## 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


2. 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:


3. \(x\) 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 \(H\_2O\) 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:


4. 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.



F*e*r*io*: No \(F\) is \(G\). Some \(H\) is \(G\). Therefore
some \(H\) is not \(F\).




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:


5. \(x\) 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 '\(\bullet\)' is permutation invariant if and
only if, for all \(A\), \(p(\bullet A)\) is true if and only if \(\bullet
p(A)\) 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.[1]


## 3. Mathematical Tools: Models and Proofs



Twentieth Century *technical* work on the notion of logical
consequence has centered on two different mathematical tools, proof
theory and model theory. Each of these can be seen as explicating
different aspects of the concept of logical consequence, backed by
different philosophical perspectives.


### 3.1 The model-theoretic account of logical consequence



We have characterized logical consequence as *necessary truth
preservation in virtue of form*. This idea can be explicated
formally. One can use mathematical structures to account for the range
of possibilities over which truth needs to be preserved. The formality
of logical consequence can be explicated formally by giving a special
role to the logical vocabulary, taken as constituting the forms of
sentences. Let us see how model theory attends to both these
tasks.



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*.



The exact structure of a model will depend on the kind of language at
hand (extensional/intensional, first/higher-order, etc.). A model for
an extensional first order language consists of a non-empty set which
constitutes the *domain*, and an *interpretation
function*, which assigns to each nonlogical term an extension over
the domain--any extension agreeing with its semantic type
(individual constants are assigned elements of the domain, function
symbols are assigned functions from the domain to itself, one-place
first-order predicates are assigned subsets of the domain, etc.).



The contemporary model-theoretic definition of logical consequence
traces back to Tarski (1936). It builds on the definition of *truth
in a model* given by Tarski in (1935). Tarski defines a *true
sentence in a model* recursively, by giving truth (or
satisfaction) conditions on the logical vocabulary. A conjunction, for
example, is true in a model if and only if both conjuncts are true in
that model. A universally quantified sentence \(\forall xFx\) 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. For detail on how this is accomplished, see the entry on
 Tarski's truth definitions.)
 Now we can define logical consequence as preservation of truth over
models: an argument is valid if 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.



The model-theoretic definition is one of the most successful
mathematical explications of a philosophical concept to date. It
promises to capture both the necessity of logical consequence--by
looking at truth over all models, and the formality of logical
consequence--by varying the interpretations of the nonlogical
vocabulary across models: an argument is valid no matter what the
nonlogical vocabulary means. Yet, models are just sets, which are
merely mathematical objects. How do they account for the range of
possibilities, or circumstances required? John Etchemendy (1990)
offers two perspectives for understanding models. On the
*representational approach*, each model is taken to represent a
possible world. If an argument preserves truth over models, we are
then guaranteed that it preserves truth over possible worlds, and if
we accept the identification of necessity with truth in all possible
worlds, we have the necessary truth preservation of logical
consequence. The problem with this approach is that it identifies
logical consequence with metaphysical consequence, and it gives no
account of the formality of logical consequence. On the
representational approach, there is no basis for a distinction between
the logical and the nonlogical vocabulary, and there is no explanation
of why the interpretations of the nonlogical vocabulary are maximally
varied. The second perspective on models is afforded by the
*interpretational approach*, by which each model assigns
extensions to the nonlogical vocabulary from the actual world: what
varies between models is not the world depicted but the meaning of the
terms. Here, the worry is that necessity isn't captured. For
instance, on the usual division of the vocabulary into logical and
nonlogical, identity is considered a logical term, and can be used to
form statements about the cardinality of the domain (e.g.,
''there are at least two things'') which are
true under every reinterpretation, but perhaps are not necessarily
true. On this approach, there is no basis for considering models with
domains other than the universe of what actually exists, and
specifically, there is no explanation of model theory's use of
domains of different sizes. Each approach, as described here, is
flawed with respect to our analysis of logical consequence as
necessary and formal. The interpretational approach, by looking only
at the actual world fails to account for necessity, and the
representational approach fails to account for formality (for details,
see Etchemendy 1990, Sher 1996, and Shapiro 1998, and for refinements
see Etchemendy 2008). A possible response to Etchemendy would be to
blend the representational and the interpretational perspectives,
viewing each model as representing a possible world under a
re-interpretation of the nonlogical vocabulary (Shapiro 1998, see also
Sher 1996 and Hanson 1997 for alternative responses).



One of the main challenges set by the model-theoretic definition of
logical consequence is to distinguish between the logical and the
nonlogical vocabulary. The logical vocabulary is defined in all models
by the recursive clauses (such as those mentioned above for
conjunction and the universal quantifier), and in that sense its
meaning is fixed. The choice of the logical vocabulary determines the
class of models considered when evaluating validity, and thus it
determines the class of the logically valid arguments. Now, while each
formal language is typically defined with a choice of a logical
vocabulary, one can ask for a more principled characterization of
logical vocabulary. Tarski left the question of a principled
distinction open in his 1936, and only gave the lines of a
relativistic stance, by which different choices of the logical
vocabulary may be admissible. Others have proposed criteria for
logicality, demanding that logical constants be appropriately formal,
general or topic neutral (for references and details, see the entry on
 logical constants).
 Note that a choice of the logical vocabulary is a special case of
setting constraints on the class of models to be used. It has been
suggested that the focus on criteria for the logical vocabulary misses
this point, and that more generally the question is which *semantic
constraints* should be adopted, limiting the *admissible
models* for a language (Sagi 2014a, Zinke 2017). 



Another challenge faced by the model-theoretic account is due to the
limitations of its set-theoretic basis. Recall that models are sets.
The worry is that truth-preservation over models might not guarantee
necessary truth preservation--moreover, it might not even
guarantee material truth preservation (truth preservation in the
actual world). The reason is that each model domain is a set, but the
actual world presumably contains all sets, and as a collection which
includes all sets is too ''large'' to be a set
(it constitutes a *proper class*), the actual world is not
accounted for by any model (see Shapiro 1987).



One way of dealing with this worry is to employ external means, such
as proof theory, in support of the model-theoretic definition. This is
done by Georg Kreisel in his "squeezing argument", which
we present in section 3.3. Kreisel's argument crucially depends
on the language in question having a sound and complete proof system.
Another option is to use set-theoretic reflection principles.
Generally speaking, reflection principles state that whatever is true
of the universe of sets, is already true in an initial segment thereof
(which is always a set). If reflection principles are accepted, then
at least as concerns the relevant language, one can argue that an
argument is valid if and only if there is no counter set-model (see
Kreisel 1967, Shapiro 1987, Kennedy & Vaananen
2017).



Finally, the explanation of logical consequence in terms of truth in
models is typically preferred by "Realists", who take
truth of sentences to be independent of what can be known. 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. Some,
however, view logical consequence as having an indispensable epistemic
component, having to do with the way we establish the conclusion on
the basis of the premises. "Anti-realists", who eschew
taking truth (or at least, correspondence-truth) as an explanatory
notion, will typically prefer explaining logical consequence in terms
of *proof*--to which we turn next.


### 3.2 The proof-theoretic account of logical consequence



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 systems, 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.



The proof-centered approach highlights epistemic aspects of logical
consequence. A proof does not merely attest to the validity of the
argument: it provides the steps by which we can establish this
validity. And so, if a reasoner has grounds for the premises of an
argument, and they infer the conclusion via a series of applications
of valid inference rules, they thereby obtain grounds for the
conclusion (see Prawitz 2012). One can go further and subscribe to
*inferentialism*, the view by which the meaning of expressions
is determined by their role in inference. The idea is that our use of
a linguistic expression is regulated by rules, and mastering the rules
suffices for understanding the expression. This gives us a preliminary
restriction on what semantic values of expressions can be: they cannot
make any distinctions not accounted for by the rules. One can then go
even further, and reject any kind of meaning that goes beyond the
rules--adopting the later Wittgensteinian slogan "meaning
is use". This view is favored by anti-realists about meaning,
since meaning on this view is fully explained by what is knowable.




The condition of necessity on logical consequence obtains a new
interpretation in the proof-centered approach. The condition can be
reformulated thus: in a valid argument, the truth of the conclusion
follows from the truth of the premises by necessity of thought
(Prawitz 2005). Let us parse this formulation. *Truth* is
understood *constructively*: sentences are true in virtue of
*potential evidence* for them, and the facts described by true
sentences are thus conceived as constructed in terms of potential
evidence. (Note that one can completely forgo reference to truth, and
instead speak of assertibility or acceptance of sentences.) Now, the
necessity of thought by which an argument is valid is explained by the
meaning of the terms involved, which compels us to accept the truth of
the conclusion given the truth of the premises. Meanings of
expressions, in turn, are understood through the rules governing their
use: the usual *truth conditions* give their way to *proof
conditions* of formulas containing an expression.



One can thus provide a *proof-theoretic semantics* for a
language (Schroeder-Heister 1991). When presenting his system of
natural deduction, Gentzen remarked that the introduction rules for
the logical expressions represent their "definitions," and
the elimination rules are consequences of those definitions (Gentzen
1933). For example, the introduction rule for conjunction dictates
that a conjunction \(A \amp B\) may be inferred from both
conjuncts \(A\) and \(B\), and this rule captures the
meaning of the connective. Conversely, the elimination rule for
conjunction says that from \(A \amp B\) one may infer both
\(A\) and \(B\). The universal quantifier rules tell us that
from the universally quantified claim \(\forall xFx\) we can
infer any instance \(Fa\), and we can infer \(\forall xFx\)
from the instance \(Fa\), provided that no other assumption has
been made involving the name \(a\). Under certain requirements,
one can show that the elimination rule is validated by the
introduction rule.



One of the main challenges for the proof-centered approach is that of
distinguishing between rules that are genuinely meaning-determining
and those that are not. Some rules for connectives, if added to a
system, would lead to triviality. Prior (1960) offered the following
rules for a connective "\(\tonk\)". Its introduction rule says
that from \(A\) one can infer \(A \tonk B\), and its
elimination rule says that from \(A \tonk B\) one can infer
\(B\). With the introduction of these rules, the system becomes
trivial so long as at least one thing is provable, since from any
assumption \(A\) one can derive any conclusion \(B\). Some
constraints have to posed on inference rules, and much of subsequent
literature has been concerned with these constraints (Belnap 1962,
Dummett 1991, Prawitz 1974).



To render the notions of proof and validity more systematized, Prawitz
has introduced the notion of a *canonical proof*. A sentence
might be proved in several different ways, but it is the direct, or
canonical proof that is constitutive of its meaning. A canonical proof
is a proof whose last step is an application of an introduction rule,
and its immediate subproofs are canonical (unless they have free
variables or undischarged assumptions--for details see Prawitz
2005). A canonical proof is conceived as giving direct evidence for
the sentence proved, as it establishes the truth of the sentence by
the rule constitutive of the meaning of its connectives. For more on
canonical proofs and the ways other proofs can be reduced to them, see
the entry on
 proof-theoretic semantics.



We have indicated how the condition of necessity can be interpreted in
the proof-centered approach. The condition of formality can be
accounted for as well. Note that on the present perspective as well,
there is a division of the vocabulary into logical and nonlogical.
This division can be used to define *substitutions* of an
argument. A substitution of an argument is an argument obtained from
the original one by replacing the nonlogical terms with terms of the
same syntactic category in a uniform manner. A definition of validity
that respects the condition of formality will entail that an argument
is valid if and only if all its substitutions are valid, and in the
present context, this is a requirement that there is a proof of all
its substitutions. This condition is satisfied in any proof system
where rules are given only for the logical vocabulary. Of course, in
the proof-centered approach as well, there is a question of
distinguishing the logical vocabulary (see the entry on
 logical constants).



Finally, it should be noted that a proof theoretic semantics can be
given for classical logic as well as a variety of non-classical
logics. However, due to the epistemic anti-realist attitude that lies
at the basis of the proof-centered approach, its proponents have
typically advocated
 intuitionistic logic
 (see Dummett 1991).



For more on the proof-centered perspective and on proof-theoretic
semantics, see the entry on
 proof-theoretic semantics.


### 3.3 Between models and proofs



The proof-theoretic and model-theoretic perspectives have been
considered as providing rival accounts of logical consequence.
However, one can also view "logical consequence" and
"validity" as expressing *cluster concepts*:
"A number of different, closely related notions go by those
names. They invoke matters of modality, meaning, effectiveness,
justification, rationality, and form" (Shapiro 2014). One can
also note that the division between the model-theoretic and the
proof-theoretic perspectives is a modern one, and it was only made
possible when tools for metamathematical investigations were
developed. Frege's *Begriffsschrift*, for instance, which
predates the development of those tools, is formulated as an axiomatic
proof system, but the meanings of the connectives are given via truth
conditions.



Once there are two different analyses of a relation of logical
consequence, one can ask about possible interactions, and we'll
do that next. One can also ask what general features such a relation
has independently of its analysis as proof-theoretic or
model-theoretic. One way of answering this question goes back to
 Tarski,
 who introduced the notion of consequence operations. For our
purposes, we note only some features of such operations. Let
\(Cn(X)\) be the consequences of \(X\). (One can think of
the operator \(Cn\) as deriving from a prior consequence relation
which, when taking \(X\) as 'input (or premise)' set,
tells you what follows from \(X\). But one can also see the
'process' in reverse, and a key insight is that
consequence relations and corresponding operations are, in effect,
interdefinable. See the entry on
 algebraic propositional logic
 for details.) Among some of the minimal conditions one might impose
on a consequence relation are the following two (from Tarski):


1. \(X\) is a subset of \(Cn(X)\).
2. \(Cn(Cn(X)) = Cn(X)\).



If you think of \(X\) as a set of claims, then the first
condition tells you that the consequences of a set of claims includes
the claims themselves. The second condition demands that the
consequences of \(X\) just are the consequences of the
consequences of \(X\). Both of these conditions can be motivated
from reflection on the model-theoretic and proof-theoretic approaches;
and there are other such conditions too. (For a general discussion,
see the entry on
 algebraic propositional logic.)
 But as with many foundation issues (e.g., 'what are the
essential features of consequence relations in general?'), even
such minimal conditions are contentious in philosophical logic and the
philosophy of logic. For example, some might take condition (2) to be
objectionable on the grounds that, for reasons of vagueness (or more),
important consequence relations over natural languages (however
formalized) are not generally transitive in ways reflected in (2).
(See Tennant 1994, Cobreros et al 2012, and Ripley 2013, for
philosophical motivations against transitive consequence.) But we
leave these issues for more advanced discussion.



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 often coincide, at least in extension. A proof system is
*sound* with respect to a model-theoretic semantics if every
argument that has a proof in the system is model-theoretically valid.
A proof system is *complete* with respect to a model-theoretic
semantics if every model-theoretically valid argument has a proof in
the system. While soundness is a principal condition on any proof
system worth its name, completeness cannot always be expected.
Admittedly, these definitions are biased towards the model-theoretic
perspective: the model-theoretic semantic sets the standard to what is
"sound" and "complete". Leaving terminological
issues aside, if a proof system is both sound and complete with
respect to a model-theoretic semantics (as, significantly, in the case
of first order predicate logic), then the proof system and the
model-theoretic semantics agree on which arguments are valid.



Completeness results can also support the adequacy of the
model-theoretic account, as in Kreisel's "squeezing
argument". We have noted a weakness of the model-theoretic
account: all models are sets, and so it might be that no model
represents the actual world. Kreisel has shown that if we have a proof
system that is "intuitively sound" and is complete with
respect to the model-theoretic semantics, we won't be missing
any models: every intuitively valid argument will have a
counter-model. Let \(L\) be a first order language. Let \(Val\) denote
the set of intuitively valid arguments in \(L\). Kreisel takes
intuitive validity to be preservation of truth across all structures
(whether sets or not). His analysis privileges the modal analysis of
logical consequence--but note that the weakness we are addressing
is that considering set-theoretic structures might not be enough. Let
\(V\) denote the set of model-theoretic validities in \(L\): arguments
that preserve truth over models. Let \(D\) be the set of deductively
valid arguments, by some accepted proof system for first order
logic. Now, any such proof system is "intuitively sound",
meaning that what is deductively valid by the system is intuitively
valid. This gives us \(D \subseteq Val\). And obviously, by the
definitions we've given, \(Val \subseteq V\), since an
argument that preserves truth over all structures will preserve truth
over set-structures.



By the completeness result for first order logic, we have: \(V\)
&subseteq; \(D\). Putting the three inclusions together (the
"squeeze"), we get that all three sets must be equal, and
in particular: \(V = Val\). In this way, we've proven
that if there is some structure that is a counterexample to a first
order argument, then there is a set-theoretic one.



Another arena for the interaction between the proof-theoretic and the
model-theoretic perspectives has to do with the definition of the
logical vocabulary. For example, one can hold a "moderate"
inferentialist view which defines the meanings of logical connectives
through their semantics (i.e. truth conditions) but demands that the
meaning of a connective be determined by inference rules. Carnap has
famously shown that the classical inference rules allow non-standard
interpretations of the logical expressions (Carnap 1943). Much recent
work in the field has been devoted to the exact nature and extent of
Carnap's categoricity problem (Raatikainen 2008, Murzi and
Hjortland 2009, Woods 2012, Garson 2013, Peregrin 2014, Bonnay and
Westerstahl 2016. See also the entry on
 sentence connectives in formal logic).



Finally, we should note that while model theory and proof theory are
the most prominent contenders for the explication of logical
consequence, there are alternative frameworks for formal semantics
such as
 algebraic semantics,
 game-theoretic semantics and
 dynamic semantics
 (see Wansig 2000).


## 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 two or more
premises and a single conclusion. In fact, Aristotle focuses on
arguments with exactly two premises (the major premise and the minor
premise), but nothing in his definition forbids arguments with three
or more premises. Surely, such arguments should be permitted: 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. On the other hand, on a common reading
of Aristotle's definition of syllogism, *one*-premise
arguments are ruled out--but this seems arbitrary, as even
Aristotle's own "conversion" inferences are thus
excluded. 



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 (2005) for a defence of multiple
conclusion consequence for Anti-realists; and see Beall (2011) for a
defence of certain sub-classical multiple-conclusion logics in the
service of non-classical solutions to paradox.)



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. (For the proof-theoretic picture, see Negri and von
Plato (2001).) 



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).