Logic: The Art of Thinking

Lesson

Rules of Inference

MODERN SYMBOLIC LOGIC

STATEMENT CONSTANTS A B C D E F G H I J...

STATEMENT VARIABLES p q r s t u v w x y...

LOGICAL OPERATORS - NEGATION "NOT"

 

& CONJUNCTION "AND"

\/ ALTERNATION "OR "

> IMPLICATION "IF...THEN"

= EQUIVALENCE "IF AND ONLY IF"

LOGICAL PUNCTUATION ( and )

STATEMENT FORMS

WELL-FORMED FORMULAS

==================================================================

TRUTH TABLES

__

p - p p &q p\/q p\/q p >q p =q

T F T T T T T T T T F T T T T T T T

F T F T F F T T F T T F T F F T F F

F F T F T T F T T F T T F F T

F F F F F F F F F F T F F T F

==================================================================

SETTING UP TRUTH TABLES (NUMBER OF THE VARIABLES)

NUMBER OF LINES IN THE TABLES = 2

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TYPES OF TRUTH TABLES

1. TAUTOLOGY ALL TRUE VALUES

2. CONTRADICTION ALL FALSE VALUES

3. CONTINGENCY SOME VALUES ARE TRUE, SOME ARE FALSE

 

LOGIC

RULES OF INFERENCE

1. MODUS PONENS 2. MODUS TOLENS 3. HYPOTHETICAL SYLLOGISM

p > q p > q p > q

p -q q > r

---------------- ------------- ---------------

:. q :. -p :. p > r

4. DISJUNCTIVE SYLLOGISM 5. ADDITION 6. SIMPLIFICATION

p \/ q p p & q

-p

------------- ---------- ------------

:. q :. p \/ q :. p

7. CONSTRUCTIVE DILEMMA 8. CONJUNCTION

p > q p

r > s q

p \/r

------------------ -------------

:. q \/s :. p & q

------------------------------------------------------------------

RULES OF EQUIVALENCE

9. DeMorgan's Rules 14. Commutation

-(p & q )= -p \/ -q p & q = q & p

-(p \/q )= -p & -q p \/q = q \/p

10. Implication 15. Distribution

p > q = -p \/ q p & (q \/ r) =(p&q)\/(p&r)

p \/(q & r) = p\/q & p\/r

11. Transposition 16. Biconditional

p > q = -q > -p (p = q) = (p > q) & (q >p)

12. Double Negation 17. Exportation

--p = p (p & q) > r = p > (q > r )

13. Association 18. Tautology

p \/ (q \/ r)=(p \/q) \/ r p \/ p = p

p & (q & r )=(p & q) & r p & p = p

-

:. q :. q \/ s

 

 

FORMAL FALLACIES

I. Syllogisms

A. Standard Categorical Syllogisms

Check for violation of the five rules

B. Non-Standard Categorical Syllogism

Violates standards try to recast

C. Non-Categorical Syllogisms

Fallacies--Affirming the Consequent/Denying the Antecedent

False Dilemma Others

------------------------------------------------------------------

Valid Forms Invalid Forms

1. Modus Ponens Affirming the consequent

p>q p>q

p q

------- ------

:. q :. p

------------------------------------------------------------------

2. Modus Tollens Denying the Antecedent

p>q p>q

-q -p

------ -------

:. -p :. -q

------------------------------------------------------------------

3. Disjunctive Syllogism

p \/ q

-p

---------------

:. q

------------------------------------------------------------------

4. Alternative Syllogism

__

p \/ q

p

----------------------

:. -q

----------------------------------------------------------------

5. Hypothetical Syllogism

p > q

q > r

-----------

:. p > r

-----------------------------------------------------------------

6. Dilemma

simple complex

p > q p > q

r > q r > s

p \/r p\/ r

----------- -----------

:. q :. q \/ s

PROPOSITIONAL CALCULUS

Evaluation of an Argument's Validity

I. Truth Table Method

1. State the Premises and Conclusion in propositional form

Use Well-Formed Formulas

2. Determine whether the premises alone are consistent,i.e. not a

contradiction. Form a single statement form by conjoining the

premises and set up a truth-table. Show that the premises can

all be true at the same time.

3. Determine whether the conclusion is a tautology. Show that the

conclusion could be false or take on a false value. An

argument with a tautology for a conclusion is not a very good

argument at all, for its conclusion is true (always) regardless

of the truth or falsity of the premises. There is also very

little need to argue for things that are always true.

4. If the premises do not constitute a contradiction and if the

conclusion is not a tautology, then proceed with the

evaluation.

5. Use the truth-table method of evaluating the argument:

P (1)

Argument schema P (2)

P (3)

:

P (n)

-------------

:. C

( P(1) & P(2) & P(3) & ... & P(n) > C )

If this statement form is a tautology the argument schema is valid.

II. Method of Natural Deduction- Propositional Calculus

Attempt to deduce the stated conclusion from the stated premises

by using ingenuity and the method of natural deduction including:

8 Rules of Inference Conditional Proof Method

10 Rules of Replacement Indirect Proof Method

III. Shortened Truth-Table Method

If the stated conclusion cannot be reached by using the

propositional calculus after a considerable effort has been made,

then check for the invalidity of the argument by using the

shortened truth-table method, i.e., assign values to the variables

in the conclusion so that the entire conclusion will have a false

value and then attempt to make all of the premises true at once

while remaining consistent with the assignment of the truth values

of the variables once assigned.

 

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