DOUBLES & IDENTITIES

Borges slides

... the ultimate laws of thought are mathematical ...
-GEORGE BOOLE
The Laws of Thought (1854)

RULES OF INFERENCE


            1. Modus Ponens             2. Modus Tollens

                    p->q                        p->q

                    p                           ~q

                    ∴ q                        ∴ ~p

            

            3. Hypothetical Syllogism   4. Disjunctive Syllogism

                    p->q                        p v q

                    q->r                        ~p

                    ∴ p->r                     ∴ q

            

            5. Constructive Dilemma     6. Absorption

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

                    p v r                       ∴ p->(p & q)

                    ∴ q v s

            

            7. Simplification           8. Conjunction

                    p & q                       p

                    ∴ p                         q

                                                ∴ p & q

            

            9. Addition

                    p

                    ∴ p v q

            

            Replacement: Any of the following logically equivalent 

            expressions can replace each other wherever they occur:

            

            10. De Morgan's Theorems    ~(p&q) = (~pv~q)

                                        ~(pvq) = (~p&~q)

            11. Commutation             (pvq) = (qvp)

                                        (p&q) = (q&p)

            12. Association             [pv(qvr)] = [(pvq)vr]

                                        [p&(q&r)] = [(p&q)&r]

            13. Distribution            [p&(qvr)] = [(p&q)v(p&r)]

                                        [pv(q&r)] = [(pvq)&(pvr)]

            14. Double Negation         p = ~~p

            15. Transposition           (p->q) = (~q->~p)

            16. Material Implication    (p->q) = (~pvq)

            17. Material Equivalence    (p=q) = [(p->q)&(q->p)]

                                        (p=q) = [(p&q)v(~p&~q)]

            18. Exportation             [(p&q)->r] = [p->(q->r)]

            19. Tautology               p = (pvp)

                                        p = (p&p)

-IRVING M. COPI
Introduction to Logic (1953)

The question whether identity is or is not a relation, and even whether there is such a concept at all, is not easy to answer. For, it may be said, identity cannot be a relation, since, where it is truly asserted, we have only one term, whereas two terms are required for a relation. And indeed identity, an objector may urge, cannot be anything at all: two terms plainly are not identical, and one term cannot be, for what is it identical with? Nevertheless identity must be something ... Identity, which occurs here, is defined as follows: x is identical with y if y belongs to every class to which x belongs, in other words, if "x is a u" implies "y is a u" for all values of u.
-BERTRAND RUSSELL
The Principles of Mathematics (1903)

CON

There is no such thing as two individuals indiscernible from each other. An ingenious gentleman of my acquaintance, discoursing with me in the presence of Her Electoral Highness, the Princess Sophia, in the garden of Herrenhausen, thought he could find two leaves perfectly alike. The princess defied him to do it, and he ran all over the garden a long time to look for some; but it was to no purpose. Two drops of water or milk, viewed with a microscope, will appear distinguishable from each other. This is an argument against atoms, which are confuted, as well as a vacuum, by the principles of true metaphysics.
-GOTTFRIED WILHELM LEIBNIZ
Fourth Letter to Samuel Clarke (1716)

A man cannot step into the same river twice.
-HERACLITUS (c540-c480)

PRO

MARIONETTES, INC.
Duplicate self or friends; new humanoid plastic 1990 models, guaranteed against all physical wear. $7,600 to our $15,000 de luxe model.
-RAY BRADBURY
Illustrated Man (1951)

One of the more fantastic possibilities is that man will be able to make biological carbon copies of himself. Through a process known as "cloning" it will be possible to grow from the nucleus of an adult cell a new organism that has the same genetic characteristics of the person contributing the cell nucleus. The resultant human "copy" would start life with a genetic endowment identical to that of the donor, although cultural differences might thereafter alter the personality or physical development of the clone.
-ALVIN TOFFLER
Future Shock (1970)

neptune

NEPTUNE




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