
to be proven: E > P
(premise)
(assumption)
(from 1 and 3, modus ponens)
(from 4, simplification)
(from 2 and 6, modus ponens)
(3-6, conditional proof)
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Indirect Proof: Reductio ad Absurdum
ƒ Reductio ad absurdum (< Latin “reduce to
absurdity”) is a proof technique in which we
prove a statement by showing how the
premises can entail a contradiction.
I.e., they entail a statement that cannot be
true, a statement of the form “P . ~P”
ƒ The technique: If we assume X and show by a
series of valid intermediate steps that it leads
to a statement of the form P . ~P, then we can
validly conclude ~X.
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Reductio ad Absurdum: An Example
1. C > ~R
2. C > A
3. A > R
4. C
5. ~R
6. A
7. R
8. R . ~R
9. C > (R . ~R)
10. ~(R . ~R)
11. ~C
(premise) to be proven: ~C
(premise)
(premise)
(assumption)
(from 1 and 4, modus ponens)
(from 2 and 4, modus ponens)
(from 3 and 6, modus ponens)
(from 8 and 5, conjunction)
(4-8, conditional proof)
(noncontradiction)
(from 9 and 10, modus tollens)
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Two Fallacies Related to Conditionals
modus ponens:
modus tollens:
P > Q; P ∴ Q
P > Q; ~Q ∴ ~P
Both of these argument forms are valid. They
can be proven to be valid using truth tables
and, moreover, they appear intuitively to be
valid. They are “basic to human thinking.”
Both of these argument forms are commonly
misused, however …
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Conditional Fallacies
Affirming the consequent: P > Q; Q ∴ P *
Consider: If you got the job, you must have
impressed the interviewer. You impressed the
interviewer. Therefore, you must have got the
job (!?)
Denying the antecedent: P > Q; ~P ∴~Q *
Consider: If Derek is in Saskatoon, then he is
in Saskatchewan. Derek is not in Saskatoon.
Therefore, he is not in Saskatchewan (!?)
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Propositional Logic and Cogency
ƒ We have only touched on the basics of formal
logic in this course: It can be pursued in much
greater depth (e.g., U of R, PHIL 250, 350,
351, MATH 221, 301; U of S, PHIL 241, 243,
343, CMPT 260, 417, … etc.)
ƒ For present purposes we should bear in mind
the connection between formal logic and the
topics in natural language (informal) logic that
we encountered in Govier chs. 1-6 …
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… we noted then that an argument is cogent
iff it satisfies all of the ARG conditions.
If an argument is deductively valid, then it
satisfies the R and (if it premises are all true)
G conditions. But just because an argument is
valid does not mean that the A condition is
satisfied.
A special case of formally valid arguments
failing the A test occurs in connection with
dilemma arguments …
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“Either employment levels will go up or there
will be a revolt. Employment levels are not
going up. Therefore, there will be a revolt.”
E v R; ~E ∴ R (valid – disjunctive syllogism)
But it the first premise really acceptable as it
stands? Does anyone have good reason to
believe that these are the only two
alternatives? If we can specify a third
alternative (a counterexample to premise 1)
we can be said to have “escaped through the
horns of the dilemma”
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