Model-Theoretic Conceptions of Logical Consequence

One sentence X is said to be a logical consequence of a set K of
sentences, if and only if, in virtue of logic alone, it is impossible
for all the sentences in the set to be true without X being true as
well. One well-known specification of this informal characterization
is the model-theoretic conception of logical consequence: a sentence X
is a logical consequence of a set K of sentences if and only if all
models of K are models of X. The model-theoretic characterization is a
theoretical definition of logical consequence. It has been argued that
this conception of logical consequence is more basic than the
characterization in terms of deducibility in a deductive system. The
correctness of the model-theoretic characterization of logical
consequence, and the adequacy of the notion of a logical constant it
utilizes are matters of contemporary debate.

1. Introduction

One sentence X is said to be a logical consequence of a set of
sentences, if and only if, in virtue of logic alone, it is impossible
for all the sentences in K to be true without X being true as well.
One well-known specification of this informal characterization, due to
Tarski (1936), is: X is a logical consequence of K if and only if
there is no possible interpretation of the non-logical terminology of
the language L according to which all the sentence in K are true and X
is false. A possible interpretation of the non-logical terminology of
L according to which sentences are true or false is a reading of the
non-logical terms according to which the sentences receive a
truth-value (that is, are either true or false) in a situation that is
not ruled out by the semantic properties of the logical constants. The
philosophical locus of the technical development of 'possible
interpretation' in terms of models is Tarski (1936). A model for a
language L is the theoretical development of a possible interpretation
of non-logical terminology of L according to which the sentences of L
receive a truth-value. The characterization of logical consequence in
terms of models is called the Tarskian or model-theoretic
characterization of logical consequence. It may be stated as follows.

X is a logical consequence of K if and only if all models of K are
models of X.

See the entry, Logical Consequence, Philosophical Considerations, for
discussion of Tarski's development of the model-theoretic
characterization of logical consequence in light of the ordinary
conception.

We begin by giving an interpreted language M. Next, logical
consequence is defined model-theoretically. Finally, the status of
this characterization is discussed, and criticisms of it are
entertained.
2. Linguistic Preliminaries: the Language M

Here we define a simple language M, a language about the McKeon
family, by first sketching what strings qualify as well-formed
formulas (wffs) in M. Next we define sentences from formulas, and then
give an account of truth in M, that is we describe the conditions in
which M-sentences are true.
a. Syntax of M

Building blocks of formulas

Terms

Individual names—'beth', 'kelly', 'matt', 'paige', 'shannon', 'evan',
and 'w1', 'w2', 'w3 ', etc.

Variables—'x', 'y', 'z', 'x1', 'y1 ', 'z1', 'x2', 'y2', 'z2', etc.

Predicates

1-place predicates—'Female', 'Male'

2-place predicates—'Parent', 'Brother', 'Sister', 'Married',
'OlderThan', 'Admires', '='.

Blueprints of well-formed formulas (wffs)

Atomic formulas: An atomic wff is any of the above n-place predicates
followed by n terms which are enclosed in parentheses and separated by
commas.

Formulas: The general notion of a well-formed formula (wff) is defined
recursively as follows:

(1) All atomic wffs are wffs.
(2) If α is a wff, so is '~α'.
(3) If α and β are wffs, so is '(α & β)'.
(4) If α and β are wffs, so is '(α v β)'.
(5) If α and β are wffs, so is '(α → β)'.
(6) If Ψ is a wff and v is a variable, then '∃vΨ' is a wff.
(7) If Ψ is a wff and v is a variable, then '∀vΨ' is a wff.
Finally, no string of symbols is a well-formed formula of M unless
the string can be derived from (1)-(7).

The signs '~', '&', 'v', and '→', are called sentential connectives.
The signs '∀' and '∃' are called quantifiers.

It will prove convenient to have available in M an infinite number of
individual names as well as variables. The strings 'Parent(beth,
paige)' and 'Male(x)' are examples of atomic wffs. We allow the
identity symbol in an atomic formula to occur in between two terms,
e.g., instead of '=(evan, evan)' we allow '(evan = evan)'. The symbols
'~', '&', 'v', and '→' correspond to the English words 'not', 'and',
'or' and 'if…then', respectively. '∃' is our symbol for an existential
quantifier and '∀' represents the universal quantifier. '∃vΨ' and
'∀vΨ' correspond to for some v, Ψ, and for all v, Ψ, respectively. For
every quantifier, its scope is the smallest part of the wff in which
it is contained that is itself a wff. An occurrence of a variable v is
a bound occurrence iff it is in the scope of some quantifier of the
form '∃v' or the form '∀v', and is free otherwise. For example, the
occurrence of 'x' is free in 'Male(x)' and in '∃y Married(y, x)'. The
occurrences of 'y' in the second formula are bound because they are in
the scope of the existential quantifier. A wff with at least one free
variable is an open wff, and a closed formula is one with no free
variables. A sentence is a closed wff. For example, 'Female(kelly)'
and '∃y∃x Married(y, x)' are sentences but 'OlderThan(kelly, y)' and
'(∃x Male(x) & Female(z))' are not. So, not all of the wffs of M are
sentences. As noted below, this will affect our definition of truth
for M.
b. Semantics for M

We now provide a semantics for M. This is done in two steps. First, we
specify a domain of discourse, that is, the chunk of the world that
our language M is about, and interpret M's predicates and names in
terms of the elements composing the domain. Then we state the
conditions under which each type of M-sentence is true. To each of the
above syntactic rules (1-7) there corresponds a semantic rule that
stipulates the conditions in which the sentence constructed using the
syntactic rule is true. The principle of bivalence is assumed and so
'not true' and 'false' are used interchangeably. In effect, the
interpretation of M determines a truth-value (true, false) for each
and every sentence of M.

Domain D—The McKeons: Matt, Beth, Shannon, Kelly, Paige, and Evan.

Here are the referents and extensions of the names and predicates of M.

Terms: 'matt' refers to Matt, 'beth' refers to Beth, 'shannon' refers
to Shannon, etc.

Predicates. The meaning of a predicate is identified with its
extension, that is the set (possibly empty) of elements from the
domain D the predicate is true of. The extension of a one-place
predicate is a set of elements from D, the extension of a two-place
predicate is a set of ordered pairs of elements from D.

The extension of 'Male' is {Matt, Evan}.

The extension of 'Female' is {Beth, Shannon, Kelly, Paige}.

The extension of 'Parent' is {<Matt, Shannon>, <Matt, Kelly>, <Matt,
Paige>, <Matt, Evan>, <Beth, Shannon>, <Beth, Kelly>, <Beth, Paige>,
<Beth, Evan>}.

The extension of 'Married' is {<Matt, Beth>, <Beth, Matt>}.

The extension of 'Sister' is {<Shannon, Kelly>, <Kelly, Shannon>,
<Shannon, Paige>, <Paige, Shannon>, <Kelly, Paige>, <Paige, Kelly>,
<Kelly, Evan>, <Paige, Evan>, <Shannon, Evan>}.

The extension of 'Brother' is {<Evan, Shannon>, <Evan, Kelly>, <Evan, Paige>}.

The extension of 'OlderThan' is {<Beth, Matt>, <Beth, Shannon>, <Beth,
Kelly>, <Beth, Paige>, <Beth, Evan>, <Matt, Shannon>, <Matt, Kelly>,
<Matt, Paige>, <Matt, Evan>, <Shannon, Kelly>, <Shannon, Paige>,
<Shannon, Evan>, <Kelly, Paige>, <Kelly, Evan>, <Paige, Evan>}.

The extension of 'Admires' is {<Matt, Beth>, <Shannon, Matt>,
<Shannon, Beth>, <Kelly, Beth>, <Kelly, Matt>, <Kelly, Shannon>,
<Paige, Beth>, <Paige, Matt>, <Paige, Shannon>, <Paige, Kelly>, <Evan,
Beth>, <Evan, Matt>, <Evan, Shannon>, <Evan, Kelly>, <Evan, Paige>}.

The extension of '=' is {<Matt, Matt>, <Beth, Beth>, <Shannon,
Shannon>, <Kelly, Kelly>, <Paige, Paige>, <Evan, Evan>}.
(I) An atomic sentence with a one-place predicate is true iff the
referent of the term is a member of the extension of the predicate,
and an atomic sentence with a two-place predicate is true iff the
ordered pair formed from the referents of the terms in order is a
member of the extension of the predicate.

The atomic sentence 'Female(kelly)' is true because, as indicated
above, the referent of 'kelly' is in the extension of the property
designated by 'Female'. The atomic sentence 'Married(shannon, kelly)'
is false because the ordered pair <Shannon, Kelly> is not in the
extension of the relation designated by 'Married'.

Let α and β be any M-sentences.

(II) '~α' is true iff α is false.
(III) '(α & β)' is true when both α and β are true; otherwise '(α &
β)' is false.
(IV) '(α v β)' is true when at least one of α and β is true;
otherwise '(α v β)' is false.
(V) '(α → β)' is true if and only if (iff) α is false or β is true.
So, '(α → β)' is false just in case α is true and β is false.

The meanings for '~' and '&' roughly correspond to the meanings of
'not' and 'and' as ordinarily used. We call '~α' and '(α & β)'
negation and conjunction formulas, respectively. The formula '(~α v
β)' is called a disjunction and the meaning of 'v' corresponds to
inclusive or. There are a variety of conditionals in English (e.g.,
causal, counterfactual, logical), each type having a distinct meaning.
The conditional defined by (V) is called the material conditional. One
way of following (V) is to see that the truth conditions for '(α → β)'
are the same as for '~(α & ~β)'.

By (II) '~Married(shannon, kelly)' is true because, as noted above,
'Married(shannon, kelly)' is false. (II) also tells us that
'~Female(kelly)' is false since 'Female(kelly)' is true. According to
(III), '(~Married(shannon, kelly) & Female(kelly))' is true because
'~Married(shannon, kelly)' is true and 'Female(kelly)' is true. And
'(Male(shannon) & Female(shannon))' is false because 'Male(shannon)'
is false. (IV) confirms that '(Female(kelly) v Married(evan, evan))'
is true because, even though 'Married(evan, evan)' is false,
'Female(kelly)' is true. From (V) we know that the sentence '(~(beth =
beth) → Male(shannon))' is true because '~(beth = beth)' is false. If
α is false then '(α → β)' is true regardless of whether or not β is
true. The sentence '(Female(beth) → Male(shannon))' is false because
'Female(beth)' is true and 'Male(shannon)' is false.

Before describing the truth conditions for quantified sentences we
need to say something about the notion of satisfaction. We've defined
truth only for the formulas of M that are sentences. So, the notions
of truth and falsity are not applicable to non-sentences such as
'Male(x)' and '((x = x) → Female(x))' in which 'x' occurs free.
However, objects may satisfy wffs that are non-sentences. We introduce
the notion of satisfaction with some examples. An object satisfies
'Male(x)' just in case that object is male. Matt satisfies 'Male(x)',
Beth does not. This is the case because replacing 'x' in 'Male(x)'
with 'matt' yields a truth while replacing the variable with 'beth'
yields a falsehood. An object satisfies '((x = x) → Female(x))' if and
only if it is either not identical with itself or is a female. Beth
satisfies this wff (we get a truth when 'beth' is substituted for the
variable in all of its occurrences), Matt does not (putting 'matt' in
for 'x' wherever it occurs results in a falsehood). As a first
approximation, we say that an object with a name, say 'a', satisfies a
wff 'Ψv' in which at most v occurs free if and only if the sentence
that results by replacing v in all of its occurrences with 'a' is
true. 'Male(x)' is neither true nor false because it is not a
sentence, but it is either satisfiable or not by a given object. Now
we define the truth conditions for quantifications, utilizing the
notion of satisfaction. The notion of satisfaction will be revisited
below when we formalize the semantics for M and give the
model-theoretic characterization of logical consequence.

Let Ψ be any formula of M in which at most v occurs free.
(VI) '∃vΨ' is true just in case there is at least one individual in
the domain of quantification (e.g. at least one McKeon) that satisfies
Ψ.
(VII) '∀vΨ' is true just in case every individual in the domain of
quantification (e.g. every McKeon) satisfies Ψ.

Here are some examples. '∃x(Male(x) & Married(x, beth))' is true
because Matt satisfies '(Male(x) & Married(x, beth))'; replacing 'x'
wherever it appears in the wff with 'matt' results in a true sentence.
The sentence '∃xOlderThan(x, x)' is false because no McKeon satisfies
'OlderThan(x, x)', that is replacing 'x' in 'OlderThan(x, x)' with the
name of a McKeon always yields a falsehood.

The universal quantification '∀x( OlderThan(x, paige) → Male(x))' is
false for there is a McKeon who doesn't satisfy '(OlderThan(x, paige)
→ Male(x))'. For example, Shannon does not satisfy '(OlderThan(x,
paige) → Male(x))' because Shannon satisfies 'OlderThan(x, paige)' but
not 'Male(x)'. The sentence '∀x(x = x)' is true because all McKeons
satisfy 'x = x'; replacing 'x' with the name of any McKeon results in
a true sentence.

Note that in the explanation of satisfaction we suppose that an object
satisfies a wff only if the object is named. But we don't want to
presuppose that all objects in the domain of discourse are named. For
the purposes of an example, suppose that the McKeons adopt a baby boy,
but haven't named him yet. Then, '∃x Brother(x, evan)' is true because
the adopted child satisfies 'Brother(x, evan)', even though we can't
replace 'x' with the child's name to get a truth. To get around this
is easy enough. We have added a list of names, 'w1', 'w2', 'w3', etc.,
to M, and we may say that any unnamed object satisfies 'Ψv' iff the
replacement of v with a previously unused wi assigned as a name of
this object results in a true sentence. In the above scenerio,
'∃xBrother(x, evan)' is true because, ultimately, treating 'w1' as a
temporary name of the child, 'Brother(w1, evan)' is true. Of course,
the meanings of the predicates would have to be amended in order to
reflect the addition of a new person to the domain of McKeons.
3. What is a Logic?

We have characterized an interpreted language M by defining what
qualifies as a sentence of M and by specifying the conditions under
which any M-sentence is true. The received view of logical consequence
entails that the logical consequence relation in M turns on the nature
of the logical constants in the relevant M-sentences. We shall regard
just the sentential connectives, the quantifiers of M, and the
identity predicate as logical constants (the language M is a
first-order language). For discussion of the notion of a logical
constant see Section 5c below.

At the start of this article, it is said that a sentence X is a
logical consequence of a set K of sentences, if and only if, in virtue
of logic alone, it is impossible for all the sentences in K to be true
without X being true as well. A model-theoretic conception of logical
consequence in language M clarifies this intuitive characterization of
logical consequence by appealing to the semantic properties of the
logical constants, represented in the above truth clauses (I)-(VII).
In contrast, a deductive-theoretic conception clarifies logical
consequence in M, conceived of in terms of deducibility, by appealing
to the inferential properties of logical constants portrayed as
intuitively valid principles of inference, that is, principles
justifying steps in deductions. See Logical Consequence,
Deductive-Theoretic Conceptions for a deductive-theoretic
characterization of logical consequence in terms of a deductive
system, and foror a discussion on the relationship between the logical
consequence relation and the model-theoretic and deductive-theoretic
conceptions of it.

Following Shapiro (1991, p. 3) define a logic to be a language L plus
either a model-theoretic or a deductive-theoretic account of logical
consequence. A language with both characterizations is a full logic
just in case the two characterizations coincide. The logic for M
developed below may be viewed as a classical logic or a first-order
theory.
4. Model-Theoretic Consequence

The technical machinery to follow is designed to clarify how it is
that sentences receive truth-values owing to interpretations of them.
We begin by introducing the notion of a structure. Then we revisit the
notion of satisfaction in order to make it more precise, and link
structures and satisfaction to model-theoretic consequence. We offer a
modernized version of the model-theoretic characterization of logical
consequence sketched by Tarski and so deviate from the details of
Tarski's presentation in his (1936).
a. Truth in a structure

Relative to our language M, a structure U is an ordered pair <D, I>.
(1) D, a non-empty set of elements, is the domain of discourse. Two
things to highlight here. First, the domain D of a structure for M may
be any set of entities, e.g. the dogs living in Connecticut, the
toothbrushes on Earth, the natural numbers, the twelve apostles, etc.
Second, we require that D not be the empty set.

(2) I is a function that assigns to each individual constant of M an
element of D, and to each n-place predicate of M a subset of Dn (that
is, the set of n-tuples taken from D). In essence, I interprets the
individual constants and predicates of M, linking them to elements and
sets of n-tuples of elements from of D. For individual constants c and
predicates P, the element IU(c) is the element of D designated by c
under IU, and IU(P) is the set of entities assigned by IU as the
extension of P.

By 'structure' we mean an L-structure for some first-order language L.
The intended structure for a language L is the course-grained
representation of the piece of the world that we intend L to be about.
The intended domain D and its subsets represent the chunk of the world
L is being used to talk about and quantify over. The intended
interpretation of L's constants and predicates assigns the actual
denotations to L's constants and the actual extensions to the
predicates. The above semantics for our language M, may be viewed, in
part, as an informal portrayal of the intended structure of M, which
we refer to as UM. That is, we take M to be a tool for talking about
the McKeon family with respect to gender, who is older than whom, who
admires whom, etc. To make things formally prim and proper we should
represent the interpretation of constants as IUM(matt) = Matt,
IUM(beth) = Beth, and so on. And the interpretation of predicates can
look like IUM(Male) = {Matt, Evan}, IUM(Female) = {Beth, Shannon,
Kelly, Paige}, and so on. We assume that this has been done.

A structure U for a language L (that is, an L-structure) represents
one way that a language can be used to talk about a state of affairs.
Crudely, the domain D and the subsets recovered from D constitute a
rudimentary representation of a state of affairs, and the
interpretation of L's predicates and individual constants makes the
language L about the relevant state of affairs. Since a language can
be assigned different structures, it can be used to talk about
different states of affairs. The class of L-structures represents all
the states of affairs that the language L can be used to talk about.
For example, consider the following M-structure U'.

D = the set of natural numbers
IU'(beth) = 2
IU'(matt) = 3
IU'(shannon) = 5
IU'(kelly) = 7
IU'(paige) = 11
IU'(evan) = 10 I U'(Male) = {d ∈ D | d is prime}
I U'(Female) = {d ∈ D | d is even}
I U'(Parent) = ∅
I U'(Married) = {<d, d'> ∈ D2 | d + 1 = d' }
I U'(Sister) = ∅
I U'(Brother) = {<d, d'> ∈ D2 | d < d' }
I U'(OlderThan) = {<d, d'> ∈ D2 | d > d' }
I U'(Admires) = ∅
I U'(=) = {<d, d'> ∈ D2 | d = d' }

In specifying the domain D and the values of the interpretation
function defined on M's predicates we make use of brace notation,
instead of the earlier list notation, to pick out sets. For example,
we write

{d ∈ D | d is even}

to say "the set of all elements d of D such that d is even." And

{<d, d'> ∈ D2 | d > d'}

reads: "The set of ordered pairs of elements d, d' of D such that d >
d'." Consider: the sentence

OlderThan(beth, matt)

is true in the intended structure UM for <IUM(beth), IUM(matt)> is in
IUM(OlderThan). But the sentence is false in U' for <IU'(beth),
IU'(matt)> is not in IU'(OlderThan) (because 2 is not greater than 3).
The sentence

(Female(beth) & Male(beth))

is not true in UM but is true in U' for IU'(beth) is in IU'(Female)
and in IU'(Male) (because 2 is an even prime). In order to avoid
confusion it is worth highlighting that when we say that the sentence
'(Female(beth) & Male(beth))' is true in one structure and false in
another we are saying that one and the same wff with no free variables
is true in one state of affairs on an interpretation and false in
another state of affairs on another interpretation.
b. Satisfaction revisited

Note the general strategy of giving the semantics of the sentential
connectives: the truth of a compound sentence formed with any of them
is determined by its component well-formed formulas (wffs), which are
themselves (simpler) sentences. However, this strategy needs to be
altered when it comes to quantificational sentences. For
quantificational sentences are built out of open wffs and, as noted
above, these component wffs do not admit of truth and falsity.
Therefore, we can't think of the truth of, say,

∃x(Female(x) & OlderThan(x, paige))

in terms of the truth of '(Female(x) & OlderThan(x, paige))' for some
McKeon x. What we need is a truth-relevant property of open formulas
that we may appeal to in explaining the truth-value of the compound
quantifications formed from them. Tarski is credited with the
solution, first hinted at in the following.

The possibility suggests itself, however, of introducing a more
general concept which is applicable to any sentential function [open
or closed wff] can be recursively defined, and, when applied to
sentences leads us directly to the concept of truth. These
requirements are met by the notion of satisfaction of a given
sentential function by given objects. (Tarski 1933, p. 189)

The needed property is satisfaction. The truth of the above
existential quantification will depend on there being an object that
satisfies both 'Female(x)' and 'OlderThan(x, paige)'. Earlier we
introduced the concept of satisfaction by describing the conditions in
which one element satisfies an open formula with one free variable.
Now we want to develop a picture of what it means for objects to
satisfy a wff with n free variables for any n ≥ 0. We begin by
introducing the notion of a variable assignment.

A variable assignment is a function g from a set of variables (its
domain) to a set of objects (its range). We shall say that the
variable assignment g is suitable for a well-formed formula (wff) Ψ of
M if every free variable in Ψ is in the domain of g. In order for a
variable assignment to satisfy a wff it must be suitable for the
formula. For a variable assignment g that is suitable for Ψ, g
satisfies Ψ in U iff the object(s) g assigns to the free variable(s)
in Ψ satisfy Ψ. Unlike the earlier first-step characterization of
satisfaction, there is no appeal to names for the entities assigned to
the variables. This has the advantage of not requiring that new names
be added to a language that does not have names for everything in the
domain. In specifying a variable assignment g, we write α/v, β/v',
χ/v", … to indicate that g(v) = α, g(v' ) = β, g(v" ) = χ, etc. We
understand

U ⊨ Ψ[g]

to mean that g satisfies Ψ in U.

UM ⊨ OlderThan(x, y)[Shannon/x, Paige/y]

This is true: the variable assignment g, identified with [Shannon/x,
Paige/y], satisfies 'Olderthan(x, y)' because Shannon is older than
Paige.

UM ⊨ Admires(x, y)[Beth/x, Matt/y]

This is false for this variable assignment does not satisfy the wff:
Beth does not admire Matt. However, the following is true because Matt
admires Beth.

UM ⊨ Admires(x, y)[Matt/x, Beth/y]

For any wff Ψ, a suitable variable assignment g and structure U
together ensure that the terms in Ψ designate elements in D. The
structure U insures that individual constants have referents, and the
assignment g insures that any free variables in Ψ get denotations. For
any individual constant c, c[g] is the element IU(c). For each
variable v, and assignment g whose domain contains v, v[g] is the
element g(v). In effect, the variable assignment treats the variable v
as a temporary name. We define t[g] as 'the element designated by t
relative to the assignment g'.
c. A formalized definition of truth for Language M

We now give a definition of truth for the language M via the detour
through satisfaction. The goal is to define for each formula α of M
and each assignment g to the free variables, if any, of α in U what
must obtain in order for U ⊨ α[g].
(I) Where R is an n-place predicate and t1, …, tn are terms, U ⊨R(t1,
…, tn)[g] if and only if (iff) the n-tuple <t1[g], …, tn[g]> is in
IU(R).

(II) U ⊨ ~α[g] iff it is not true that U ⊨ α[g].

(III) U ⊨ (α & β)[g] iff U ⊨ α[g] and U ⊨ β[g].

(IV) U ⊨ (α v β)[g] iff U ⊨ α[g] or U ⊨ β[g].

(V) U ⊨ (α → β)[g] iff either it is not true that U ⊨ α[g] or U ⊨ β[g].

Before going on to the (VI) and (VII) clauses for quantificational
sentences, it is worthwhile to introduce the notion of a variable
assignment that comes from another. Consider

∃y(Female(x) & OlderThan(x, y)).

We want to say that a variable assignment g satisfies this wff if and
only if there is a variable assignment g' differing from g at most
with regard to the object it assigns to the variable y such that g'
satisfies '(Female(x) & OlderThan(x, y))'. We say that a variable
assignment g' comes from an assignment g when the domain of g' is that
of g and a variable v, and g' assigns the same values as g with the
possible exception of the element g' assigns to v. In general, we
represent an extension g' of an assignment g as follows.

[g, d/v]

This picks out a variable assignment g' which differs at most from g
in that v is in its domain and g'(v) = d, for some element d of the
domain D. So, it is true that

UM ⊨∃y(Female(x) & OlderThan(x, y)) [Beth/x]

since

UM ⊨ (Female(x) & OlderThan(x, y)) [Beth/x, Paige/y].

What this says is that the variable assignment that comes from the
assignment of Beth to 'x' by adding the assignment of Paige to 'y'
satisfies '(Female(x) & OlderThan(x, y))' in UM. This is true for Beth
is a female who is older than Paige. Now we give the satisfaction
clauses for quantificational sentences. Let Ψ be any formula of M.
(VI) U ⊨∃vΨ[g] iff for at least one element d of D, U ⊨ Ψ[g, d/v].

(VII) U ⊨ ∀vΨ[g] iff for all elements d of D, U ⊨ Ψ[g, d/v].

If α is a sentence, then it has no free variables and we write U ⊨
α[g∅] which says that the empty variable assignment satisfies α in U.
The empty variable assignment g∅ does not assign objects to any
variables. In short: the definition of truth for language L is

A sentence α is true in U if and only if U ⊨ α[g∅], that is the
empty variable assignment satisfies α in U.

The truth definition specifies the conditions in which a formula of M
is true in a structure by explaining how the semantic properties of
any formula of M are determined by its construction from semantically
primitive expressions (e.g., predicates, individual constants, and
variables) whose semantic properties are specified directly. If every
member of a set of sentences is true in a structure U we say that U is
a model of the set. We now work through some examples. The reader will
be aided by referring when needed to the clauses (I)-(VII).

It is true that UM ⊨ ~Married(kelly, kelly))[g∅], that is, by (II) it
is not true that UM ⊨ Married(kelly, kelly))[g∅], because <kelly[g∅],
kelly[g∅]> is not in IUM(Married). Hence, by (IV)

UM ⊨ (Married(shannon, kelly) v ~Married(kelly, kelly))[g∅].

Our truth definition should confirm that

∃x∃y Admires(x, y)

is true in UM. Note that by (VI) UM ⊨∃yAdmires(x, y)[g∅, Paige/x]
since UM ⊨ Admires(x, y)[g∅, Paige/x, kelly/y]. Hence, by (VI)

UM ⊨∃x∃y Admires(x, y)[g∅] .

The sentence, '∀x∃y(Older(y, x) → Admires(x, y))' is true in UM . By
(VII) we know that

UM ⊨ ∀x∃y(Older(y, x) → Admires(x, y))[g∅]

if and only if

for all elements d of D, UM ⊨∃y(Older(y, x) → Admires(x, y))[g∅, d/x].

This is true. For each element d and assignment [g∅, d/x], UM ⊨
(Older(y, x) → Admires(x, y))[g∅, d/x, d'/y], that is, there is some
element d' and variable assignment g differing from [g∅, d/x] only in
assigning d' to 'y', such that g satisfies '(Older(y, x) → Admires(x,
y))' in UM .
d. Model-theoretic consequence defined

For any set K of M-sentences and M-sentence X, we write

K ⊨ X

to mean that every M-structure that is a model of K is also a model of
X, that is, X is a model-theoretic consequence of K.

(1) OlderThan(paige, matt)
(2) ∀x(Male(x) → OlderThan(paige, x))

Note that both (1) and (2) are false in the intended structure UM . We
show that (2) is not a model theoretic consequence of (1) by
describing a structure which is a model of (1) but not (2). The above
structure U' will do the trick. By (I) it is true that U' ⊨
OlderThan(paige, matt)[g∅] because <(paige)[g∅], (matt)[g∅]> is in
IU'(OlderThan) (because 11 is greater than 3). But, by (VII), it is
not the case that

U' ⊨ ∀x(Male(x) → OlderThan(paige, x))[g∅]

since the variable assignment [g∅, 13/x] doesn't satisfy '(Male(x) →
OlderThan(paige, x))' in U' according to (V) for U' ⊨ Male(x)[g∅,
13/x] but not U' ⊨ OlderThan(paige, x))[g∅, 13/x]. So, (2) is not a
model-theoretic consequence of (1). Consider the following sentences.

(3) (Admires(evan, paige) → Admires(paige, kelly))
(4) (Admires(paige, kelly) → Admires(kelly, beth))
(5) (Admires(evan, paige) → Admires(kelly, beth))

(5) is a model-theoretic consequence of (3) and (4). For assume
otherwise. That is assume, that there is a structure U" such that

(i) U" ⊨ (Admires(evan, paige) → Admires(paige, kelly))[g∅]

and

(ii) U" ⊨ (Admires(paige, kelly) → Admires(kelly, beth))[g∅]

but not

(iii) U" ⊨ (Admires(evan, paige) → Admires(kelly, beth))[g∅].

By (V), from the assumption that (iii) is false, it follows that U" ⊨
Admires(evan, paige)[g∅] and not U" ⊨ Admires(kelly, beth)[g∅]. Given
the former, in order for (i) to hold according to (V) it must be the
case that U" ⊨ Admires(paige, kelly))[g∅]. But then it is true that U"
⊨ Admires(paige, kelly))[g∅] and false that U" ⊨ Admires(kelly,
beth)[g∅], which, again appealing to (V), contradicts our assumption
(ii). Hence, there is no such U", and so (5) is a model-theoretic
consequence of (3) and (4).

Here are some more examples of the model-theoretic consequence
relation in action.

(6) ∃xMale(x)
(7) ∃xBrother(x, shannon)
(8) ∃x(Male(x) & Brother(x, shannon))

(8) is not a model-theoretic consequence of (6) and (7). Consider the
following structure U"'.

D = {1, 2, 3}

For all M-individual constants c, IU"'(c) = 1.

IU"'(Male) = {2}, IU"'(Brother) = {<3, 1>}. For all other
M-predicates P, IU"'(P) = ∅.

Appealing to the satisfaction clauses (I), (III), and (VI), it is
fairly straightforward to see that the structure U"' is a model of (6)
and (7) but not of (8). For example, U"' is not a model of (8) for
there is no element d of D and assignment [d/x] such that

U"' ⊨ (Male(x) & Brother(x, shannon))[g∅, d/x].

Consider the following two sentences

(9) Female(shannon)
(10) ∃x Female(x)

(10) is a model-theoretic consequence of (9). For an arbitrary
M-structure U, if U ⊨ Female(shannon)[g∅], then by satisfaction clause
(I), shannon[g∅] is in IU(Female), and so there is at least one
element of D, shannon[g∅], in IU(Female). Consequently, by (VI), U ⊨∃x
Female(x)[g∅].

For a sentence X of M, we write

⊨ X.

to mean that X is a model-theoretic consequence of the empty set of
sentences. This means that every M-structure is a model of X. Such
sentences represent logical truths; it is not logically possible for
them to be false. For example,

⊨ (∀x Male(x) → ∃x Male(x))

is true. Here's one explanation why. Let U be an arbitrary
M-structure. We now show that

U ⊨ (∀x Male(x) → ∃x Male(x))[g∅].

If U ⊨ ∀x Male(x) [g∅] holds, then by (VII) for every element d of the
domain D, U ⊨ Male(x)[g∅, d/x]. But we know that D is non-empty, by
the requirements on structures (see the beginning of Section 4.1), and
so D has at least one element d. Hence for at least one element d of
D, U ⊨ Male(x)[g∅, d/x], that is by (VI), U ⊨∃x Male(x))[g∅]. So, if U
⊨ (∀x Male(x)[g∅] then U ⊨∃x Male(x))[g∅], and, therefore according to
(V),

U ⊨ (∀x Male(x) → ∃x Male(x))[g∅].

Since U is arbitrary, this establishes

⊨ (∀x Male(x) → ∃x Male(x)).

If we treat '=' as a logical constant and require that for all
M-structures U, IU(=) = {<d, d'> ∈ D2| d = d'}, then M-sentences
asserting that identity is reflexive, symmetrical, and transitive are
true in every M-structure, that is the following hold.

⊨ ∀x(x = x)
⊨ ∀x∀y((x = y) → (y = x))
⊨ ∀x∀y∀z(((x = y) & (y = z)) → (x = z))

Structures which assign {<d, d'> ∈ D2| d = d'} to the identity symbol
are sometimes called normal models. Letting 'Ψ(v)' be any wff in which
just variable v occurs free,

∀x∀y((x = y) → (Ψ(x) → Ψ(y)))

is an instance of the principle that identicals are indiscernible—if x
= y then whatever holds of x holds of y—and it is true in every
M-structure U that is a normal model. Treating '=' as a logical
constant (which is standard) requires that we restrict the class of
M-structures appealed to in the above model-theoretic definition of
logical consequence to those that are normal models.
5. The Status of the Model-Theoretic Characterization of Logical Consequence

Logical consequence in language M has been defined in terms of the
model-theoretic consequence relation. What is the status of this
definition? We answered this question in part in Logical Consequence,
Deductive-Theoretic Conceptions: Section 5a. by highlighting Tarski's
argument for holding that the model-theoretic conception of logical
consequence is more basic than any deductive-system account of it.
Tarski points to the fact that there are languages for which valid
principles of inference can't be represented in a deductive-system,
but the logical consequence relation they determine can be represented
model-theoretically. In what follows, we identify the type of
definition the model-theoretic characterization of logical consequence
is, and then discuss its adequacy.
a. The model-theoretic characterization is a theoretical definition of
logical consequence

In order to determine the success of the model-theoretic
characterization, we need to know what type of definition it is.
Clearly it is not intended as a lexical definition. As Tarski's
opening passage in his (1936) makes clear, a theory of logical
consequence need not yield a report of what 'logical consequence'
means. On other hand, it is clear that Tarski doesn't see himself as
offering just a stipulative definition. Tarski is not merely stating
how he proposes to use 'logical consequence' and 'logical truth' (but
see Tarski 1986) any more than Newton was just proposing how to use
certain words when he defined force in terms of mass and acceleration.
Newton was invoking a fundamental conceptual relationship in order to
improve our understanding of the physical world. Similarly, Tarski's
definition of 'logical consequence' in terms of model-theoretic
consequence is supposed to help us formulate a theory of logical
consequence that deepens our understanding of what Tarski calls the
common concept of logical consequence. Tarski thinks that the logical
consequence relation is commonly regarded as necessary, formal, and a
priori . As Tarski (1936, p. 409) says, "The concept of logical
consequence is one of those whose introduction into a field of strict
formal investigation was not a matter of arbitrary decision on the
part of this or that investigator; in defining this concept efforts
were made to adhere to the common usage of the language of everyday
life."

Let's follow this approach in Tarski's (1936) and treat the
model-theoretic definition as a theoretical definition of 'logical
consequence'. The questions raised are whether the Tarskian
model-theoretic definition of logical consequence leads to a good
theory and whether it improves our understanding of logical
consequence. In order to sketch a framework for thinking about this
question, we review the key moves in the Tarskian analysis. In what
follows, K is an arbitrary set of sentences from a language L, and X
is any sentence from L. First, Tarski observes what he takes to be the
commonly regarded features of logical consequence (necessity,
formality, and a prioricity) and makes the following claim.

(1) X is a logical consequence of K if and only if (a) it is not
possible for all the K to be true and X false, (b) this is due to the
forms of the sentences, and (c) this is known a priori.

Tarski's deep insight was to see the criteria, listed in bold, in
terms of the technical notion of truth in a structure. The key step in
his analysis is to embody the above criteria (a)-(c) in terms of the
notion of a possible interpretation of the non-logical terminology in
sentences. Substituting for what is in bold in (1) we get

(2) X is a logical consequence of K if and only if there is no
possible interpretation of the non-logical terminology of the language
according to which all the sentences in K are true and X is false.

The third step of the Tarskian analysis of logical consequence is to
use the technical notion of truth in a structure or model to capture
the idea of a possible interpretation. That is, we understand there is
no possible interpretation of the non-logical terminology of the
language according to which all of the sentences in K are true and X
is false in terms of: Every model of K is a model of X, that is, K ⊨
X.

To elaborate, as reflected in (2), the analysis turns on a selection
of terms as logical constants. This is represented model-theoretically
by allowing the interpretation of the non-logical terminology to
change from one structure to another, and by making the interpretation
of the logical constants invariant across the class of structures.
Then, relative to a set of terms treated as logical, the Tarskian,
model-theoretic analysis is committed to

(3) X is a logical consequence of K if and only if K ⊨ X.

and

(4) X is a logical truth, that is, it is logically impossible for
X to be false, if and only if ⊨ X.

As a theoretical definition, we expect the ⊨-relation to reflect the
essential features of the common concept of logical consequence. By
Tarski's lights, the ⊨-consequence relation should be necessary,
formal, and a priori. Note that model theory by itself does not
provide the means for drawing a boundary between the logical and the
non-logical. Indeed, its use presupposes that a list of logical terms
is in hand. For example, taking Sister and Female to be logical
constants, the consequence relation from (A) 'Sister(kelly, paige)' to
(B) 'Female(kelly)' is necessary, formal and a priori. So perhaps (B)
should be a logical consequence of (A). The fact that (B) is not a
model-theoretic consequence of (A) is due to the fact that the
interpretation of the two predicates can vary from one structure to
another. To remedy this we could make the interpretation of the two
predicates invariant so that '∀x(∃y Sister(x, y) → Female(x))' is true
in all structures, and, therefore if (A) is true in a structure, (B)
is too. The point here is that the use of models to capture the
logical consequence relation requires a prior choice of what terms to
treat as logical. This is, in turn, reflected in the identification of
the terms whose interpretation is constant from one structure to
another.

So in assessing the success of the Tarskian model-theoretic definition
of logical consequence for a language L, two issues arise. First, does
the model-theoretic consequence relation reflect the salient features
of the common concept of logical consequence? Second, is the boundary
in L between logical and non-logical terms correctly drawn? In other
words: what in L qualifies as a logical constant? Both questions are
motivated by the adequacy criteria for theoretical definitions of
logical consequence. They are central questions in the philosophy of
logic and their significance is at least partly due to the prevalent
use of model theory in logic to represent logical consequence in a
variety of languages. In what follows, I sketch some responses to the
two questions that draw on contemporary work in philosophy of logic. I
begin with the first question.
b. Does the model-theoretic consequence relation reflect the salient
features of the common concept of logical consequence?

The ⊨-consequence relation is formal. Also, a brief inspection of the
above justifications that K ⊨ X obtain for given K and X reveals that
the ⊨-consequence relation is a priori. Does the ⊨-consequence
relation capture the modal element in the common concept of logical
consequence? There are critics who argue that the model-theoretic
account lacks the conceptual resources to rule out the possibility of
there being logically possible situations in which sentences in K are
true and X is false but no structure U such that U ⊨ K and not U ⊨ X.
Kneale (1961) is an early critic, and Etchemendy (1988, 1999) offers a
sustained and multi-faceted attack. We follow Etchemendy. Consider the
following three sentences.

(1) (Female(shannon) & ~Married(shannon, matt))
(2) (~Female(matt) & Married(beth, matt))
(3) ~Female(beth)

(3) is neither a logical nor a model-theoretic consequence of (1) and
(2). However, in order for a structure to make (1) and (2) true but
not (3) its domain must have at least three elements. If the world
contained, say, just two things, then there would be no such structure
and (3) would be a model-theoretic consequence of (1) and (2). But in
this scenario, (3) would not be a logical consequence of (1) and (2)
because it would still be logically possible for the world to be
larger and in such a possible situation (1) and (2) can be interpreted
true and (3) false. The problem raised for the model-theoretic account
of logical consequence is that we do not think that the class of
logically possible situations varies under different assumptions as to
the cardinality of the world's elements. But the class of structures
surely does since they are composed of worldly elements. This is a
tricky criticism. Let's look at it from a slightly different
vantagepoint.

We might think that the extension of the logical consequence relation
for an interpreted language such as our language M about the McKeons
is necessary. For example, it can't be the case that for some K and X,
even though X isn't a logical consequence of a set K of sentences, X
could be. So, on the supposition that the world contains less, the
extension of the logical consequence relation should not expand.
However, the extension of the model-theoretic consequence does expand.
For example, (3) is not, in fact, a model-theoretic consequence of (1)
and (2), but it would be if there were just two things. This is
evidence that the model-theoretic characterization has failed to
capture the modal notion inherent in the common concept of logical
consequence.

In defense of Tarski (see Ray 1999 and Sher 1991 for defenses of the
Tarskian analysis against Etchemendy), one might question the force of
the criticism because it rests on the supposition that it is possible
for there to be just finitely many things. How could there be just two
things? Indeed, if we countenance an infinite totality of necessary
existents such as abstract objects (e.g., pure sets), then the class
of structures will be fixed relative to an infinite collection of
necessary existents, and the above criticism that turns on it being
possible that there are just n things for finite n doesn't go through
(for discussion see McGee 1999). One could reply that while it is
metaphysically impossible for there to be merely finitely many things
it is nevertheless logically possible and this is relevant to the
modal notion in the concept of logical consequence. This reply
requires the existence of primitive, basic intuitions regarding the
logical possibility of there being just finitely many things. However,
intuitions about possible cardinalities of worldly individuals—not
informed by mathematics and science—tend to run stale. Consequently,
it is hard to debate this reply: one either has the needed logical
intuitions, or not.

What is clear is that our knowledge of what is a model-theoretic
consequence of what in a given L depends on our knowledge of the class
of L-structures. Since such structures are furniture of the world, our
knowledge of the model-theoretic consequence relation is grounded on
knowledge of substantive facts about the world. Even if such knowledge
is a priori, it is far from obvious that our a priori knowledge of the
logical consequence relation is so substantive. One might argue that
knowledge of what follows from what shouldn't turn on worldly matters
of fact, even if they are necessary and a priori (see the discussion
of the locked room metaphor in Logical Consequence, Philosophical
Considerations: Section 2.2.1). If correct, this is a strike against
the model-theoretic definition. However, this standard logical
positivist line has been recently challenged by those who see logic
penetrated and permeated by metaphysics (e.g., Putnam 1971, Almog
1989, Sher 1991, Williamson 1999). We illustrate the insight behind
the challenge with a simple example. Consider the following two
sentences.

(4) ∃x(Female(x) & Sister(x, evan))
(5) ∃x Female(x)

(5) is a logical consequence of (4), that is, there is no domain for
the quantifiers and no interpretation of the predicates and the
individual constant in that domain which makes (4) true and not (5).
Why? Because on any interpretation of the non-logical terminology, (4)
is true just in case the intersection of the set of objects that
satisfy Female(x) and the set of objects that satisfy Sister(x, evan)
is non-empty. If this obtains, then the set of objects that satisfy
Female(x) is non-empty and this makes (5) true. The basic metaphysical
truth underlying the reasoning here is that for any two sets, if their
intersection is non-empty, then neither set is the empty set. This
necessary and a priori truth about the world, in particular about its
set-theoretic part, is an essential reason why (5) follows from (4).
This approach, reflected in the model-theoretic consequence relation
(see Sher 1996), can lead to an intriguing view of the formality of
logical consequence reminiscent of the pre-Wittgensteinian views of
Russell and Frege. Following the above, the consequence relation from
(4) to (5) is formal because the metaphysical truth on which it turns
describes a formal (structural) feature of the world. In other words:
it is not possible for (4) to be true and (5) false because

For any extensions of P, P', if an object α satisfies '(P(v) &
P'(v, n))', then α satisfies 'P(v)'.

According to this vision of the formality of logical consequence, the
consequence relation between (4) and (5) is formal because what is in
bold expresses a formal feature of reality. Russell writes that,
"Logic, I should maintain, must no more admit a unicorn than zoology
can; for logic is concerned with the real world just as truly as
zoology, though with its more abstract and general features" (Russell
1919, p. 169). If we take the abstract and general features of the
world to be its formal features, then Russell's remark captures the
view of logic that emerges from anchoring the necessity, formality and
a priority of logical consequence in the formal features of the world.
The question arises as to what counts as a formal feature of the
world. If we say that all set-theoretic truths depict formal features
of the world, including claims about how many sets there are, then
this would seem to justify making

∃x∃y~(x = y)

(that is, there are at least two individuals) a logical truth since it
is necessary, a priori, and a formal truth. To reflect
model-theoretically that such sentences, which consist just of logical
terminology, are logical truths we would require that the domain of a
structure simply be the collection of the world's individuals. See
Sher (1991) for an elaboration and defense of this view of the
formality of logical truth and consequence. See Shapiro (1993) for
further discussion and criticism of the project of grounding our
logical knowledge on primitive intuitions of logical possibility
instead of on our knowledge of metaphysical truths.

Part of the difficulty in reaching a consensus with respect to whether
or not the model-theoretic consequence relation reflects the salient
features of the common concept of logical consequence is that
philosophers and logicians differ over what the features of the common
concept are. Some offer accounts of the logical consequence relation
according to which it is not a priori (e.g., see Koslow 1999, Sher
1991 and see Hanson 1997 for criticism of Sher) or deny that it even
need be strongly necessary (Smiley 1995, 2000, section 6). Here we
illustrate with a quick example.

Given that we know that a McKeon only admires those who are older
(that is, we know that (a) ∀x∀y(Admires(x, y) → OlderThan(y, x))),
wouldn't we take (7) to be a logical consequences of (6)?

(6) Admires(paige, kelly)
(7) OlderThan(kelly, paige)

A Tarskian response is that (7) is not a consequence of (6) alone, but
of (6) plus (a). So in thinking that (7) follows from (6), one assumes
(a). A counter suggestion is to say that (7) is a logical consequence
of (6) for if (6) is true, then
necessarily-relative-to-the-truth-of-(a) (7) is true. The modal notion
here is a weakened sense of necessity: necessity relative to the truth
of a collection of sentences, which in this case is composed of (a).
Since (a) is not a priori, neither is the consequence relation between
(6) and (7). The motive here seems to be that this conception of
modality is inherent in the notion of logical consequence that drives
deductive inference in science, law, and other fields outside of the
logic classroom. This supposes that a theory of logical consequence
must not only account for the features of the intuitive concept of
logical consequence but also reflect the intuitively correct deductive
inferences. After all, the logical consequence relation is the
foundation of deductive inference: it is not correct to deductively
infer B from A unless B is a logical consequence of A. Referring to
our example, in a conversation where (a) is a truth that is understood
and accepted by the conversants, the inference from (6) to (7) seems
legit. Hence, this should be supported by an accompanying concept of
logical consequence. This idea of construing the common concept of
logical consequence in part by the lights of basic intuitions about
correct inferences is reflected in the Relevance logician's objection
to the Tarskian account. The Relevance logician claims that X is not a
logical consequence of K unless K is relevant to X. For example,
consider the following pairs of sentences.

(1) (Female(evan) & ~Female(evan)) (1) Admires(kelly, paige)
(2) Admires(kelly, shannon) (2) (Female(evan) v ~Female(evan))

In the first pair, (1) is logically false, and in the second, (2) is a
logical truth. Hence it isn't possible for (1) to be true and (2)
false. Since this seems to be formally determined and a priori, for
each pair (2) is a logical consequence of (1) according to Tarski.
Against this Anderson and Belnap write, "the fancy that relevance is
irrelevant to validity [that is logical consequence] strikes us as
ludicrous, and we therefore make an attempt to explicate the notion of
relevance of A to B" (Anderson and Belnap 1975, pp. 17-18). The
typical support for the relevance conception of logical consequence
draws on intuitions regarding correct inference, e.g. it is
counterintuitive to think that it is correct to infer (2) from (1) in
either pair for what does being a female have to do with who one
admires? Would you think it correct to infer, say, that Admires(kelly,
shannon) on the basis of (Female(evan) & ~Female(evan))? For further
discussion of the different types of relevance logic and more on the
relevant philosophical issues see Haack (1978, pp. 198-203) and Read
(1995, pp. 54-63). The bibliography in Haack (1996, pp. 264-265) is
helpful. For further discussion on relevance logic, see Logical
Consequence, Deductive-Theoretic Conceptions: Section 5.2.1.

Our question is, does the model-theoretic consequence relation reflect
the essential features of the common concept of logical consequence?
Our discussion illustrates at least two things. First, it isn't
obvious that the model-theoretic definition of logical consequence
reflects the Tarskian portrayal of the common concept. One option, not
discussed above, is to deny that the model-theoretic definition is a
theoretical definition and argue for its utility simply on the basis
that it is extensionally equivalent with the common concept (see
Shapiro 1998). Our discussion also illustrates that Tarski's
identification of the essential features of logical consequence is
disputed. One reaction, not discussed above, is to question the
presupposition of the debate and take a more pluralist approach to the
common concept of logical consequence. On this line, it is not so much
that the common concept of logical consequence is vague as it is
ambiguous. At minimum, to say that a sentence X is a logical
consequence of a set K of sentences is to say that X is true in every
circumstance (that is logically possible situation) in which the
sentences in K are true. "Different disambiguations of this notion
arise from taking different extensions of the term 'circumstance' "
(Restall 2002, p. 427). If we disambiguate the relevant notion of
'circumstance' by the lights of Tarski, 'Admires(kelly, paige)' is a
logical consequence of '(Female(evan) & ~Female(evan))'. If we follow
the Relevance logician, then not. There is no fact of the matter about
whether or not the first sentence is a logical consequence of the
second independent of such a disambiguation.
c. What is a logical constant?

We turn to the second, related issue of what qualifies as a logical
constant. Tarski (1936, 418-419) writes,

No objective grounds are known to me which permit us to draw a
sharp boundary between [logical and non-logical terms]. It seems
possible to include among logical terms some which are usually
regarded by logicians as extra-logical without running into
consequences which stand in sharp contrast to ordinary usage.

And at the end of his (1936), he tells us that the fluctuation in the
common usage of the concept of consequence would be accurately
reflected in a relative concept of logical consequence, that is a
relative concept "which must, on each occasion, be related to a
definite, although in greater or less degree arbitrary, division of
terms into logical and extra logical" (p. 420). Unlike the relativity
described in the previous paragraph, which speaks to the features of
the concept of logical consequence, the relativity contemplated by
Tarski concerns the selection of logical constants. Tarski's
observations of the common concept do not yield a sharp boundary
between logical and non-logical terms. It seems that the sentential
connectives and the quantifiers of our language M about the McKeons
qualify as logical if any terms of M do. We've also followed many
logicians and included the identity predicate as logical. (See Quine
1986 for considerations against treating '=' as a logical constant.)
But why not include other predicates such as 'OlderThan'?

(1) OlderThan(kelly, paige) (3) ~OlderThan(kelly, kelly)
(2) ~OlderThan(paige, kelly)

Then the consequence relation from (1) to (2) is necessary, formal,
and a priori and the truth of (3) is necessary, formal and also a
priori. If treating 'OlderThan' as a logical constant does not do
violence to our intuitions about the features of the common concept of
logical consequence and truth, then it is hard to see why we should
forbid such a treatment. By the lights of the relative concept of
logical consequence, there is no fact of the matter about whether (2)
is a logical consequence of (1) since it is relative to the selection
of 'OlderThan' as a logical constant. On the other hand, Tarski hints
that even by the lights of the relative concept there is something
wrong in thinking that B follows from A and B only relative to taking
'and' as a logical constant. Rather, B follows from A and B we might
say absolutely since 'and' should be on everybody's list of logical
constants. But why do 'and' and the other sentential connectives,
along with the identity predicate and the quantifiers have more of a
claim to logical constancy than, say, 'OlderThan'? Tarski (1936)
offers no criteria of logical constancy that help answer this
question.

On the contemporary scene, there are three general approaches to the
issue of what qualifies as a logical constant. One approach is to
argue for an inherent property (or properties) of logical constancy
that some expressions have and others lack. For example, topic
neutrality is one feature traditionally thought to essentially
characterize logical constants. The sentential connectives, the
identity predicate, and the quantifiers seem topic neutral: they seem
applicable to discourse on any topic. The predicates other than
identity such as 'OlderThan' do not appear to be topic neutral, at
least as standardly interpreted, e.g., 'OlderThan' has no application
in the domain of natural numbers. One way of making the concept of
topic neutrality precise is to follow Tarski's suggestion in his
(1986) that the logical notions expressed in a language L are those
notions that are invariant under all one-one transformations of the
domain of discourse onto itself. A one-one transformation of the
domain of discourse onto itself is a one-one function whose domain and
range coincide with the domain of discourse. And a one-one function is
a function that always assigns different values to different objects
in its domain (that is, for all x and y in the domain of f, if f(x) =
f(y), then x = y).

Consider 'Olderthan'. By Tarski's lights, the notion expressed by the
predicate is its extension, that is the set of ordered pairs <d, d'>
such that d is older than d'. Recall that the extension is:

{<Beth, Matt>, <Beth, Shannon>, <Beth, Kelly>, <Beth, Paige>,
<Beth, Evan>, <Matt, Shannon>, <Matt, Kelly>, <Matt, Paige>, <Matt,
Evan>, <Shannon, Kelly>, <Shannon, Paige>, <Shannon, Evan>, <Kelly,
Paige>, <Kelly, Evan>, <Paige, Evan>}.

If 'OlderThan' is a logical constant its extension (the notion it
expresses) should be invariant under every one-one transformation of
the domain of discourse (that is the set of McKeons) onto itself. A
set is invariant under a one-one transformation f when the set is
carried onto itself by the transformation. For example, the extension
of 'Female' is invariant under f when for every d, d is a female if
and only if f(d) is. 'OlderThan' is invariant under f when <d, d'> is
in the extension of 'OlderThan' if and only if <f(d), f(d')> is.
Clearly, the extensions of the Female predicate and the Olderthan
relation are not invariant under every one-one transformation. For
example, Beth is older than Matt, but f(Beth) is not older than
f(Matt) when f(Beth) = Evan and f(Matt) = Paige. Compare the identity
relation: it is invariant under every one-one transformation of the
domain of McKeons because it holds for each and every McKeon. The
invariance condition makes precise the concept of topic neutrality.
Any expression whose extension is altered by a one-one transformation
must discriminate among elements of the domain, making the relevant
notions topic-specific. The invariance condition can be extended in a
straightforward way to the quantifiers and sentential connectives (see
McCarthy 1981 and McGee 1997). Here I illustrate with the existential
quantifier. Let Ψ be a well-formed formula with 'x' as its only free
variable. '∃x Ψ' has a truth-value in the intended structure UM for
our language M about the McKeons. Let f be an arbitrary one-one
transformation of the domain D of McKeons onto itself. The function f
determines an interpretation I' for Ψ in the range D' of f. The
existential quantifier satisfies the invariance requirement for UM ⊨∃x
Ψ if and only if U ⊨∃x Ψ for every U derived by a one-one
transformation f of the domain D of UM (we say that the U's are
isomorphic with UM ).

For example, consider the following existential quantification.

∃x Female(x)

This is true in the intended structure for our language M about the
McKeons (that is, UM ⊨∃x Female(x)[g∅]) ultimately because the set of
elements that satisfy 'Female(x)' on some variable assignment that
extends g∅ is non-empty (recall that Beth, Shannon, Kelly, and Paige
are females). The cardinality of the set of McKeons that satisfy an
M-formula is invariant under every one-one transformation of the
domain of McKeons onto itself. Hence, for every U isomorphic with UM,
the set of elements from DU that satisfy 'Female(x)' on some variable
assignment that extends g∅ is non-empty and so

U ⊨∃x Female(x)[g∅].

Speaking to the other part of the invariance requirement given at the
end of the previous paragraph, clearly for every U isomorphic with UM,
if U ⊨∃x Female(x)[g∅], then UM ⊨∃x Female(x)[g∅] (since U is
isomorphic with itself). Crudely, the topic neutrality of the
existential quantifier is confirmed by the fact that it is invariant
under all one-one transformations of the domain of discourse onto
itself.

Key here is that the cardinality of the subset of the domain D that
satisfies an L-formula under an interpretation is invariant under
every one-one transformation of D onto itself. For example, if at
least two elements from D satisfy a formula on an interpretation of
it, then at least two elements from D' satisfy the formula under the
I' induced by f. This makes not only 'All' and 'Some' topic neutral,
but also any cardinality quantifier such as 'Most', 'Finitely many',
'Few', 'At least two', etc. The view suggested in Tarski (1986, p.
149) is that the logic of a language L is the science of all notions
expressible in L which are invariant under one-one transformations of
L's domain of discourse. For further discussion, defense of, and
extensions of the Tarskian invariance requirement on logical
constancy, in addition to McCarthy (1981) and McGee (1997), see Sher
(1989, 1991).

A second approach to what qualifies as a logical constant is not to
make topic neutrality a necessary condition for logical constancy.
This undercuts at least some of the significance of the invariance
requirement. Instead of thinking that there is an inherent property of
logical constancy, we allow the choice of logical constants to depend,
at least in part, on the needs at hand, as long as the resulting
consequence relation reflects the essential features of the intuitive,
pre-theoretic concept of logical consequence. I take this view to be
very close to the one that we are left with by default in Tarski
(1936). The approach is suggested in Prior (1976) and developed in
related but different ways in Hanson (1996) and Warmbrod (1999). It
amounts to regarding logic in a strict sense and loose sense. Logic in
the strict sense is the science of what follows from what relative to
topic neutral expressions, and logic in the loose sense is the study
of what follows from what relative to both topic neutral expressions
and those topic centered expressions of interest that yield a
consequence relation possessing the salient features of the common
concept.

Finally, a third approach the issue of what makes an expression a
logical constant is simply to reject the view of logical consequence
as a formal consequence relation, thereby nullifying the need to
distinguish logical terminology in the first place (see Etchemendy
1983 and Bencivenga 1999). We just say, for example, that X is a
logical consequence of a set K of sentences if the supposition that
all of the K are true and X false violates the meaning of component
terminology. Hence, 'Female(kelly)' is a logical consequence of
'Sister(kelly, paige)' simply because the supposition otherwise
violates the meaning of the predicates. Whether or not 'Female' and
'Sister' are logical terms doesn't come into play.
6. Conclusion

Using the first-order language M as the context for our inquiry, we
have discussed the model-theoretic conception of the conditions that
must be met in order for a sentence to be a logical consequence of
others. This theoretical characterization is motivated by a distinct
development of the common concept of logical consequence. The issue of
the nature of logical consequence, which intersects with other areas
of philosophy, is still a matter of debate. Any full coverage of the
topic would involve study of the logical consequence relation between
sentence from other types of languages such as modal languages
(containing necessity and possibility operators) (see Hughes and
Cresswell 1996) and second-order languages (containing variables that
range over properties) (see Shapiro 1991). See also the entries,
Logical Consequence, Philosophical Considerations, and Logical
Consequence, Deductive-Theoretic Conceptions, in the encyclopedia.
7. References and Further Reading

* Almog, J. (1989): "Logic and the World", pp. 43-65 in Themes
From Kaplan, ed. J. Almog, J. Perry, and H. Wettstein. New York:
Oxford University Press.
* Anderson, A.R., and N. Belnap (1975): Entailment: The Logic of
Relevance and Necessity. Princeton: Princeton University Press.
* Bencivenga, E. (1999): "What is Logic About?", pp. 5-19 in Varzi (1999).
* Etchemendy, J. (1983): "The Doctrine of Logic as Form",
Linguistics and Philosophy 6, pp. 319-334.
* Etchemendy, J. (1988): "Tarski on truth and logical
consequence", Journal of Symbolic Logic 53, pp. 51-79.
* Etchemendy, J. (1999): The Concept of Logical Consequence.
Stanford: CSLI Publications.
* Haack, S. (1978): Philosophy of Logics. Cambridge: Cambridge
University Press.
* Haack, S. (1996): Deviant Logic, Fuzzy Logic. Chicago: The
University of Chicago Press.
* Hanson, W. (1997): "The Concept of Logical Consequence", The
Philosophical Review 106, pp. 365-409.
* Hughes, G. E. and M.J Cresswell (1996): A New Introduction to
Modal Logic. London: Routledge.
* Kneale, W. (1961): "Universality and Necessity", British Journal
for the Philosophy of Science 12, pp. 89-102.
* Kneale, W. and M. Kneale (1986): The Development of Logic.
Oxford: Clarendon Press.
* Koslow, A. (1999): "The Implicational Nature of Logic: A
Structuralist Account", pp. 111-155 in Varzi (1999).
* McCarthy, T. (1981): "The Idea of a Logical Constant", Journal
of Philosophy 78, pp. 499-523.
* McCarthy, T. (1998): "Logical Constants", pp. 599-603 in
Routledge Encyclopedia of Philosophy, vol. 5, ed. E. Craig. London:
Routledge.
* McGee, V. (1999): "Two Problems with Tarski's Theory of
Consequence", Proceedings of the Aristotelean Society 92, pp. 273-292.
* Priest. G. (1995): "Etchemendy and Logical Consequence",
Canadian Journal of Philosophy 25, pp. 283-292.
* Prior, A. (1976): "What is Logic?", pp. 122-129 in Papers in
Logic and Ethics ed. P.T. Geach and A. Kenny. Amherst: University of
Massachusetts Press.
* Putnam, H. (1971): Philosophy of Logic. New York: Harper & Row.
* Quine, W.V. (1986): Philosophy of Logic, 2nd ed. Cambridge:
Harvard University Press.
* Ray, G. (1996): "Logical Consequence: A Defense of Tarski",
Journal of Philosophical Logic 25, pp. 617-677.
* Read, S. (1995): Thinking About Logic. Oxford: Oxford University Press.
* Restall, G. (2002): "Carnap's Tolerance, Meaning, And Logical
Pluralism", Journal of Philosophy 99, pp. 426-443.
* Russell, B. (1919): Introduction to Mathematical Philosophy.
London: Routledge, 1993 printing.
* Shapiro, S. (1991): Foundations without Foundationalism: A Case
For Second-order Logic. Oxford: Clarendon Press.
* Shapiro, S. (1993): "Modality and Ontology", Mind 102, pp. 455-481.
* Shapiro, S. (1998): "Logical Consequence: "Models and Modality",
pp. 131-156 in The Philosophy of Mathematics Today, ed. Matthias
Schirn. Oxford, Clarendon Press.
* Sher, G. (1989): "A Conception of Tarskian Logic", Pacific
Philosophical Quarterly 70, pp. 341-368.
* Sher, G. (1991): The Bounds of Logic: A Generalized Viewpoint.
Cambridge, Mass: MIT Press.
* Sher, G. (1996): "Did Tarski Commit 'Tarski's Fallacy'?" Journal
of Symbolic Logic 61, pp. 653-686.
* Sher, G. (1999): "Is Logic a Theory of the Obvious?", pp.207-238
in Varzi (1999).
* Smiley, T. (1995): "A Tale of Two Tortoises", Mind 104, pp. 725-36.
* Smiley, T. (1998): "Consequence, Conceptions of", pp. 599-603 in
Routledge Encyclopedia of Philosophy, vol. 2, ed. E. Craig. London:
Routledge.
* Tarski, A. (1933): "Pojecie prawdy w jezykach nauk
dedukeycyjnych", translated as "On the Concept of Truth in Formalized
Languages", pp. 152-278 in Tarski (1983).
* Tarski, A. (1936): "On the Concept of Logical Consequence", pp.
409-420 in Tarski (1983).
* Tarski, A. (1983): Logic, Semantics, Metamathematics 2nd ed.
Indianapolis: Hackett Publishing.
* Tarski, A. (1986): "What are Logical Notions?" History and
Philosophy of Logic 7, pp. 143-154.
* Varzi, A., ed. (1999): European Review of Philosophy, vol. 4,
The Nature of Logic. Stanford: CSLI Publications.
* Warbrod, K., (1999): "Logical Constants", Mind 108, pp. 503-538.