**Rules of Inference**

Sentential/propositional logic

: examines the relationship between sentences that pertain to reasoning (Bonevace, 2003, p. 36).

Our Focus

more fine grained branches...

Modal logic

: includes modalty (i.e. possibility & necessity).

Quantification logic

: includes quantifiers (i.e. all & there exists).

Rule:

Conjunction

1. P

2. Q

3. P & Q

1. Bob has a rat tail.

2. Sue has a car.

3. Therefore, Bob has a rat tail & Sue has a car.

1. If Bob cuts off his rat tail, he'll get a job.

2. Because Sue has a car, she has a job.

3. Therefore, if Bob cuts off his rat tail, he'll get a job; and because Sue has a car, she has a job.

&

is the symbol of

conjunction

. It's read as 'and'.

But it symbolizes any conjunction.

Note

: Any sentences can be joined by

&

.

For example:

They're skinning,

nevertheless

they're fast.

He's stupid,

but

he's kind.

Aryn is hip,

whereas

Matt is nerdy.

&

P

Q

Each sentence can be represented like this.--->

Rule:

Simplification

1. P & Q

2. P

1. P & Q

2. Q

1. Andy is skiing, and Shannon is biking.

2. Therefore, Shannon is biking.

Here's why this rule is useful.

Suppose you need P itself to draw a conclusion. If you have (P&Q), then you can deduce P.

Rule:

Addition

1. P

2. P v Q

1. Carson is a retired neurosurgeon.

2. Therefore, Carson is a retired neurosurgeon, or Sanders is going to becompose into candy corn.

1. Julie is a woman.

2. Therefore, Julie is a woman, or Bob doesn't exist.

The symbol:

v

, is read 'or'.

It can be used for symbolizing disjunction (i.e. sentences composed of two sentences connected by

or

).

Usefulness:

1. A

2. (A v B)

→

C

3. (A v B)

4. C

Rule:

Modus Ponens

Rule:

Modus Tollens

1. If x, then y

2. x

3. y

1. If Bob is in Seattle, then he is in WA.

2. Bob is in Seattle.

3. Therefore, Bob is in WA.

1. If Sara is human, she deserves respect.

2. Sara is human.

3. Sara deserves respect.

BEWARE

Formal fallacy lurking in the neighborhood

Q

P

AKA: Affirming the

antecedent

Fallacy ID: Affirming the

consequent

AKA: Denying the

consequent

.

1. If A, then B

2. Not B

3. Not A

1. If Bonnie Dunbar is an astronaut, then Bonnie Dunbar is a member of NASA.

2. Bonnie Dunbar is not a member of NASA.

3. Bonnie Dunbar is not an astronaut.

1. If Michelle runs a marathon, then Michelle will run at least 20 miles.

2. Michelle will not run at least 20 miles.

3. Michelle will not run a marathon.

WARNING

Formal fallacy nearby!!!

P

Q

Fallacy: Denying the

antecedent

.

Class discussion

Topic: Axiological Argument for Theism

Rule:

Absorption

REVIEW

Discussion Q

Can we put the main point of this video in the form of either modus ponens or modus tollens?

Video

: SNL 'Don't Buy Stuff..."

http://www.nbc.com/saturday-night-live/video/dont-buy-stuff/n12020

Rule:

Hypothetical Syllogism

1. If (a) Carlos is President, then (b) Carlos lives in the White House.

2. Therefore, if (a) Carlos is President, then (a & b) Carlos is President and Carlos lives in the White House.

1. If Jan can drive, then Jan can take me to the store.

2. Therefore, if Jan can drive, then Jan can drive and Jan can take me to the store.

Potential use:

1. If your first priority is to be happy, then you'll seek happiness according to inadequate means.

2. If you seek happiness according to inadequate means, then you won't be happy.

3. Therefore, if your first priority is to be happy, then you won't be happy.

1. If (

p

) Carly does her homework, then (

q

) she'll get an A.

2. If (

q

) Carly gets an A, then (

r

) she'll graduate.

3. If (

p

) Carly does her homework, then (

r

) she'll graduate.

Additional example of

Hypothetical Syllogism

The

Paradox

of

Hedonism

1.

p

-->

q

2.

q

-->

r

3.

p

-->

r

Rule:

Disjunctive Syllogism

1. Billy is alive or Billy is dead.

2. Billy is not alive.

3. Billy is dead.

1. Billy is alive or Billy is dead.

2. Billy is not dead.

3. Billy is alive.

Note 1

: the logical form of premise 1 is different than its verbal form. We'd typically say what's said in premise 1 as follows: "Billy is alive or dead."

Note 2

: In this example, the options given are mutually exclusive--Billy can't be alive and dead!

However

, premise 1 could have included sentence that are not mutually exclusive. For example, suppose we knew the following: a. Billy is skinny, or b. Billy is Australian. If we also knew Billy is not Australian, we could know Billy must be skinny.

The or (i.e. 'v') is simply saying one of these is true.

Rule:

Constructive Dilemma

1. If Sandra applies, she'll get the job; and if Emilio gets fired, he'll move to CO.

2. Sandra will apply or Emilio will get fired.

3. Sandra will get the job or Emilio will move to CO.

1. If Barky is a black lab, he is worth money; if Sue sells barky, she'll regret her decision.

2. Barky is a black lab or Sue will sell barky.

3. Barky is worth money or Sue will regret her decision.

Note:

Constructive Dilemma is often put in this form.

1. Sandra will apply or Emilio will get fired.

2. If Sandra applies, she'll get the job.

3. If Emilio gets fired, he'll move to CO.

4. Sandra will get the job or Emilio will move to CO.

We can reword our first example accordingly...

Question 1: What type of argument is this?

Question 2: Is it valid?

Question 3: What rule of inference is used & is it properly used?