# Logical Consequence

*First published Fri Jan 7, 2005*

A good argument is one whose conclusions follow from its premises;
its conclusions are *consequences* of its premises. But in what
sense do conclusions *follow from* premises? What is it for a
conclusion to be a *consequence* of premises? Those questions,
in many respects, are at the heart of logic (as a philosophical
discipline). Consider the following argument:

- If we charge high fees for university, only the rich will enroll.

We charge high fees for university.

Therefore, only the rich will enroll.

There are many different things one can say about this argument, but
many agree that if we do not equivocate (if the terms mean the same
thing in the premises and the conclusion) then the argument is
*valid*, that is, the conclusion follows deductively from the
premises. This does not mean that the conclusion is true. Perhaps the
premises are not true. However, if the premises are true, then the
conclusion is also true, as a matter of logic. This entry is about the
relation between premises and conclusions in valid arguments.

Contemporary analyses of the concept of consequence — of the
*follows from* relation — take it to be both
*necessary* and *formal*, with such answers often being
explicated via *proofs* or *models* (or, in some cases,
both). Our aim in this article is to provide a brief characterisation
of some of the notions that play a central role in contemporary
accounts of logical consequence.

We should note that we only highlight a few of the
*philosophical* aspects of logical consequence, leaving out
almost all technical details, and also leaving out a large number of
philosophical debates about the topic. Our rationale for doing as much
is that one will get the technical details, and the particular
philosophical issues that motivated them, from looking at *specific
logics* — specific theories of logical consequence (e.g.,
relevant logics, substructural logics, non-monotonic logics, dynamic
logics, modal logics, theories of quantification, and so on).
(Moreover, debates about almost any feature of language —
structure versus form of sentences, propositions, context sensitivity,
meaning, even truth — are relevant to debates about logical
consequence, making an exhaustive discussion practically impossible.)
Our aim here is simply to touch on a few of the very basic issues that
are central to logical consequence.

- 1. Deductive and Inductive Consequence
- 2. Formal and Material Consequence
- 3. Proofs and Models
- 4. Premises and Conclusions
- 5. One or Many
- Bibliography
- Other Internet Resources
- Related Entries

## 1. Deductive and Inductive Consequence

Some arguments are such that the (joint) truth of the premises is
*necessarily sufficient* for the truth of the conclusions. In
the sense of *logical consequence* central to the current
tradition, such “necessary sufficiency” distinguishes
deductive validity from *inductive* validity. In inductively
valid arguments, the (joint) truth of the premises is *very likely
(but not necessarily) sufficient* for the truth of the conclusion.
An inductively valid argument is such that, as it is often put, its
premises make its conclusion more likely or more reasonable (even
though the conclusion may well be untrue given the joint truth of the
premises). The argument

- All swans observed so far have been white.

Smoothy is a swan.

Therefore, Smoothy is white.

is not deductively valid because the premises are not necessarily sufficient for the conclusion. Smoothy may well be a black swan.

Distinctions can be drawn between different inductive arguments.
Some inductive arguments seem quite reasonable, and others are less so.
There are many different ways to attempt to analyse inductive
consequence. We might consider the degree to which the premises make
the conclusion *more likely* (a probabilistic reading), or we
might check whether the most *normal* circumstances in which the
premises are true render the conclusion true as well. (This leads to
some kinds of default or non-monotonic inference.) The field of
inductive consequence is difficult and important, but we shall leave
that topic here and focus on *deductive* validity.

(See the entries on inductive logic and non-monotonic logic for more information on these topics.)

The constraint of *necessity* is not sufficient to settle the
notion of deductive validity, for the notion of *necessity* may
also be fleshed out in a number of ways. To say that a conclusion
necessarily follows from the premises is to say that the argument is
somehow *exceptionless*, but there are many different ways to
make that idea precise.

A first stab at the notion might use what we now call metaphysical
necessity. Perhaps an argument is valid if it is *(metaphysically)
impossible* for the premises to be true and the conclusion to be
untrue, valid if — holding fixed the interpretations of premises
and conclusion — in every possible world in which the premises
hold, so does the conclusion. This constraint is plausibly thought to
be a necessary condition for logical consequence (if it *could
be* that the premises are true and the conclusion isn't,
then there is no doubt that the conclusion does not follow from the
premises); however, on most accounts of logical consequence, it is not
a sufficient condition for validity. Many admit the existence of *a
posteriori* necessities, such as the claim that water is
H_{2}O. If that claim is necessary, then the argument:

*x*is water.

Therefore,*x*is H_{2}O.

is necessarily truth preserving, but it seems a long way from being
deductively valid. It was a genuine discovery that water is
H_{2}O, one that required significant empirical investigation.
While there may be genuine discoveries of valid arguments that we had
not previously recognised as such, it is another thing entirely to
think that these discoveries require empirical investigation.

An alternative line on the requisite sort of *necessity*
turns to *conceptual necessity*. On this line, the conclusion of
(3) is not a consequence of its premise given that it is not a
conceptual truth that water is H_{2}O. The concept
*water* and the concept *H _{2}O* happen to pick
out the same property, but this agreement is determined partially by
the world.

A similar picture of logic takes consequence to be a matter of what
is *analytically* true, and it is not an analytic truth that
water is H_{2}O. The word “water” and the formula
“H_{2}O” agree in extension (and necessarily so)
but they do not agree in *meaning*.

If metaphysical necessity is too coarse a notion to determine logical consequence (since it may be taken to render too many arguments deductively valid), an appeal to conceptual or analytic necessity might seem to be a better route. The trouble, as Quine argued, is that the distinction between analytic and synthetic (and similarly, conceptual and non-conceptual) truths is not as straightforward as we might have thought in the beginning of the 20th Century. (See the entry on the analytic/synthetic distinction.) Furthermore many arguments seem to be truth-preserving on the basis of analysis alone:

- Peter is Greg's mother's brother's son.

Therefore, Peter is Greg's cousin.

One can understand that the conclusion follows from the premises, on
the basis of one's understanding of the concepts involved. One
need not know anything about the identity of Peter, Greg's
cousin. Still, many have thought that (4) is not deductively valid,
despite its credentials as truth-preserving on analytic or conceptual
grounds. It is not quite as general as it could be because it is not as
*formal* as it could be. The argument succeeds only because of
the particular details of family concepts involved.

A further possibility for carving out the distinctive notion of
*necessity* grounding logical consequence is the notion of
*apriority*. Deductively valid arguments, whatever they are, can
be known to be so without recourse to experience, so they must be
knowable *a priori*. A constraint of apriority certainly seems
to rule argument (3) out as deductively valid, and rightly so. However,
it will not do to rule out argument (4). If we take arguments like (4)
to turn not on matters of deductive validity but something else, such
as an *a priori* knowable definition, then we must look
elsewhere for a characterisation of logical consequence.

## 2. Formal and Material Consequence

The strongest and most widespread proposal for finding a narrower
criterion for logical consequence is the appeal to *formality*.
The step in (4) from “Peter is Greg's mother's
brother's son” to “Peter is my cousin” is a
*material* consequence and not a formal one, because to make the
step from the premise to the conclusion we need more than the
*structure* or *form* of the claims involved: we need to
understand their *contents* too.

What could the distinction between *form* and
*content* mean? We mean to say that consequence is formal if it
depends on the *form* and not the *substance* of the
claims involved. But how is that to be understood? We will give at most
a sketch, which, again, can be filled out in a number of ways.

The obvious first step is to notice that all presentations of the
*rules* of logical consequence rely on
schemes.
Aristotle's syllogistic is a proud
example.

Ferio: NoFisG. SomeHisG. Therefore someHis notF.

Inference schemes, like the one above, display the structure of
valid arguments. Perhaps to say that an argument is formally valid is
to say that it falls under some general scheme of which every instance
is valid, such as F*e*r*io*.

That, too, is an incomplete specification of formality. The material argument (4) is an instance of:

*x*is*y*'s mother's brother's son.

Therefore,*x*is*y*'s cousin.

every instance of which is valid. We must say more to explain why
some schemes count as properly formal (and hence a sufficient ground
for logical consequence) and others do not. A general answer will
articulate the notion of
logical form,
which is an important issue in its own right (involving the notion
of
logical constants,
among other
things). Instead of exploring the details of different candidates for
logical form, we will mention different proposals about the
*point* of the exercise.

What is the point in demanding that validity be underwritten by a notion of logical form? There are at least three distinct proposals for the required notion of formality, and each provides a different kind of answer to that question.

We might take the formal rules of logic to be totally
*neutral* with respect to particular features of
*objects*. Laws of logic, on this view, must abstract away from
particular features of objects. Logic is formal in that it is totally
*general*. One way to characterise what counts as a totally
*general notion* is by way of permutations. Tarski proposed
(1986) that an operation or predicate on a domain counted as general
(or logical) if it was invariant under permutations of objects. (A
permutation of a collection of objects assigns for each object a unique
object in that collection, such that no object is assigned more than
once. A permutation of {*a*, *b*, *c*, *d*}
might, for example, assign *b* to *a*, *d* to
*b*, *c* to *c* and *a* to *d*.) A
*2*-place predicate *R* is invariant under permutation if
for any permutation *p*, whenever *Rxy* holds,
*Rp*(*x*)*p*(*y*) holds too. You can see
that the *identity* relation is permutation invariant – if
*x = y* then *p*(*x*) = *p*(*y*)
– but the *mother-of* relation is not. We may have
permutations *p* such that even though *x* is the mother
of *y*, *p*(*x*) is not the mother of
*p*(*y*). We may use permutation to characterise
logicality for more than predicates too: we may say that a one-place
sentential connective ‘•’ is permutation invariant if
and only if *p*(•*A*) is true if and only if
•*p*(*A*) is true. Defining this rigorously requires
establishing how permutations operate on sentences, and this takes us
beyond the scope of this article. Suffice to say, an operation such as
negation passes the test of invariance, but an operation such as
‘JC believes that’ fails.

A closely related analysis for formality is that formal rules are
totally *abstract*. They abstract away from the semantic
*content* of thoughts or claims, to leave only semantic
structure. The terms ‘mother’ and ‘cousin’
enter essentially into argument (5). On this view, expressions such as
propositional connectives and quantifiers do not add new semantic
content to expressions, but instead add only ways to combine and
structure semantic content. Expressions like ‘mother’ and
‘cousin’, by contrast, add new semantic content.

Another way to draw the distinction (or to perhaps to draw a
different distinction) is to take the formal rules of logic to be
*constitutitive norms* for thought, regardless of its subject
matter. It is plausible to hold that no matter what we think about, it
makes sense to conjoin, disjoin and negate our thoughts to make new
thoughts. It might also make sense to quantify. The behaviour, then, of
logical vocabulary may be used to structure and regulate *any*
kind of theory, and the norms governing logical vocabulary apply
totally universally. The norms of valid argument, on this picture, are
those norms that apply to thought irrespective of the particular
content of that thought.

(This section owes much to the work of John MacFarlane, and his
thesis
*What Does it Mean to Say that Logic is Formal?*
MacFarlane distinguished the three kinds of formality
at which we have merely waved here, and he provides a detailed
discussion of the notions, making many distinctions over which we have
passed.)

## 3. Proofs and Models

Twentieth Century *technical* work on the notion of logical
consequence has centered on two different techniques, one explaining
the notion in terms of *proofs* and the other via
*models*.

On the *proof-centered approach* to logical consequence, the
validity of an argument amounts to there being a *proof* of the
conclusions from the premises. Exactly what proofs *are* is a
big issue but the idea is fairly plain (at least if you have been
exposed to some proof system or other). Proofs are made up of small
steps, the primitive inference principles of the proof system. The 20th
Century has seen very many different kinds of proof system, from
so-called Hilbert proofs, with simple rules and complex axioms, to
natural deduction systems, with few (or even no) axioms and very many
rules. In natural deduction proofs, the rules are plausibly thought to
somehow constitute (or display) the meaning of the connectives. For
example, the rules for conjunction dictate that a conjunction *A
& B* may be inferred from both conjuncts *A* and
*B*, and conversely, from *A & B* one may infer both
*A* and *B*. The universal quantifier rules tell us that
from the universally quantified claim ∀*x**Fx*
we can infer any instance *Fa*, and we can infer
∀*x**Fx* from the instance *Fa*, provided
that no other assumption has been made involving the name
*a*.

The *model-centered approach* to logical consequence takes
the validity of an argument to be *absence of counterexample*. A
*counterexample* to an argument is, in general, some way of
manifesting the manner in which the premises of the argument
*fail* to lead to a conclusion. One way to do this is to provide
an argument of the same form for which the premises are clearly true
and the conclusion is clearly false. Another way to do this is to
provide a *circumstance* in which the premises are true and the
conclusion is false. In the contemporary literature the intuitive idea
of a counterexample is developed into a theory of *models*.
Models are abstract mathematical structures that provide possible
interpretations for each of the non-logical primitives in a formal
language. Given a model for a language one is able to define what it is
for a sentence in that language to be true (according to that model) or
not. So, the intuitive idea of logical consequence in terms of
counterexamples is then formally rendered as follows: an argument is
valid if and only if there is no model according to which the premises
are true and the conclusion is not true. Put in positive terms: in any
*model* in which the premises are true (or in any
*interpretation* of the premises according to which they are
true), the conclusion is true too. (This is Tarski's definition of
logical consequence from 1936.) Here, the behavior of the logical
vocabulary is explained by their (recursive) truth or satisfaction
conditions relative to a model. A conjunction, for example, is true in
a model if and only if both conjuncts are true in that model. A
universally quantified sentence ∀*x**Fx* is true
in a model if and only if each instance is true in the model. (Or, on
the Tarskian account of satisfaction, if and only if the open sentence
*Fx* is satisfied by every object in the domain of the model.)
The distinctive logical vocabulary is purely formal, on this picture,
because no matter what we say about the semantics of the non-logical
parts of the vocabulary, we can determine the truth (or satisfaction)
of complex formulas involving conjunction, quantifiers, etc., without
knowing anything else about the domain or model. (For detail on how
this is accomplished, see the entry on
Tarski's truth definitions.)

Just as one can ask after the philosophical import of proof systems, so too one can (and philosophers often do) ask about the philosophical import of the model-centered approach. How, for example, are we to understand the “nature” of models? How are we to understand variation of truth-values across models? John Etchemendy (1990) discusses the philosophical ramifications of taking such variation to be “re-interpretation” of (non-logical) vocabulary versus taking it to reflect variation of “possible worlds”. On one account, models simply model different possible worlds (and so, logical consequence defined by those models is a model of necessary truth preservation). On the other, models provide different interpretations of the non-logical vocabulary of our language (and so, logical consequence is not necessary truth preservation, but rather, truth preservation on the basis of the meanings of the logical vocabulary.)

The two pictures of logical consequence are quite different, and
they are used for different philosophical purposes.
“Realists” typically prefer explaining logical consequence
in terms of truth in models, and “Anti-realists” typically
prefer explaining logical consequence in terms of proof. There are
different reasons for these preferences. Explaining logical consequence
in terms of truth in models is rather close to explaining logical
consequence in terms of *truth*, and the analysis of
truth-in-a-model is sometimes taken to be an explication of truth in
terms of correspondence, a typically Realist notion. On the other hand,
explaining logical consequence in a proof-centred way seems to require
none of this. If the analysis of logical consequence starts with a
definition of proof in terms of simple inference rules, then it seems
like an attractive possibility to take these inference principles as
basic — as *definitions* of the terms involved. If this
kind of strategy is successful, then it seems that one is able to give
an account of logical consequence in terms acceptable to the
Anti-realist, who eschews taking truth (or at least,
correspondence-truth) as an explanatory notion. This approach has
proponents as different as Prawitz (1985) and Brandom (1994).

While the philosophical divide between Realists and Anti-realists
remains vast, proof-centered and model-centered accounts of consequence
have been united (at least with respect to extension) in many cases.
The great soundness and completeness theorems for different proof
systems (or, from the other angle, for different model-theoretic
semantics) show that, in an important sense, the two approaches
coincide, at least in extension. Intuitively, if soundness and
completeness have been established for a particular proof system and a
given model-centered account of consequence, then the two accounts
agree with each other: there is a proof of an argument if and only if
there is no counterexample to it. On the other hand,
*extensional* agreement does not make for the same
*notion* of consequence, and the cases in which such agreement
can be achieved comes at expressive costs. The full philosophical
significance of so-called adequacy results (soundness, completeness)
remains an open issue.

## 4. Premises and Conclusions

There has also been dissent, even in Aristotle's day, as to the “shape” of logical consequence. In particular, there is no settled consensus on the number of premises or conclusions appropriate to “tie together” the consequence relation.

In Aristotle's syllogistic, a syllogism relates a pair of
premises (the major premise and the minor premise) and a single
conclusion. No other kinds of arguments are countenanced. This is
clearly a narrowing of a wider notion of logical consequence. If, for
example, we have one syllogism from two premises *A* and
*B* to a conclusion *C*, and we have another from the
premises *C* and *D* to the conclusion *E*, then
in some sense, the longer argument from premises *A*, *B*
and *D* to conclusion *E* is a good one. It is found by
chaining together the two smaller arguments. If the two original
arguments are formally valid, then so too is the longer argument from
three premises. Aristotle's definition of syllogism also rules
out *one*-premise arguments, including his own
“conversion” inferences.

For such reasons, many have taken the relation of logical
consequence to pair an arbitrary (possibly infinite)
*collection* of premises with a single conclusion. This account
has the added virtue of having the special case of an empty collection
of premises. Arguments to a conclusion from no premises whatsoever are
those in which the conclusion is true by logic alone. Such
“conclusions” are *logical truths* (sometimes
*tautologies*) or, on the proof-centered approach,
*theorems*.

Perhaps there is a reason to allow the notion of logical consequence
to apply even more broadly. In Gentzen's proof theory for
classical logic, a notion of consequence is defined to hold between
multiple premises and multiple conclusions. The argument from a set
*X* of premises to a set *Y* of conclusions is valid if
the truth of every member of *X* guarantees (in the relevant
sense) the truth of some member of *Y*. There is no doubt that
this is formally perspicuous, but the philosophical applicability of
the multiple premise — multiple conclusion sense of logical
consequence remains an open philosophical issue. In particular, those
anti-Realists who take logical consequence to be defined in terms of
*proof* (such as Michael Dummett) reject a multiple conclusion
analysis of logical consequence. For an Anti-realist, who takes good
inference to be characterised by the way *warrant* is
transmitted from premise to conclusion, it seems that a multiple
conclusion analysis of logical consequence is out of the question. In a
multiple conclusion argument from *A* to *B*, *C*,
any warrant we have for *A* does not necessarily transmit to
*B* or *C*: the only conclusion we are warranted to draw
is the disjunction *B or C*, so it seems for an analysis of
consequence in terms of warrant we need to understand some logical
vocabulary (in this case, disjunction) in order to understand the
consequence relation. This is unacceptable if we hope to use logical
consequence as a tool to *define* that logical vocabulary. No
such problems appear to arise in a single conclusion setting. (However,
see Restall (forthcoming) for a defence of multiple conclusion
consequence for Anti-realists.)

Another line along which the notion has been broadened (or along
which some have sought to broaden it) involves recent work on
substructural logic.
The proposal
here is that we may consider doing without some of the standard rules
governing the way that premises (or conclusions) of an argument may be
combined. Structural rules deal with the *shape* or
*structure* of an argument in the sense of the way that the
premises and conclusions are collected together, and not the way that
those statements are constructed. The structural rule of
*weakening* for example, states that if an argument from some
collection of premises *X* to a conclusion *C* is valid,
then the argument from *X* together with another premise
*A* to the conclusion *C* is also valid. This rule has
seemed problematic to some (chiefly on the grounds that the extra
premise *A* need not be used in the derivation of the conclusion
*C* and hence, that *C* does not follow *from* the
premises *X,A* in the appropriate sense).
*relevant logics*
are designed to
respect this thought, and do without the structural rule of
weakening.

Other structural rules are also a called into question. Another
possible application of substructural logic is found in the analysis of
paradoxes such as
Curry's paradox.
A
crucial move in the reasoning in Curry's paradox and other paradoxes
like it seems to require the step reducing two applications of an
assumption to a single one (which is then discharged). According to
some, this step is problematic, and so, they must distinguish an
argument from *A* to *B* and an argument from *A*,
*A* to *B*. The rule of *contraction* is
rejected.

In yet other examples, the *order* in which premises are used
is important and an argument from *A*, *B* to *C*
is to be distinguished from an argument from *B*, *A* to
*C*. (For more details, consult the entry on
substructural logics.)
There is no
doubt that the formal systems of substructural logics are elegant and
interesting, but the case for the philosophical importance and
applicability of substructural logics is not closed.

## 5. One or Many?

We have touched only on a few central aspects of the notion of
logical consequence, leaving further issues, debates and, in
particular, details to emerge from particular accounts (accounts that
are well-represented in this encyclopedia). But even a quick glance at
the *related links* section (below) will attest to a fairly
large number of different logical theories, different accounts of what
(logically) follows from what. And that observation raises a question
with which we will close: Is there one notion of logical consequence
that is the target of all such theories, or are there many?

We all agree that there are many different formal techniques for
studying logical consequence, and very many different formal systems
that each propose different relations of logical consequence. But given
a particular argument, is the question as to whether it is deductively
valid an all-or-nothing affair? The orthodoxy, logical *monism*,
answers affirmatively. There is one relation of deductive consequence,
and different formal systems do a better or worse job of modelling that
relation. (See, for example, Priest 1999 for a defence of monism.) The
logical *contextualist* or *relativist* says that the
validity of an argument depends on the subject matter or the frame of
reference or some other context of evaluation. (For example, a use of
the law of the excluded middle might be valid in a classical
mathematics textbook, but not in an intuitionistic mathematics
textbook, or in a context where we reason about fiction or vague
matters.) The logical *pluralist*, on the other hand, says that
of one and the same argument, in one and the same context, there are
sometimes different things one should say with respect to its validity.
For example, perhaps one ought say that the argument from a
contradictory collection of premises to an unrelated conclusion is
*valid* in the sense that in virtue of its form it is not the
case that the premises are true an the conclusion untrue (so it is
valid in one precise sense) but that nonetheless, in another sense the
form of the argument does not ensure that the truth of the premises
*leads to* the truth of the conclusion. The monist or the
contextualist holds that in the case of the one argument a single
answer must be found for the question of its validity. The pluralist
denies this. The pluralist holds that the notion of logical consequence
itself may be made more precise in more than one way, just as the
original idea of a “good argument” bifurcates into
deductive and inductive validity (see Beall and Restall 2000 for a
defence of pluralism).

## Bibliography

### History of Logical Consequence

#### Expositions

- Coffa, J. Alberto, 1993,
*The Semantic Tradition from Kant to Carnap*, Edited by Linda Wessels, Cambridge: Cambridge University Press.- An historical account of the Kantian origins of the rise of analytic philosophy and its development from Bolzano to Carnap.

- Kneale, W. and Kneale, M., 1962
*The Development of Logic*, Oxford: Oxford University Press, reprinted 1984.- The classic text on the history of logic until the middle 20th Century.

#### Source Material

- Ewald, William, 1996,
*From Kant to Hilbert: a source book in the foundations of mathematics*Volumes I and II, Oxford: Oxford University Press.- Reprints and translations of important Texts, including Bolzano on logical consequence.

- van Heijenoort, Jean, 1967,
*From Frege to Gödel: a sourcebook in mathematical logic 1879–1931*, Cambridge, MA: Harvard University Press.- Reprints and translations of central texts in the development of logic.

- Husserl, Edmund, 1900/2001,
*Logical Investigations*Volumes 1 and 2. Translation by J. N. Findlay, Introduction by Dermot Moran, London: Routledge. - Mill, John Stuart, 1872/1973,
*A System of Logic*(8th edition) in J. M. Robson (Ed.)*Collected works of John Stuart Mill*(Volumes 7 & 8), Toronto: University of Toronto Press.

### 20th Century Developments

- Anderson, A.R., and Belnap, N.D., 1975,
*Entailment: The Logic of Relevance and Necessity*, Volume I. Princeton: Princeton University Press. - Anderson, A.R., Belnap, N.D. Jr., and Dunn, J.M., 1992,
*Entailment*, Volume II, Princeton: Princeton University Press- This book and the previous one summarise the work in relevant logic in the Anderson–Belnap tradition. Some chapters in these books have other authors, such as Robert K. Meyer and Alasdair Urquhart.

- Dummett, Michael, 1991
*The Logical Basis of Metaphysics*, Cambridge, MA: Harvard University Press.- Groundbreaking use of natural deduction proof to provide an anti-realist account of logical consequence as the central plank of a theory of meaning.

- Mancosu, Paolo, 1998,
*From Brouwer to Hilbert*, Oxford: Oxford University Press.- Reprints and translations of source material concerning the constructivist debates in the foundations of mathematics in the 1920s.

- Shoesmith D. J. and Smiley, T. J., 1978
*Multiple-Conclusion Logic*, Cambridge: Cambridge University Press.- The first full-scale exposition and defence of the notion that logical consequence relates multiple premises and multiple conclusions.

- Gentzen, Gerhard, 1969,
*The Collected Papers of Gerhard Gentzen*, edited by M. E. Szabo, Amsterdam: North Holland. - Restall, Greg, 2000,
*An Introduction to Substructural Logics*, Routledge. (Précis available online)- An introduction to the field of substructural logics.

- Tarski, Alfred, 1956,
*Logic, Semantics, Metamathematics: papers from 1923 to 1938*, Translated by J. H. Woodger, Oxford: Oxford University Press.

### Philosophy of Logical Consequence

There are many (many) other works on this topic, but the bibliographies of the following will serve as a suitable resource for exploring the field.

- Beall, JC and Restall, Greg, 2000, “Logical Pluralism,”
*Australasian Journal of Philosophy*, 78:457–493. - Brandom, Robert, 1994,
*Making It Explicit*, Cambridge, MA: Harvard University Press. [See especially Chapters 5 and 6 on the account of logical consequence according to which truth is not a fundamental explanatory notion.] - Etchemendy, John, 1990,
*The Concept of Logical Consequence*, Cambridge, MA: Harvard University Press. - Gomez-Torrente, Mario, 1996, “Tarski on Logical
Consequence,”
*Notre Dame Journal of Formal Logic*, 37:125–151. - McGee, Vann, 1992, “Two Problems with Tarski's Theory of
Consequence”,
*Proceedings of the Aristotelian Society*, 92:273–292. - Prawitz Dag, 1985, “Remarks on some approaches to the concept
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## Other Internet Resources

- MacFarlane, John, 2000,
*What Does it Mean to Say that Logic is Formal?*, Ph. D. Dissertation, University of Pittsburgh.

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