# Deductive Arguments and Inference Rules

```Deductive Arguments and Inference
Rules
Terminology:
 Valid Argument:
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truth of the premises guarantees the truth of the
conclusion
It would be contradictory for the premises to be
true and the conclusion false
Deductive Arguments and Inference
Rules

Sound Argument: A Valid argument with true
premises. A sound argument is one for which
both of the following are true:
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it would be contradictory for the premises to be
true, but the conclusion false
The premises are all true
Deductive Arguments and Inference
Rules
Use correct terminology:
 Arguments are valid or invalid, sound or
unsound
 Sentences (premises) are true or false
Deductive Arguments and Inference
Rules
Common Valid Forms:
1.Modus Ponens (mode of putting)
If p then q
p
Therefore, q
Example:
If the death penalty is abolished,
murder rates will increase.
The death penalty will be abolished
Murder rates will increase
Deductive Arguments and Inference
Rules
2. Modus Tollens (mode of taking)
If p then q
Not q
Therefore, not p
Example:
If determinism is false, then some events do not
have a cause
There is no event without a cause
Therefore, determinism is true
Deductive Arguments and Inference
Rules
More terminology:
Antecedent: the “If” position
Consequent: the “Then” position
Example:
If I wake up on time, I’ll make the 7 a.m. train
I wake up on time= antecedent
I’ll make the 7 a.m. train=consequent
Deductive Arguments and Inference
Rules
IF p Then q

p is a sufficient condition for q
–

a sufficient condition for ___ is a condition which, if met,
guarantees that ___ is the case
q is a necessary condition for p
–
a necessary condition for ___ is a condition that must be
met in order for ___ to be the case
Deductive Arguments and Inference
Rules
Evaluating arguments:
Example: If the death penalty is abolished,
murder rates will increase. The death penalty
will be abolished. Therefore, murder rates will
be increased
STEP 1: make a dictionary
d=death penalty is abolished; m=murder rates
will increase
Deductive Arguments and Inference
Rules
STEP #2: Put the argument into logical form
P1. If d THEN m
P2. d
Therefore, m
STEP #3: Is this a valid form?
Yes—modus ponens
Deductive Arguments and Inference
Rules
STEP #4: Is the argument Sound?
 Is the argument valid (is it in a valid form)?
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
Our example: yes
Are all the premises true??? Which premise would
you challenge?
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P1. If the death penalty is abolished, murder rates will
increase
or
P2. The death penalty will be abolished
Deductive Arguments and Inference
Rules
“only”?
The phrase that immediately
follows the word ‘only’ is the
consequent
Deductive Arguments and Inference
Rules
3. Hypothetical Syllogism
If p then q
If q then r
Therefore, If p then r
Example
If we establish strict gun laws, there will be fewer
guns on the street
If there are fewer guns on the street, there will be
less crime
Therefore, if we establish strict gun laws, there will
be less crime
Deductive Arguments and Inference
Rules
4. Disjunctive Syllogism
p or q
Not-p
Therefore, q
Example:
Either Gil is a Cubs fan or Gil is a fool.
Gil is not a Cubs fan
Gil is a fool
Deductive Arguments and Inference
Rules
Disjunctive Syllogisms are tricky:
The word “or” can have two meanings:
 Inclusive meaning: at least one of p or q is
true and possibly both
 Exclusive meaning: either p or q is true but
not both
Deductive Arguments and Inference
Rules
Why is this important??
We need to know what we can infer:
P1. p or q
P2. p
Inclusive sense:
nothing can be inferred from P1 and P2
Exclusive sense:
can infer not-q
Exercises

Either the White Sox will win the world series
or Chicago’s southsiders will be
disappointed. The White Sox will win the
world series. So, Chicago’s southsiders will
not be disappointed.
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Make a dictionary
P1. Either W or D
P2. W
Therefore, not D
exclusive or inclusive???
More Exercices

If the President does not raise taxes to save Social
Security, then Social Security will collapse. The
President has decided to raise taxes to save Social
Security. Therefore, Social Security will not collapse
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Make a dictionary
P1. If not R then C
P2. R
Therefore, Not C
Invalid
More exercises
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Either Mugabe will leave office or there will be continued
unrest in Zimbabwe. If Mugabe leaves office, the US will
send millions in aid to Zimbabwe. There will not be
continued unrest in Zimbabwe. Thus, the US will send
millions in aid to Zimbabwe. (use M, U, A).
P1. Either M or U
If M then A
Not U
Therefore, A
Valid
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