## 1. What is a Schema?



A schema is a complex system consisting of


1. a *template-text* or *schema-template*: a syntactic
string composed of significant words and/or symbols and also of
placeholders (letters, blanks, circled numbers, ellipses, ordinal
number expressions like 'the first' and 'the
second', etc.), and
2. a *side condition* specifying how the placeholders are to
be filled to obtain instances, and also, sometimes, how the
significant words or symbols are to be understood (Tarski 1933/1983:
155; Church 1956: 172). In particular, the side condition specifies
the language, whether natural or formal, to which the instances of the
schema are to belong.



Among the best-known schemas is Tarski's schema T, whose
template-text is the eight-word two-ellipsis string:



> 
> ...is a true sentence if and only if....
> 



The side condition requires that the second blank is to be filled in
with a (declarative) sentence of English and the first blank is to be
filled in by a name of that sentence (Tarski 1933/1983: 155). The
following string is an instance: 



> 
> 'zero is one' is a true sentence if and only if zero is
> one.
> 



More revealing instances are obtained by using a sentence not known to
be true and not known to be false: 



> 
> 'every perfect number is even' is a true sentence if and
> only if every perfect number is even.
> 



The fourteen-word sentence



> 
> Either zero is even or it is not the case that zero is even.
> 



is an instance of the *excluded-middle sentence schema for
English*, which involves the template 



> 
> Either \(A\) or it is not the case that \(A\).
> 



The side condition is that the two occurrences of '\(A\)'
are to be filled by occurrences of the same well-formed English
declarative sentence, that the discontinuous expression 'Either
... or ... '; expresses classical non-exclusive
disjunction, and that the six-word sentence-prefix 'it is not
the case that'; expresses classical negation. Notice that this
schema-template is not an English sentence. It would be strictly
speaking incoherent to use it as a sentence in an attempted
assertion. It would also be wrong to call it true or false, though it
can be characterized as valid or invalid, in appropriate senses of
these ambiguous words. 



Some logicians seem to identify the schema with the template alone.
(Tarski's wording at 1933/1983: 155-6 suggests this
identification, while Church's at 1956: 149 seems calculated to
avoid it.) But one and the same schema-template may be a component of
any number of different schemata depending on the side condition or
the underlying language. Furthermore, since different characters or
strings can be used as placeholders (see above) and since even one
notational change produces a different syntactic string in the strict
sense (Corcoran et al. 1974), one and the same set of instances may be
determined by different schema-template/side-condition pairings even
given a fixed language. It may be this fact that leads some authors to
write as though the schema is to be identified with the set of
instances. For many purposes it is the set of specified instances that
is of primary importance and the question of exactly what is involved
in specifying it is considered a mere technicality.



Sometimes (as in the excluded-middle schema, above) the placeholders
in a schema-template are marked by letters. It is important to keep in
mind the distinction between, on one hand, an open sentence, such as
'\((x + y) = (y + x)\)'; whose object-language numerical
variables '\(x\)' and '\(y\)' range over the
numbers and, on the other, a schema such as *the number-theoretic
commutativity schema* whose template-text is '\((X + Y) = (Y
+ X)\)'; and whose side condition is that the two occurrences of
'\(X\)' are to be replaced by two occurrences of one and
the same numeral, and likewise for the two occurrences of
'\(Y\)'. The numerals belong to the object language, while
the place-holders belong to the metalanguage. The variables in the
object-language range over a domain of objects, while the 'dummy
letters' in the template-text are just placeholders for syntactic
substituends. (For a careful exposition of the distinction, see Quine
1945: sec. 1.)



Schemas may be classed by the syntactic type of their instances as
sentence schemas, subsentential schemas, or argument-text schemas. We
have already seen two examples of sentence schemas. The string



> 
> the successor of \(A\)
> 



is the template-text for a *subsentential schema*, where the
side condition specifies that the letter '\(A\)' be
replaced by an arabic numeral. The definite description 



> 
> the successor of 9
> 



would be an instance. Note that this schema is very different from the
open term 



> 
> the successor of \(x\),
> 



where the '\(x\)' is an object-language variable. The
schema is essentially a recipe for generating syntactic instances. The
'dummy letter' '\(A\)' in its template-text is
just a placeholder for substituends (here, numerals). The
'\(x\)' in the open term, by contrast, is a variable
ranging over objects (here, numbers).




An *argument-text schema* is a schema whose instances are
argument-texts. An *argument-text* is a two part system
composed of a set of sentences called the premises and a single
sentence called the conclusion. (An *argument* is that which is
expressed by an argument-text, as a proposition is that which is
expressed by a sentence.) Of the various ways of presenting an
argument-text perhaps the one least open to misinterpretation is the
premises-line-conclusion format which consists in listing the premises
followed by a line followed by the conclusion. For example:



\[
\begin{align}
& \textrm{Every circle is a polygon.}\\
& \textrm{Every triangle is a circle.}\\
& \textrm{Every square is a triangle.}\\\hline
& \textrm{Every square is a polygon.}
\end{align}
\]


An example of an argument-text schema is the inference rule *modus
ponens*:

 

\[
\begin{align}
& A\\
& \textrm{if } A \textrm{ then }B\\\hline
& B
\end{align}
\]


The side condition specifies that '\(A\)' and
'\(B\)' be replaced with declarative sentences of English,
and that both occurrences of '\(A\)' (and likewise of
'\(B\)') be replaced by the same sentence or formula.




Axiom schemas can be thought of as zero-premise argument-text
schemas.


## 2. Uses of Schemas



Schemas are used in the formalization of logic, mathematics, and
semantics. In logic, they are used to specify the axioms and inference
rules of a system. For example, one formalization of first-order logic
(in Shapiro 1991: 65) states that



> 
> 
> Any formula obtained by substituting formulas for the Greek letters is
> an *axiom*:
> 
>  
> 
> \[ \begin{align} \Phi & \rightarrow(\Psi \rightarrow \Phi)\\ (\Phi
> \rightarrow(\Psi \rightarrow \Xi)) & \rightarrow((\Phi \rightarrow
> \Psi) \rightarrow(\Phi \rightarrow \Xi))\\ (\neg \Phi \rightarrow \neg
> \Psi) & \rightarrow(\Psi \rightarrow \Phi)\\ \forall x\Phi(x)
> & \rightarrow \Phi(t) \end{align} \]
> 
> 
> where \(t\) is a term free for \(x\) in \(\Phi\),
> 
> 
> 



and that any inference of the form 

 

\[
\begin{align}
& \Phi\\
& \Phi \rightarrow \Psi\\\hline
& \Psi\\
\end{align}
\]


or (where \(x\) does not occur free in \(\Phi)\) 

 

\[
\begin{align}
 \Phi & \rightarrow \Psi(x)\\\hline
 \Phi & \rightarrow \forall x\Psi(x),
\end{align}
\]


is valid. 



Some mathematical theories can be finitely axiomatized in a
first-order language, but certain historically important number
theories and set theories cannot. The axioms of these theories can
sometimes be specified using schemata. For example, in first-order
number theory the induction principle is specified using the
schema

 

\[ [F(0) \mathbin{\&} \forall x((\textit{Num}(x) \mathbin{\&}
F(x)) \rightarrow F(sx)] \rightarrow \forall x(\textit{Num}(x)
\rightarrow F(x)) \]


where the two blanks marked '\(F(x)\)' are to be filled
with a first-order formula having one or more free occurrences of the
variable '\(x\)', the blank marked '\(F(0)\)'
is to be filled with the same formula after each free occurrence of
'\(x\)' has been replaced by an occurrence of
'0', and the blank labeled '\(F(sx)\)' is to
be filled with the same formula after each free occurrence of
'\(x\)' has been replaced by an occurrence of
'\(sx\)'. 



For example, if we fill the two blanks marked '\(F(x)\)'
with '\(x\ne s x\)', we have: 

 

\[ [ 0\ne s0 \mathbin{\&} \forall x ((\textit{Num} (x)
\mathbin{\&} x\ne s x) \rightarrow s x\ne ss x)] \rightarrow
\forall x (\textit{Num} (x) \rightarrow x\ne s x) \]


Using English as the underlying object language, the following
template-text could be used.



If zero is \(F\) and the successor of every number that is \(F\) is
also \(F\), then every number is \(F\), 



where the four occurrences of '\(F\)' are to be filled in
with one and the same arithmetic predicate (e.g., 'smaller than
some prime'). 



In a second-order formalization of number theory, by contrast, a
single induction axiom can be given: 




\[ \forall F \{[F (0) \mathbin{\&} \forall x ((\textit{Num} (x)
 \mathbin{\&} F (x)) \rightarrow F(sx)] \rightarrow \forall x
 (\textit{Num} (x) \rightarrow F(x))\} \]


For every \(F\), if zero is \(F\) and the successor of every number
that is \(F\) is also \(F\), then every number is \(F\).




Here '\(F\)' is not a placeholder in a schema, but a
genuine variable ranging over properties or classes (or, on some
interpretations, ranging plurally over individuals). For comparisons
between first-order and second-order logic, see Corcoran 1998.



The orthographic similarities between the first-order induction schema
and the second-order induction axiom have an unfortunate tendency to
obscure the important differences between them. The latter is a
sentence in the language, whereas the former is just a recipe for
generating sentences. Nor are they inferentially equivalent: the set
of instances of the first-order induction schema is logically weaker
than the second-order induction axiom. That is, there are sentences of
first-order arithmetic that can be deduced from the second-order
induction axiom (together with the other axioms of arithmetic, which
are common to first-order and second-order arithmetic) but not from
the instances of the first-order induction schema (see Shapiro 1991:
110).



Schemas have also played a prominent role in semantics. Tarski held
that an instance of his 'T-schema' (which he calls a
'scheme') could be regarded as a "partial definition
of truth", or rather of "true sentence":



> 
> 
> The general scheme of this kind of sentence can be depicted in the
> following way:
> 
> 
> * (2) \(x\) is a true
> sentence if and only if \(p\).
> 
> 
> 
> In order to obtain concrete definitions we substitute in the place of
> the symbol '\(p\)' in this scheme any sentence, and in the
> place of '\(x\)' any individual name of this sentence.
> (Tarski 1933/1983: 155-6)
> 
> 
> 



He took it to be a criterion of adequacy for a definition of
'true sentence' for a language that it have all such
'partial definitions' as consequences (Tarski 1933/1983:
187-8). 


## 3. Ontological Status of Schemas



It is important to be clear about the mixed ontological status of
schemas. The template-text of the schema is a syntactic object, a
string of characters, and has the same ontological presuppositions as
numerals, words, formulas, and the like. For example, the
template-text for the English naming schema--'The
expression ... names the entity ...'-- is a
forty-character expression involving twenty-seven letter-occurrences,
six occurrences of the space, and seven occurrences of the period. On
the other hand, the side condition is an intensional entity comparable
to a proposition.



A schema-template is a string type having indefinitely many tokens in
Peirce's sense (Peirce 1906; Corcoran et al. 1974: 638 n. 5).
But none of the tokens of a schema-template are instances of the
schema. In fact, every instance of a schema is a string type having
its own tokens. The word 'instance' is a relation noun for
a relation certain string types bear to certain schemas. The word
'token' is a relation noun for a relation certain
macroscopic physical objects bear to certain abstract objects. Neither
a schema nor a schema-template is a common noun denoting the
instances, and neither is a proper name of a set of instances.



Some philosophers emphasize the ontological economies possible by
using schemas rather than second-order axioms (e.g., Quine 1970/1986).
But rarely if ever do these philosophers present a full and objective
discussion of the "ontological commitments" entailed by
the use of schemas. For example, number theory *per se*
presupposes the existence of numbers, and perhaps numerical functions
and numerical properties, but it does not presuppose the existence of
mathematical notation and it *a fortiori* does not presuppose
the existence of the vast, intricate notational system that we call
the language of number theory. Sometimes the use of schemas may
decrease the ontological commitments of the object language while
increasing those of the metalanguage, or at least not achieving any
net savings.


## 4. Schemas in the History of Logic



The Greek word 'schema'; was used in Plato's Academy
for "[geometric] figure" and in Aristotle's Lyceum
for "[syllogistic] figure". Although Aristotle's
syllogistic figures or "schemata" were not schemas in the
modern sense, Aristotle's moods were. For example, the
template-text of the mood BARBARA is

 

\[
\begin{align}
& P \textrm{ belongs-to-every }M.\\
& M \textrm{ belongs-to-every }S.\\\hline
& P \textrm{ belongs-to-every }S.
\end{align}
\]


The associated side condition is that (1) both occurrences of
'\(P\)' are to be filled with occurrences of one and the
same common noun, (2) both occurrences of '\(M\)' are to
be filled with occurrences of one and the same common noun other than
the one used for '\(P\)', (3) both occurrences of
'\(S\)' are to be filled with occurrences of one and the
same common noun other than the ones used for '\(P\)' and
'\(M\)', and that (4) the expression
'belongs-to-every' is taken to express universal
affirmative predication as in the *Prior analytics*. The rules
of the Stoic propositional logic have been taken to be schemas.



It is hard to date self-conscious use of the word
'schema'; in the modern sense. Russell's
*Introduction to Mathematical Philosophy* (1919) uses it
casually to describe propositional functions: 



> 
> 
> A propositional function ... may be taken to be a mere schema, a
> mere shell, an empty receptacle for meaning, not something already
> significant. (1919: 157) 
> 
> 
> 



But propositional functions are not syntactic schemas in the modern
sense. Tarski's 1933 truth-definition paper (Tarski 1933/1983:
157, 160, 172) was one of the first prominent publications to use the
word 'scheme' in a sense close to that of this article
(Tarski 1933/1983: 155, 156). Tarski also uses the word
'schema', and its plural 'schemata', in the
pre-World-War II period (1983: 63-64, 114, 310, 386, 423).



Early twentieth-century formalizations of logic used what were called
"substitution rules" with finite set of axioms instead of
schemata that specified infinitely many axioms. These
"substitution rules" were not the familiar rules for
"substituting equals for equals"; rather they were closer
to what are called instantiation rules today. The intuitive motivation
for "substitution rules" was very simple but the syntactic
details for implementing them were "intolerably
complex"--to use the words of Paul Rosenbloom (1950: 109).
In fact several first-rate logicians were led to make embarrassing
errors, as Rosenbloom documents in the place just cited. Church (1956:
158) credits von Neumann with "the device of using axiom
schemata", which rendered the (notoriously difficult to state)
substitution rule unnecessary.



As Church has emphasized (e.g., 1956: 59), metamathematical treatment
of schemas requires use of formalized or logically perfect languages
and an axiomatized theory of strings as found for the first time in
Tarski's 1933 truth-definition paper (1933/1983: 152-256).
For more on the history, philosophy, and mathematics of this important
but somewhat neglected field, see Corcoran *et al*. 1974;
Corcoran 2006).