Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

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PRIMITIVE PROPOSITIONS

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Primitive propositions. Some propositions must be assumed without proof,
since all inference proceeds from propositions previously asserted. These, as
far as they concern the functions of propositions mentioned above, will be
found stated in *1, where the formal and continuous exposition of the subject
commences. Such propositions will be called "primitive propositions." These,
like the primitive ideas, are to some extent a matter of arbitrary choice;
though, as in the previous case, a logical system grows in importance
according as the primitive propositions are few and simple. It will be
found that owing to the weakness of the imagination in dealing with simple
abstract ideas no very great stress can be laid upon their obviousness. They
are obvious to the instructed mind, but then so are many propositions which
cannot be quite true, as being disproved by their contradictory consequences.
The proof of a logical system is its adequacy and its coherence. That is:
(1) the system must embrace among its deductions all those propositions
which we believe to be true and capable of deduction from logical premisses
alone, though possibly they may require some slight limitation in the form of
an increased stringency of enunciation; and (2) the system must lead to no
contradictions, namely in pursuing our inferences we must never be led to
assert both p and not-p, i.e. both " F. p" and " F. ~p" cannot legitimately
appear.
The following are the primitive propositions employed in the calculus
of propositions. The letters " Pp " stand for " primitive proposition."
(1) Anything implied by a true premiss is true Pp.
This is the rule which justifies inference.
(2) F:pvp.).p Pp,
i.e. if p or p is true, then p is true.
(3) F:q.).pvq Pp,
i.e. if q is true, then p or q is true.
(4) I:pvq.D.qvp Pp,
i.e. if p or q is true, then q or p is true.
(5) F:pv(qvr)..qv(pvr) Pp,
i.e. if either p is true or "q or r" is true, then either q is true or "p or r"
is true.
(6) F:.q)r.):pvq.).pvr Pp,
i.e. if q implies r, then "p or q " implies "p or r."
(7) Besides the above primitive propositions, we require a primitive
proposition called " the axiom of identification of real variables." When we
have separately asserted two different functions of x, where x is undetermined,
it is often important to know whether we can identify the x in one