Logical Consequence

For a given language, a sentence is said to be a logical consequence
of a set of sentences, if and only if, in virtue of logic alone, the
sentence must be true if every sentence in the set were to be true.
This corresponds to the ordinary notion of a sentence "logically
following" from others. Logicians have attempted to make the ordinary
concept more precise relative to a given language L by sketching a
deductive system for L, or by formalizing the intended semantics for
L. Any adequate precise characterization of logical consequence must
reflect its salient features such as those highlighted by Alfred
Tarski: (1) that the logical consequence relation is formal, that is,
depends on the forms of the sentences involved, (2) that the relation
is a priori, that is, it is possible to determine whether or not it
holds without appeal to sense-experience, and (3) that the relation
has a modal element.

1. Introduction

Logical consequence is arguably the central concept of logic. The
primary aim of logic is to tell us what follows logically from what.
In order to simplify matters we take the logical consequence relation
to hold for sentences rather than for abstract propositions, facts,
state of affairs, etc. Correspondingly, logical consequence is a
relation between a given class of sentences and the sentences that
logically follow. One sentence 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 the sentences in the set to be all true without the
other sentence being true as well. If sentence X is a logical
consequence of a set of sentences K, then we may say that K implies or
entails X, or that one may correctly infer the truth of X from the
truth of the sentences in K. For example, Kelly is not at work is a
logical consequence of Kelly is not both at home and at work and Kelly
is at home. However, the sentence Kelly is not a football fan does not
follow from All West High School students are football fans and Kelly
is not a West High School student. The central question to be
investigated here is: What conditions must be met in order for a
sentence to be a logical consequence of others?

One popular answer derives from the work of Alfred Tarski, one of the
preeminent logicians of the twentieth century, in his famous 1936
paper, "The Concept of Logical Consequence." Here Tarski uses his
observations of the salient features of what he calls the common
concept of logical consequence to guide his theoretical development of
it. Accordingly, we begin by examining the common concept focusing on
Tarski's observations of the criteria by which we intuitively judge
what follows from what and which Tarski thinks must be reflected in
any theory of logical consequence. Then two theoretical definitions of
logical consequence are introduced: the model theoretic and the
deductive theoretic definitions. They represent two major approaches
to making the common concept of logical consequence more precise. The
article concludes by highlighting considerations relevant to
evaluating model theoretic and deductive theoretic characterizations
of logical consequence. For more comprehensive presentations of the
two definitions of logical consequence, as well as further critical
discussion, see the entries Logical Consequence, Model-Theoretic
Conceptions and Logical Consequence, Deductive-Theoretic Conceptions.
2. The Concept of Logical Consequence
a. Tarski's characterization of the common concept of logical consequence

Tarski begins his article, "On the Concept of Logical Consequence," by
noting a challenge confronting the project of making precise the
common concept of logical consequence.

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. But these efforts have been
confronted with the difficulties which usually present themselves in
such cases. With respect to the clarity of its content the common
concept of consequence is in no way superior to other concepts of
everyday language. Its extension is not sharply bounded and its usage
fluctuates. Any attempt to bring into harmony all possible vague,
sometimes contradictory, tendencies which are connected with the use
of this concept, is certainly doomed to failure. We must reconcile
ourselves from the start to the fact that every precise definition of
this concept will show arbitrary features to a greater or less degree.
(Tarski 1936, p. 409)

Not every feature of the technical account will be reflected in the
ordinary concept, and we should not expect any clarification of the
concept to reflect each and every deployment of it in everyday
language and life. Nevertheless, despite its vagueness, Tarski
believes that there are identifiable, essential features of the common
concept of logical consequence.

…consider any class K of sentences and a sentence X which follows
from this class. From an intuitive standpoint, it can never happen
that both the class K consists of only true sentences and the sentence
X is false. Moreover, since we are concerned here with the concept of
logical, that is, formal consequence, and thus with a relation which
is to be uniquely determined by the form of the sentences between
which it holds, this relation cannot be influenced in any way by
empirical knowledge, and in particular by knowledge of the objects to
which the sentence X or the sentences of class K refer. The
consequence relation cannot be affected by replacing designations of
the objects referred to in these sentences by the designations of any
other objects. (Tarski 1936, pp. 414-415)

According to Tarski, the logical consequence relation as it is
employed by typical reasoners is (1) necessary, (2) formal, and (3)
not influenced by empirical knowledge. I now elaborate on (1)-(3) in
order to shape two preliminary characterizations of logical
consequence.
i. The logical consequence relation has a modal element

Tarski countenances an implicit modal notion in the common concept of
logical consequence. If X is a logical consequence of K, then not only
is it the case that not all of the elements of K are true and X is
false, but also this is necessarily the case. That is, X follows from
K only if it is not possible for all of the sentences in K to be true
with X false. For example, the supposition that All West High School
students are football fans and that Kelly is not a West High School
student does not rule out the possibility that Kelly is a football
fan. Hence, the sentences All West High School students are football
fans and Kelly is not a West High School student do not entail Kelly
is not a football fan, even if she, in fact, isn't a football fan.
Also, Most of Kelly's male classmates are football fans does not
entail Most of Kelly's classmates are football fans. What if the
majority of Kelly's class is composed of females who are not fond of
football?

We said above that Kelly is not both at home and at work and Kelly is
at home jointly imply Kelly is not at work. Note that it doesn't seem
possible for the first two sentences to be true and Kelly is not at
work false. But it is hard to see what this comes to without further
clarification of the relevant notion of possibility. For example,
consider the following pairs of sentences.
Kelly kissed her sister at 2:00pm.
2:00pm is not a time during which Kelly
and her sister were 100 miles apart. Kelly is a female.
Kelly is not the US President.
There is a chimp in Paige's house.
There is a primate in Paige's house. Ten is a prime number.
Ten is greater than nine.

For each pair of sentences, there is a sense in which it is not
possible for the first to be true and the second false. At the very
least an account of logical consequence must distinguish logical
possibility from other types of possibility. Should truths about
physical laws, US political history, zoology, and mathematics
constrain what we take to be possible in determining whether or not
the first sentence of each pair could logically be true with the
second sentence false? If not, then this seems to mystify logical
possibility (e.g., how could ten be a prime number?). To paraphrase
questions asked by G.E. Moore (1959, pp. 231-238), given that I know
that George W. Bush is US President and that he is not a female named
Kelly, isn't it inconsistent for me to grant the logical possibility
of the truth of Kelly is a female and the falsity of Kelly is not the
US President? Or should I ignore my present state of knowledge in
considering what is logically possible? Tarski does not derive a clear
notion of logical possibility from the common concept of logical
consequence. Perhaps there is none to be had, and we should seek the
help of a proper theoretical development in clarifying the notion of
logical possibility. Towards this end, let's turn to the other
features of logical consequence highlighted by Tarski, starting with
the formality criterion of logical consequence.
ii. The logical consequence relation is formal

Tarski observes that logical consequence is a formal consequence
relation. And he tells us that a formal consequence relation is a
consequence relation that is uniquely determined by the form of the
sentences between which it holds. Consider the following pair of
sentences

(1) Some children are both lawyers and peacemakers.
(2) Some children are peacemakers

Intuitively, (2) is a logical consequence of (1). It appears that this
fact does not turn on the subject matter of the sentences. Replace
'children', 'lawyers', and 'peacemakers' in (1) and (2) with the
variables S, M, and P to get the following.

(1′) Some S are both M and P
(2′) Some S are P

(1′) and (2′) are forms of (1) and (2), respectively. Note that there
is no interpretation of S, M, and P according to which the sentence
that results from (1′) is true and the resulting instance of (2′) is
false. Hence, (2) is a formal consequence of (1) and on each
interpretation of S, M, and P the resulting (2′) is a formal
consequence of the sentence that results from (1′) (e.g., Some clowns
are sad is a formal consequence of Some clowns are both lonely and
sad). Tarski's observation is that for any sentence X and set K of
sentences, X is a logical consequence of K only if X is a formal
consequence of K. The formality criterion of logical consequence can
work in explaining why one sentence doesn't entail another in cases
where it seems impossible for the first to be true and the second
false. For example, (3) is false and (4) is true.

(3) Ten is a prime number
(4) Ten is greater than nine

Does (4) follow from (3)? One might think that (4) does not follow
from (3) because being a prime number does not necessitate being
greater than nine. However, this does not require one to think that
ten could be a prime number and less than or equal to nine, which is
probably a good thing since it is hard to see how this is possible.
Rather, we take

(3′) a is a P
(4′) a is R than b

to be the forms of (5) and (6) and note that there are interpretations
of 'a', 'b', 'P', and 'R' according to which the first is true and the
second false (e.g. let 'a' and 'b' name the numbers two and ten,
respectively, and let 'P' mean prime number, and 'R' greater). Note
that the claim here is not that formality is sufficient for a
consequence relation to qualify as logical but only that it is a
necessary condition. I now elaborate on this last point by saying a
little more about forms of sentences (that is, sentential forms) and
formal consequence.

Distinguishing between a term of a sentence replaced with a variable
and one held constant determines a form of the sentence. In Some
children are both lawyers and peacemakers we may replace 'Some' with a
variable and treat all the other terms as constant. Then

(1") D children are both lawyers and peacemakers

is a form of (1), and each sentence generated by assigning a meaning
to D shares this form with (1). For example, the following three
sentences are instances of (1"), produced by interpreting D as 'No',
'Many', and 'Few'.

No children are both lawyers and peacemakers
Many children are both lawyers and peacemakers
Few children are both lawyers and peacemakers

Whether X is a formal consequence of K then turns on a prior selection
of terms as constant and others replaced with variables. Relative to
such a determination, X is a formal consequence of K if and only if
(iff) there is no interpretation of the variables according to which
each of the K are true and X is false. So, taking all the terms,
except for 'Some', in (1) Some children are both philosophers and
peacemakers and in (2) Some children are peacemakers as constants
makes the following forms of (1) and (2).

(1") D children are both lawyers and peacemakers
(2") D children are peacemakers

Relative to this selection, (2) is not a formal consequence of (1)
because replacing 'D' with 'No' yields a true instance of (1") and a
false instance of (2").

Consider the following pair.

(5) Kelly is female
(6) Kelly is not US President

(6) is a formal consequence of (5) relative to replacing 'Kelly' with
a variable. Given current U.S. political history, there is no
individual whose name yields a true (5) and a false (6) when it
replaces 'Kelly'. This is not, however, sufficient reason for seeing
(6) as a logical consequence of (5). There are two ways of thinking
about why, a metaphysical consideration and an epistemological one.
First the metaphysical consideration. It seems possible for (5) to be
true and (6) false. The course of U.S. political history could have
turned out differently. One might think that the current US President
could–logically–have been a female named, say, 'Sally'. Using 'Sally'
as a replacement for 'Kelly' would yield in that situation a true (5)
and a false (6). Also, it seems possible that in the future there will
be a female US President. In order for a formal consequence relation
from K to X to qualify as logical it has to be the case that it is
necessary that there is no interpretation of the variables in K and X
according to which the K-sentences are true and X is false.

The epistemological consideration is that one might think that
knowledge that X follows logically from K should not essentially
depend on being justified by experience of extra-linguistic states of
affairs. Clearly, the determination that (6) follows formally from (5)
essentially turns on empirical knowledge, specifically knowledge about
the current political situation in the US. This leads to the final
highlight of Tarski's rendition of the intuitive concept of logical
consequence: that logical consequence cannot be influenced by
empirical knowledge.
iii. The logical consequence relation is a priori

Tarski says that by virtue of being formal, knowledge that X follows
logically from K cannot be affected by knowledge of the objects that X
and the sentences of K are about. Hence, our knowledge that X is a
logical consequence of K cannot be influenced by empirical knowledge.
However, as noted above, formality by itself does not insure that the
extension of a consequence relation is not influenced by empirical
knowledge. So, let's view this alleged feature of logical consequence
as independent of formality. We characterize empirical knowledge in
two steps as follows. First, a priori knowledge is knowledge "whose
truth, given an understanding of the terms involved, is ascertainable
by a procedure which makes no reference to experience" (Hamlyn 1967,
p. 141). Empirical, or a posteriori, knowledge is knowledge that is
not a priori, that is, knowledge whose validation necessitates a
procedure that does make reference to experience. We can safely read
Tarski as saying that a consequence relation is logical only if
knowledge that something falls in its extension is a priori, that is,
only if the relation is a priori. Knowledge of physical laws, a
determinant in people's observed sizes, is not a priori and such
knowledge is required to know that there is no interpretation of k, h,
and t according to which (7) is true and (8) false.

(7) k kissed h at time t
(8) t is not a time during which k and h were 100 miles apart

So (8) cannot be a logical consequence of (7). However, my knowledge
that Kelly is not Paige's only friend follows from Kelly is taller
than Paige's only friend is a priori since I know a priori that nobody
is taller than herself.

Let's summarize and tie things together. We began by asking, for a
given language L, what conditions must be met in order for a sentence
X of L to be a logical consequence of a class K of L-sentences? Tarski
thinks that an adequate response must reflect the common concept of
logical consequence, that is, the concept as it is ordinarily
employed. By the lights of this concept, an adequate account of
logical consequence must reflect the formality and necessity of
logical consequence, and must also reflect the fact that knowledge of
what follows logically from what is a priori. Tying the criteria
together, in order to fix what follows logically from what in a given
language L, we must select a class of constants that determines a
formal consequence relation that is both necessary and known, if at
all, a priori. Such constants are called logical constants, and we say
that the logical form of a sentence is a function of the logical
constants that occur in the sentence and the pattern of the remaining
expressions. As was illustrated above, the notion of formality does
not presuppose a criterion of logical constancy. A consequence
relation based on any division between constants and terms replaced
with variables will automatically be formal with respect to the
latter.
b. Logical and non-logical terminology

Tarski's basic move from his rendition of the common concept of
logical consequence is to distinguish between logical terms and
non-logical terms and then say that X is a logical consequence of K
only if there is no possible interpretation of the non-logical terms
of the language L that makes all of the sentences in K true and X
false. The choice of the right terms as logical will reflect the modal
element in the concept of logical consequence, that is, will insure
that there is no 'possible' interpretation of the variable,
non-logical terms of the language L that makes all of the K true and X
false, and will insure that this is known a priori. Of course, we have
yet to spell out the modal notion in the concept of logical
consequence. Tarski pretty much left this underdeveloped in his
(1936). Lacking such an explanation hampers our ability to clarify the
rationale for a selection of terms to serve as the logical ones.

Traditionally, logicians have regarded sentential connectives such as
and, not, or, if…then, the quantifiers all and some, and the identity
predicate '=' as logical terms. Remarking on the boundary between
logical and non-terms, Tarski (1936, p. 419) writes the following.

Underlying this characterization of logical consequence is the
division of all terms of the language discussed into logical and
extra-logical. This division is not quite arbitrary. If, for example,
we were to include among the extra-logical signs the implication sign,
or the universal quantifier, then our definition of the concept of
consequence would lead to results which obviously contradict ordinary
usage. On the other had, no objective grounds are known to me which
permit us to draw a sharp boundary between the two groups of terms. It
seems to be possible to include among logical terms some which are
usually regarded by logicians as extra-logical without running into
consequences which stands in sharp contrast to ordinary usage.

Tarski seems right to think that the logical consequence relation
turns on the work that the logical terminology does in the relevant
sentences. It seems odd to say that Kelly is happy does not logically
follow from All are happy because the second is true and the first
false when All is replaced with Few. However, by Tarski's version of
the ordinary concept of logical consequence there is no reason not to
treat say taller than as a logical term along with not and, therefore,
no reason not to take Kelly is not taller than Paige as following
logically from Paige is taller than Kelly. Also, it seems plausible to
say that I know a priori that there is no possible interpretation of
Kelly and is mortal according to which it is necessary that Kelly is
mortal is true and Kelly is mortal is false. This makes Kelly is
mortal a logical consequence of it is necessary that Kelly is mortal.
Given that taller than and it is necessary that, along with other
terms, were not generally regarded as logical terms by logicians of
Tarski's day, the fact that they seem to be logical terms by the
common concept of logical consequence, as observed by Tarski,
highlights the question of what it takes to be a logical term. Tarski
says that future research will either justify the traditional boundary
between the logical and the non-logical or conclude that there is no
such boundary and the concept of logical consequence is a relative
concept whose extension is always relative to some selection of terms
as logical (p. 420). For further discussion of Tarski's views on
logical terminology and contemporary views see Logical Consequence,
Model-Theoretic Conceptions: Section 5.3.

How, exactly, does the terminology usually regarded by logicians as
logical work in making it the case that one sentence follows from
others? In the next two sections two distinct approaches to
understanding the nature of logical terms are sketched. Each approach
leads to a unique way of characterizing logical consequence and thus
yields a unique response to the above question.
i. The nature of logical constants explained in terms of their
semantic properties

Consider the following metaphor, borrowed from Bencivenga (1999).

The locked room metaphor

Suppose that you are locked in a dark windowless room and you know
everything about your language but nothing about the world outside. A
sentence X and a class K of sentences are presented to you. If you can
determine that X is true if all the sentences in K are, X is a logical
consequence of K.

Ignorant of US politics, I couldn't determine the truth of Kelly is
not US President solely on the basis of Kelly is a female. However,
behind such a veil of ignorance I would be able to tell that Kelly is
not US President is true if Kelly is female and Kelly is not US
President is true. How? Short answer: based on my linguistic
competence; longer answer: based on my understanding of the semantic
contribution of and to the determination of the truth conditions of a
sentence of the form P and Q. For any sentences P and Q, I know that P
and Q is true just in case P is true and Q is true. So, I know, a
priori, if P and Q is true, then Q is true. As noted by one
philosopher, "This really is remarkable since, after all, it's what
they mean, together with the facts about the non-linguistic world,
that decide whether P or Q are true" (Fodor 2000, p.12).

Taking not and and to be the only logical constants in (9) Kelly is
not both at home and at work, (10) Kelly is at home, and (11) Kelly is
not at work, we formalize the sentences as follows, letting k mean
Kelly, H mean is at home, and W mean is at work.

(9′) not-(Hk and Wk)
(10′) Hk
(11′) not-Wk

There is no interpretation of k, H, and W according to which (9′) and
(10′) are true and (11′) is false. The reason why turns on the
semantic properties of and and not, which are knowable a priori.
Suppose (9′) and (10′) are true on some interpretation of the variable
terms. Then the meaning of not in (9′) makes it the case that Hk and
Wk is false, which, by the meaning of and requires that Hk is false or
Wk is false. Given (10′), it must be that Wk is false, that is, not-Wk
is true. So, there can't be an interpretation of the variable terms
according to which (9′) and (10′) are true and (11′) is false, and, as
the above reasoning illustrates, this is due exclusively to the
semantic properties of not and and. So the reason that it is
impossible that an interpretation of k, H, and W make (9′) and (10′)
true and (11′) false is that the supposition otherwise is inconsistent
with the semantic functioning of not and and. Compare: the supposition
that there is an interpretation of k according to which k is a female
is true and k is not US President is false does not seem to violate
the semantic properties of the constant terms. If we identify the
meanings of the predicates with their extensions in all possible
worlds, then the supposition that there is a female U.S. President
does not violate the meanings of female and US President for surely it
is possible that there be a female US President. But, supposing that
(9′) and (10′) could be true with (11′) false on some interpretation
of k, H, and W, violates the semantic properties of either and or not.

In sum, our first-step characterization of logical consequence is the
following. For a given language L,

X is a logical consequence of K if and only if there is no
possible interpretation of the non-logical terminology of L according
to which all the sentence in K are true and X is false.

A possible interpretation of the non-logical terminology of the
language 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, is 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. Models have become standard tools for
characterizing the logical consequence relation, and the
characterization of logical consequence in terms of models is called
the Tarskian or model-theoretic characterization of logical
consequence. We say that X is a model-theoretic consequence of K if
and only if all models of K are models of X. This relation may be
represented as K ⊨ X. If model-theoretic consequence is adequate as a
representation of logical consequence, then it must reflect the
salient features of the common concept, which, according to Tarski
means that it must be necessary, formal and a priori.

For further discussion of this conception of logical consequence, see
the article, Logical Consequence, Model-Theoretic Conceptions.
ii. The nature of logical constants explained in terms of their
inferential properties

We now turn to a second approach to understanding logical constants.
Instead of understanding the nature of logical constants in terms of
their semantic properties as is done on the model-theoretic approach,
on the second approach we appeal to their inferential properties
conceived of in terms of principles of inference, that is, principles
justifying steps in deductions. We begin with a remark made by
Aristotle. In his study of logical consequence, Aristotle comments
that

A syllogism is discourse in which, certain things being stated,
something other than what is stated follows of necessity from their
being so. I mean by the last phrase that they produce the consequence,
and by this, that no further term is required from without in order to
make the consequence necessary. (Prior Analytics, p. 24b)

Adapting this to our X and K, we may say that X is a logical
consequence of K when the sentences of K are sufficient to produce X.
How are we to think of a sentence being produced by others? One way of
developing this is to appeal to a notion of an actual or possible
deduction. X is a deductive consequence of K if and only if there is a
deduction of X from K. In such a case, we say that X may be correctly
inferred from K or that it would be correct to conclude X from K. A
deduction is associated with a pair ; the set K of sentences is the
basis of the deduction, and X is the conclusion. A deduction from K to
X is a finite sequence S of sentences ending with X such that each
sentence in S (that is, each intermediate conclusion) is derived from
a sentence (or more) in K or from previous sentences in S in
accordance with a correct principle of inference.

For example, intuitively, the following inference seems correct.
Kelly is not both at home and at work
Kelly is at home
(therefore) Kelly is not at work

The set K of sentences above the line is the basis of the inference
and the sentence X below is the conclusion. We represent their logical
forms, again, as follows.
(9′) not-(Hk and Wk)
(10′) Hk
(therefore) (11′) not-Wk

Consider the following deduction of (11′) from (10′) and (9′).

Deduction: Assume that (12′) Wk. Then from (10′) and (12′) we may
deduce that (13′) Hk and Wk. (13′) contradicts (9′) and so (12′), our
initial assumption, must be false. We have deduced not-Wk from not-(Hk
and Wk) and Hk.

Since the deduction of not-Wk from not-(Hk and Wk) and Hk did not
depend on the interpretation of k, W, and H, the deductive relation is
formal. Furthermore, my knowledge of this is a priori because my
knowledge of the underlying principles of inference in the above
deduction is not empirical. For example, letting P and Q be any
sentences, we know a priori that P and Q may be inferred from the set
K={P, Q} of basis sentences. This principle grounds the move from
(10′) and (12′) to (13′). Also, the deduction appeals to the principle
that if we deduce a contradiction from an assumption, then we may
infer that the assumption is false. The correctness of this principle
seems to be an a priori matter. Let's look at another example of a
deduction.
(1) Some children are both lawyers and peacemakers
(therefore) (2) Some children are peacemakers

The logical forms are, again, the following.
(1′) Some S are both M and P
(therefore) (2′) Some S are P

Again, intuitively, (2′) is deducible from (1′).

Deduction: The basis tells us that at least one S–let's call this
S 'a'–is both an M and a P. Clearly, a is a P may be deduced from a is
both an M and a P. Since we've assumed that a is an S, what we derive
with respect to a we derive with respect to some S. So our derivation
of a is a P is a derivation of Some S is a P, which is our desired
conclusion.

Since the deduction is formal, we have shown not merely that (2) can
be correctly inferred from (1), but we have shown that for any
interpretation of S, M, and P it is correct to infer (2′) from (1′).

Typically, deductions leave out steps (perhaps because they are too
obvious), and they usually do not justify each and every step made in
moving towards the conclusion (again, obviousness begets brevity). The
notion of a deduction is made precise by describing a mechanism for
constructing deductions that are both transparent and rigorous (each
step is explicitly justified and no steps are omitted). This mechanism
is a deductive system (also known as a formal system or as a formal
proof calculus). A deductive system D is a collection of rules that
govern which sequences of sentences, associated with a given , are
allowed and which are not. Such a sequence is called a proof in D (or,
equivalently, a deduction in D) of X from K. The rules must be such
that whether or not a given sequence associated with qualifies as a
proof in D of X from K is decidable purely by inspection and
calculation. That is, the rules provide a purely mechanical procedure
for deciding whether a given object is a proof in D of X from K.

We say that a deductive system D is correct when for any K and X,
proofs in D of X from K correspond to intuitively valid deductions.
For example, intuitively, there are no correct principles of inference
according to which it is correct to conclude

Some animals are both mammals and reptiles

on the basis of the following two sentences.

Some animals are mammals
Some animals are reptiles

Hence, a proof in a deductive system of the former sentence from the
latter two is evidence that the deductive system is incorrect. The
point here is that a proof in D may fail to represent a deduction if D
is incorrect.

A rich variety of deductive systems have been developed for
registering deductions. Each system has its advantages and
disadvantages, which are assessed in the context of the more specific
tasks the deductive system is designed to accomplish. Historically,
the general purpose of the construction of deductive systems was to
reduce reasoning to precise mechanical rules (Hodges 1983, p. 26).
Some view a deductive system defined for a language L as a
mathematical model of actual or possible chains of correct reasoning
in L. Sundholm (1983) offers a thorough survey of three main types of
deductive systems. For a shorter, excellent introduction to the
concept of a deductive system see Henkin (1967). A deductive system is
developed in detail in the accompanying article, Logical Consequence,
Deductive-Theoretic Conceptions.

If there is a proof of X from K in deductive system D, then we may say
that X is a deductive consequence in D of K, which is sometimes
expressed as K ⊢D X. Relative to a correct deductive system D, we
characterize logical consequence in terms of deductive consequence as
follows.

X is a logical consequence of K if and only if X is a deductive
consequence in D of K, that is, there is an actual or possible proof
in D of X from K.

This is called the deductive-theoretic (or proof-theoretic)
characterization of logical consequence.
3. Model-Theoretic and Deductive-Theoretic Conceptions of Logic

We began with Tarski's observations of the common or ordinary concept
of logical consequence that we employ in daily life. According to
Tarski, if X is a logical consequence of a set of sentences, K, then,
in virtue of the logical forms of the sentences involved, if all of
the members of K are true, then X must be true, and furthermore, we
know this a priori. The formality criterion makes the logical
constants the essential determinant of the logical consequence
relation. The logical consequence relation is fixed exclusively in
terms of the nature of the logical terminology. We have highlighted
two different approaches to the nature of a logical constant: (1) in
terms of its semantic contribution to sentences in which it occurs and
(2) in terms of its inferential properties. The two approaches yield
distinct conceptions of the notion of necessity inherent in the common
concept of logical consequence, and lead to the following
characterizations of logical consequence.

(1) 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.

(2) X is a logical consequence of K if and only if X is deducible from K.

We make the notions of possible interpretation in (1) and deducibility
in (2) precise by appealing to the technical notions of model and
deductive system. This leads to the following theoretical
characterizations of logical consequence.

(1) The model-theoretic characterization of logical consequence: X
is a logical consequence of K iff all models of K are models of X.

(2) The deductive- theoretic characterization of logical
consequence: X is a logical consequence of K iff there is a deduction
in a correct deductive system of X from K.

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 both characterizations coincide. A soundness proof
establishes K ⊢D X only if K ⊨ X, and a completeness proof establishes
K ⊢D X if K ⊨ X. These proofs together establish that the two
characterizations coincide, and in such a case the deductive system D
is said to be complete and sound with respect to the model-theoretic
consequence relation defined for the relevant language L.

We said that the primary aim of logic is to tell us what follows
logically from what. These two characterizations of logical
consequence lead to two different orientations or conceptions of logic
(see Tharp 1975, p. 5).

Model-theoretic approach: Logic is a theory of possible
interpretations. For a given language the class of situations that
can–logically–be described by that language.

Deductive-theoretic approach: Logic is a theory of formal
deductive inference.

The article now concludes by highlighting three considerations
relevant to evaluating a particular deployment of the model-theoretic
or deductive-theoretic definition in defining logical consequence.
These considerations emerge from the above development of the two
theoretic definitions from the common concept of logical consequence.
4. Conclusion

The two theoretical characterizations of logical consequence do not
provide the means for drawing a boundary in a language L between
logical and non-logical terms. Indeed, their use presupposes that a
list of logical terms is in hand. Hence, in evaluating a
model-theoretic or deductive-theoretic definition of logical
consequence for a language L the issue arises whether or not the
boundary in L between logical and non-logical terms has been correctly
drawn. This requires a response to a central question in the
philosophy of logic: what qualifies as a logical constant? Tarski
gives a well-reasoned response in his (1986). (For more recent
discussion see McCarthy 1981 and 1998, Hanson 1997, and Warbrod 1999.)

A second thing to consider in evaluating a theoretical account of
logical consequence is whether or not its characterization of the
logical terminology is accurate. For example, model-theoretic and
deductive accounts of logical consequence are inadequate unless they
reflect the semantic and inferential properties of the logical terms,
respectively. So a model-theoretic account is inadequate unless it
gets right the semantic contributions of the logical terms to the
truth conditions of the sentences formed using them. For a particular
deductive system D, the question arises whether or not D's rules of
inference reflect the inferential properties of the logical terms.
(For further discussion of the semantic and inferential properties of
logical terms see Haack 1978 and 1996, Read 1995, and Quine 1986.)

A third consideration in assessing the success of a theoretical
definition of logical consequence is whether or not the definition,
relative to a selection of terms as logical, reflects the salient
features of the common concept of logical consequence. There are
criticisms of the theoretical definitions that claim that they are
incapable of reflecting the common concept of logical consequence.
Typically, such criticisms are used to question the status of the
model-theoretic and deductive-theoretic approaches to logic.

For example, there are critics who question the model-theoretic
approach to logic by arguing that any model-theoretic account lacks
the conceptual resources to reflect the notion of necessity inherent
in the common concept of logical consequence because such an account
does not rule out the possibility of there being logically possible
situations in which sentences in K are true and X is false even though
every model of K is a model of X. Kneale (1961) is an early critic,
Etchemendy (1988, 1999) offers a sustained and multi-faceted attack.
Also, it is argued that the model-theoretic approach to logic makes
knowledge of what follows from what depend on knowledge of the
existence of models, which is knowledge of worldly matters of fact.
But logical knowledge should not depend on knowledge about the
extra-linguistic world (recall the locked room metaphor in 2.2.1).
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).

The status of the deductive-theoretic approach to logic is not clear
for, as Tarski argues in his (1936), deductive-theoretic accounts are
unable to reflect the fact that, according to the common concept,
logical consequence is not compact. Relative to any deductive system
D, the ⊢D-consequence relation is compact if and only if for any
sentence X and set K of sentences, if K ⊢D X, then K' ⊢D X, where K'
is a finite subset of sentences from K. But there are intuitively
correct principles of inference according to which one may infer a
sentence X from a set K of sentences, even though it is incorrect to
infer X from any finite subset of K. This suggests that the intuitive
notion of deducibility is not completely captured by any compact
consequence relation. We need to weaken

X is a logical consequence of K if and only if there is a proof in
a correct deductive system of X from K,

given above, to

X is a logical consequence of K if there is a proof in a correct
deductive system of X from K.

In sum, the issue of the nature of logical consequence, which
intersects with other areas of philosophy, is still a matter of
debate. Tarski's analysis of the concept is not universally accepted;
philosophers and logicians differ over what the features of the common
concept are. For example, 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). The entry Logical Consequence, Model-Theoretic Conceptions
gives a model-theoretic definition of logical consequence. For a
detailed development of a deductive system see the entry Logical
Consequence, Deductive-Theoretic Conceptions. The critical discussion
in both articles deepens and extends points made in the conclusion of
this article.
5. References and Further Reading

* Almog, J. (1989): "Logic and the World", pp. 43-65 in Themes
From Kaplan, ed. J. Almog, J. Perry, J., and H. Wettstein. New York:
Oxford UP.
* Aristotle. (1941): Basic Works, ed. R. McKeon. New York: Random House.
* 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.
* Fodor, J. (2000): The Mind Doesn't Work That Way. Cambridge: The
MIT Press.
* Gabbay, D. and F. Guenthner, eds. (1983): Handbook of
Philosophical Logic, Vol 1. Dordrecht: D. Reidel Publishing Company.
* Haack, S. (1978): Philosophy of Logics . Cambridge: Cambridge
University Press.
* Haack, S. (1996): Deviant Logic, Fuzzy Logic. Chicago: The
University of Chicago Press.
* Hodges, W. (1983): "Elementary Predicate Logic", in Gabbay, D.
and F. Guenthner (1983).
* Hamlyn, D.W. (1967): "A Priori and A Posteriori", pp.105-109 in
The Encyclopedia of Philosophy, Vol. 1, ed. P. Edwards. New York:
Macmillan & The Free Press.
* Hanson, W. (1997): "The Concept of Logical Consequence", The
Philosophical Review 106, pp. 365-409.
* Henkin, L. (1967): "Formal Systems and Models of Formal
Systems", pp. 61-74 in The Encyclopedia of Philosophy, Vol. 8, ed. P.
Edwards. New York: Macmillan & The Free Press.
* Kneale, W. (1961): "Universality and Necessity", British Journal
for the Philosophy of Science 12, pp. 89-102.
* 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.
* Moore, G.E., (1959): "Certainty", pp. 227-251 in Philosophical
Papers. London: George Allen & Unwin.
* Priest. G. (1995): "Etchemendy and Logical Consequence",
Canadian Journal of Philosophy 25, pp. 283-292.
* Putnam, H. (1971): Philosophy of Logic. New York: Harper & Row.
* Quine, W.V. (1986): Philosophy of Logic, 2nd ed.. Cambridge: Harvard UP.
* Read, S. (1995): Thinking About Logic. Oxford: Oxford UP.
* 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.
* Shapiro, S. (2000): Thinking About Mathematics , Oxford: Oxford
University 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, MA: The 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.
* Sundholm, G. (1983): "Systems of Deduction", in Gabbay and
Guenthner (1983).
* 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.
* Tharp, L. (1975): "Which Logic is the Right Logic?" Synthese 31, pp. 1-21.
* Warbrod, K., (1999): "Logical Constants" Mind 108, pp. 503-538.
* Williamson, T. (1999): "Existence and Contingency", Proceedings
of the Aristotelian Society Supplementary Vol. 73, pp. 181-203.
* Varzi, A., ed. (1999): European Review of Philosophy, Vol. 4:
The Nature of Logic, Stanford: CSLI Publications.