## Notes to Alfred Tarski

1. Hodges (1985/6) claims that no notion defined by Tarski in the monograph on truth possesses all the crucial features of the notion of truth in a structure, and thus that this notion is not really defined by Tarski at this time. (See also Milne 1999. Gómez-Torrente 2001 contains a critique of Hodges’s claim.)

2.
The claim that in the 1936 paper Tarski is primarily thinking of
languages with the conventions of LAr is controversial. It is defended
in Gómez-Torrente (1996), (2009) and disputed e.g. in Mancosu (2006), (2010).
According to Mancosu, Tarski is paradigmatically thinking of languages
like LAr*: LAr* is like LAr except that quantifications need not be
relativized to “*N*”, and the range of the variables is the set
of all individuals of some underlying type theory. See the next
footnote.

3.
A very similar definition can be given for languages like LAr* (see
the preceding footnote): *an interpretation <A, a, R> of LAr*
satisfies the formula function X with respect to a sequence f*
(that assigns arbitrary values from the set of individuals of an
underlying type theory to the original variables of LAr*) if and only
if:

(i)

*X*is*P**x*_{n}(for some) and*n**f*(*x*_{n})∈*A*; or*X*is*P**y*and*a*∈*A*; or(ii)

*X*is*Y**x*_{n}*x*_{m}(for some*m*and*n*) and <*f*(*x*_{n}),*f*(*x*_{m})>∈*R*; or*X*is*Y**y**x*_{n}(for some*n*) and <*a*,*f*(*x*_{n})>∈*R*; or*X*is*Y**x*_{n}*y*(for some*n*) and <*f*(*x*_{n}),*a*>∈*R*; or*X*is*Y**y**y*and <*a*,*a*>∈*R*; or- there is a formula function
*Y*such that*X*is ¬*Y*and <*A*,*a*,*R*> does not satisfy*Y*with respect to sequence*f*; or - there are formula functions
*Y*and*Z*such that*X*is (*Y*→*Z*) and either <*A*,*a*,*R*> does not satisfy*Y*with respect to sequence*f*or <*A*,*a*,*R*> satisfies*Z*with respect to sequence*f*; or, finally, - there is a formula function
*Z*and a number*n*such that*X*is ∀*x*_{n}*Z*and every sequence*g*that assigns arbitrary values to the (original) variables of LAr and that differs from*f*at most in what it assigns to*x*_{n}is such that <*A*,*a*,*R*> satisfies*Z*with respect to*g*.