## 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:

(a) (i) *X* is *P**x*_{n} (for some
* n*) and

*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

(b) 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

(c) 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,

(d) 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*.