... 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 (c540c480)
PRO
MARIONETTES, INC.
Duplicate self or friends; new humanoid plastic 1990 models,
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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)
