Contents

philosophy

# Contents

## Idea

In philosophy, the statement that given any $A$, then $A$ equals $A$

$A = A$

(i.e. reflexivity) has been called (see the references below) the first law of thought.

In terms of type theory this is the judgement that exhibits the term introduction for the reflector in an identity type

$A \colon Type \;\vdash\; refl_A \;\colon A = A\; \,.$

In fact, identity types may be regarded as inductive types generated by just this reflector (Licata 11, Shulman 12, slide 35, see there).

## References

In Gottfried Leibniz‘s unpublished but famous manuscript on logic (from some time in 1683-1716), reproduced in the Latin original in

• K. Gerhard (ed.), Section XIX, p. 228 in: Die philosophischen Schriften von Gottfried Wilhelm Leibniz, Siebenter Band, Weidmannsche Buchhandlung (1890) [archive.org]

and English translation in

• Clarence I. Lewis, Appendix (p. 373) of: A Survey of Symbolic Logic, University of California (1918) [pdf]

it says, after statement of the identity of indiscernibles and then the indiscernibility of identicals, that

$A$ and $A$ are, of course, said to be the same

Later,

calls “$A = A$” “den ersten, schlechthin unbedingent Grundsatz” hence the “absolutely imperative principle”. (In fact Fichte there highlights the importance of the antecedent $A \colon Type$ to arrive at the consequent $A = A$.)

in its §875 calls it the first original law of thought:

The first original law of thought (WdL §875): everything is identical with itself (WdL §863), no two things are like each other (WdL §903).

• Thomas Hughes, The Ideal Theory of Berkeley and the Real World, (1865) Part II, Section XV,

in Footnote, p. 38 calls it the first of the four primary laws of thought.

For the understanding of identity types as inductive types genrated just by the reflector $x = x$ see