An argument in the philosophical sense is a set of sentences
consisting of (at least) one sentence stating a conclusion
and (at least) one sentence stating a premise which is or are supposed
to support the conclusion. Arguments are of central
importance to philosophy not only as a subject of systematic study, but
also methodologically as the means to criticise or support philosophical
claims and theories. More generally, arguments are an indispensable part
of any responsible rational discourse; to give an argument for a claim
is to give a reason for it and to set out this reason for oneself and
for others to scrutinize.
The analysis, development, and critique of arguments are some of the
most important tasks performed by contemporary philosophers working in
the analytic tradition. The process of formalization is an important
step in any one of these tasks since it makes arguments amenable to the
application of formal methods, such as those of model theory or of proof
theory. These methods give us precise and objective quality-criteria for
arguments, including in particular criteria for their logical
validity.
Assuming that we have identified the premises and conclusion of an
argument, its formalization will require us to make a number of choices,
including those captured by the following four interrelated
questions:
- Which kind of inferential support do the premises lend to the
conclusion of the argument?
- Into which formal language should we translate the argument’s
premises and conclusion?
- What makes such a translation into a particular formal language
adequate?
- Which formalisms can be used to evaluate the quality of the
argument?
The remainder of this introduction is structured around these four
questions about the formalization of arguments. It starts out with a
brief discussion of each of these questions in the following four
sections, briefly discussing some answers given in the literature and
providing some references for further reading. The main aim of this
introductory part of this paper is to give readers who are not familiar
with the relevant literature a partial look at the more general
discussion to which the papers collected in this special issue
contribute. This overview is neither comprehensive, nor authoritative.
The last two sections of the introduction contain some information about
the genesis of the special issue and the editor’s acknowledgements and a
brief overview of the content of the papers published in this special
issue.
Inferential Support
A standard classification of arguments individuates kinds of
arguments in terms of the kind of inferential support which its
premises lend to an argument’s conclusion. We may accordingly
distinguish between, among others, abductive, statistical, inductive,
deductive arguments and arguments from analogy. The sort of arguments we
encounter in everyday life, e.g. in discussions with neighbours and
friends or in political debates, rarely fit into just one of these
categories. Rather, they might consist, for example, of an abductive
argument for a conclusion which in turn serves as a premise among others
in a deductive argument, whose conclusion in turn is used to argue for
another claim by analogy, and so on. They may of course also involve
particular forms of reasoning which do not neatly fit into the
classificatory scheme which one finds in philosophy books, e.g. because
they draw on particular non-verbal aspects of a particular discussion,
or positively contribute to a debate in a particular context, even
though they have the form of a logical fallacy (e.g. an appeal to
authority). One might hence argue that theoretical engagement with “real
world” arguments require different, perhaps more permissive approaches
than those covered in introductory books and courses on logic and
critical thinking. Still, many such arguments, or at
least parts of them, can be broken down into smaller segments which
exemplify one of the canonical argument types.
Deductive arguments enjoy a special status in philosophy due to the
particularly strict way in which the premises of a deductive argument
supports its conclusion. Consider for example the following
argument:
If the train runs late, its passengers will miss their
connections.
The train runs late.
\(\therefore\) Its passengers
will miss their connections.
The conclusion of this argument, which in schemas of this sort will
be marked by the prefixed symbol “\(\therefore\)” throughout this text, like
that of any valid deductive argument, is logically entailed by its
premisses. But what is logical entailment? In contemporary logic, there
are two fundamental accounts of what it means for a sentence to be
logically entailed by another. The first is the syntactic account which
characterizes logical entailment proof-theoretically in terms of
derivability or provability in a logical system. Considering the formal
language of first-order logic, the core idea of this account is that a
sentence \(s\) of language is logically
entailed by a set of sentences \(\Delta\) of the same language if, and only
if, there is a proof of \(s\) which can
be constructed in a formal calculus, e.g. using the introduction- and
elimination-rules of the logical constants in case of the natural
deduction calculus, and taking at most the sentences in \(\Delta\) as hypotheses. The
second account is the semantic account, which characterizes entailment
in model-theoretic terms. Its core idea is that, focusing again on the
language of first order logic, a sentence \(s\) (i.e. a well-formed formula of that
formal language) is logically entailed by a set of sentences \(\Delta\) if, and only if, for all models
\(\mathfrak{M}\) for this language, if
all sentences in \(\Delta\) are true in
\(\mathfrak{M}\), \(s\) is true in \(\mathfrak{M}\), where a model is a
set-theoretical construction used to semantically interpret all
well-formed sentences of the language. As
is well-known, the two relations characterized by these accounts
coincide for sound and complete logics, such as classical first-order
logic, in the sense that they render exactly the same entailments valid.
The term “logical consequence” is usually reserved for the latter,
semantic notion and I will follow this convention in the remainder of
this section.
It is important to distinguish the question of the validity of an
argument from that of its soundness. An argument is sound if,
and only if, it is both valid, i.e. if its conclusion is logically
entailed by its premises, and if its premises are true. Neither the
proof-theoretic, nor the model-theoretic approach just described is
concerned with the truth of an argument’s premises. Both approaches
target the notion of validity.
The proof-theoretic characterization of deductive entailment is
intrinsically linked to particular formal systems which characterize
logical expressions like that of negation, conjunction, or the
quantifiers in terms of introduction- and elimination-rules which tell
us under which conditions we can either introduce or eliminate formulas
containing such an expression in the context of a proof. The totality of
these rules fix what is provable in such a system and a fortiori give us
the sort of syntactic characterization of logical entailment which
interests us in the current context. One important philosophical
question about introduction- and elimination-rules in a formal system
concerns the relation between the two kinds of rules. It was forcefully
raised in Prior
(1960), who argued against the idea that the meaning of logical
expressions is completely fixed by their introduction- and
elimination-rules by introducing the connective “tonk” whose associated
pair of rules permit us to derive absolutely any sentence from any
sentence. An influential idea for how the problem raised by “tonk” and
similarly problematic connectives can be avoided is that such
connectives violate a harmony-constraint which is supposed to govern the
relation between a logical expression’s introduction- and its
elimination-rules. But even if it turned out that such
a constraint can be formulated, Prior’s argument could still be taken to
show that, as Prawitz puts it, “ordinary proof theory has nothing to
offer an analysis of logical consequence” (2005, 683). A
suitable notion of harmony may give us a way of guarding a formal system
against incoherence and a fortiori allow us to accept its harmonious
introduction- and elimination-rules as constitutive of the meaning of
its logical expressions within that system. Even so, there still would
remain an explanatory gap between a formal-system-relative harmonious
notion of provability and the general, formal-system-independent notion
of logical consequence. One proposal for a way to close this gap is due
to Dummett and Prawitz, who argue that logical consequence can be
characterized using proof-theoretic means and the notion of canonical
proof (see e.g.
Dummett 1976; Prawitz 1974, 2005).
Concerning the semantic characterization, many contributors to the
recent literature have focused on two different properties which might
be used to characterize or define logical consequence, that of being
necessarily truth-preserving and that of being
formal.
That logical consequence is closely linked to necessity is a
well-established idea in analytic philosophy. In
the contemporary debate, this connection is usually spelled out in terms
of necessary truth-preservation: If a sentence \(s\) is a logical consequence of a set of
sentences \(\Gamma\), then it is
necessary that if the sentences in \(\Gamma\) are true, so is \(s\). Or, to put it differently, it is
impossible for the sentences in \(\Gamma\) to be true, but for \(s\) not to be.
The property of being necessarily truth preserving distinguishes
deductive from inductive arguments, such as the following:
Every dog which has been observed up until now likes to chase
cats.
Bella is a dog.
\(\therefore\) Bella is a dog
who likes to chase cats.
Clearly, the fact that every dog observed up until now likes to chase
cats does not guarantee that absolutely every dog, including (possibly
unobserved) Bella, likes to chase cats. The truth of the premises of
this argument, and of those of any inductive argument in general, does
not necessitate the truth of its conclusion. The
focus of this special issue and of the following parts of this
introduction is on deductive arguments.
While necessary truth preservation plausibly gives us a necessary
condition for an argument’s being deductive, i.e. for its conclusion to
be a logical consequence of its premises, there are reasons to doubt
that the notion of logical consequence can be adequately explained,
characterized, or defined in terms of this property. An important open
question in this regard is what kind of necessity the property of
necessarily preserving truth involves. The seemingly obvious claim that
it is the notion of logical necessity would lead us into an explanatory
circle, since logical necessity is plausibly explainable in terms of
logical consequence. It is furthermore not clear whether other kinds of
necessity, such as for example analyticity, a priority, or metaphysical
necessity, can serve this purpose (see Beall, Restall and Sagi 2019, sec.
1).
The second property which is much discussed in the literature on
logical consequence is the notion’s formality. Intuitively
speaking, this property distinguishes logical inferences from material
entailments such as:
The ball is red.
\(\therefore\) The ball is
coloured.
Or:
Some dog sees some cat.
\(\therefore\) Some cat is seen
by some dog.
While these arguments reflect intuitively correct inferences, their
conclusions are not logical consequences of their premises. This is
because the entailments from (7) to (8) and from (9) to (10) obtain due
to the material content of these sentences, i.e. due to what the
sentences are about, not due to their form: That (8) is entailed by (7)
is guaranteed by the meanings of “is red” and of “is coloured” and that
(10) is entailed by (9) is guaranteed by the meanings of “sees” and “is
seen by.”
The validity of a deductive argument in contrast depends solely on
the logical form of its premises and conclusion.
The logical form of a sentence in turn is determined by the logical
expressions it contains and the way they combine with the contained
non-logical expressions. That deductive logic is formal in this sense is
uncontroversial, but it is hard to say what “formal” means without just
defining it ostensively by referring to examples of sentences which we
assume to share the same logical form. Can we define the notion of
formality in other terms, giving us a systematic criterion to
distinguish between the logical and the non-logical expressions of a
language? There are several answers to this question two of which will
now be briefly introduced. Before this is done,
it should be noted that while the focus in the current section is on the
notion of logical consequence, most of the discussion of formality
focuses on the use of this notion to distinguish logical from
non-logical expressions of languages.
There is a direct connection between these two loci of formality, since
the logical expressions in a sentence determine its logical form and it
is in turn the logical form of sentences which ensure that they stand in
the relation of logical consequence.
One approach to formality proposed in the literature says that
formality can be understood in terms of topic neutrality (see e.g. Ryle
1954, 115ff; Haack 1978, 5–6). The idea is that logical
entailments hold irrespective of what the entailed and the entailing
sentences are about. What distinguishes the logical expressions of a
language is that they, unlike predicates like “is red” and “is coloured”
or individual constants, are not about any thing in particular, but that
their meaning is rather tied to certain schematic patterns of
application which are universally applicable. This criterion for
formality gives us a simple and plausible explanation of why the
entailment from (7) to (8) is not formal and thus not logical. The main
problem noted even by those like Haack who are sympathetic to it is that
topic neutrality only gives us a vague criterion for demarcating logical
from non-logical expressions: Why could we for example not count the
inference from (9) to (10) as formal? After all, it might appear that we
can extract a schematic pattern of the following form from this
entailment:
\(x\) \(\Phi\)s \(y\).
\(\therefore\) \(y\) is \(\Phi\)ed by \(x\).
Putting complications about surface grammar aside which the schema
ignores (e.g. “sees” and “is seen by”), one may on the one hand
argue against its formality by pointing out that the correctness of the
inference seems to depend on the seemingly material fact that “\(\Phi\)s” and “is \(\Phi\)ed by” are converse relations. On the
other hand, one might argue that the two converses are really identical
(see Williamson
1985) and then claim that (11) and (12) are just the same
sentence in different guises. After stripping away these guises, the
inference would really just be a trivial inference from one sentence to
itself, instantiating an inference schema which holds irrespective of
what the sentence involved means. The point here is of course only that
as a criterion for logicality, topic neutrality leaves room for
disagreement about particular cases, giving us at best a vague account
of what formality is.
The second account of formality is provided by Tarski’s classical
permutation-invariance-based characterization of logicality (see 1986). This
account could be seen as a way to make the topic-neutrality-based
account of formality more precise. Its core idea is that the
distinguishing feature of logical expressions is that their meaning is
invariant under all permutations of the domain of objects of a model. A
model in the model-theoretic sense is a set-theoretical
construction based on a domain of objects which is designed to enable us
to semantically interpret sentences of a formal language in
set-theoretic terms with respect to that domain. A permutation of
the domain of a model is a function which maps each object in that
domain to a unique object from the same domain. Within a model,
first-order predicates can e.g. be interpreted as sets of objects and
first-order relational predicates accordingly as sets of tuples of
objects. Logical expressions are also given a set-theoretic
interpretation, so that first-order quantifiers can e.g. be interpreted
in terms of relations between predicates, i.e. sets of tuples of sets of
objects. The sets corresponding to material predicates in a model, such
as e.g. the relational predicate “is larger than” in a model which is
used to interpret a fragment of natural language involving the
predicate, vary under at least some permutations of a model’s domain.
There will e.g. be a permutation which maps two objects a and
b which stand in this relation to other objects from the domain
which do not (e.g. simply to b and a, respectively).
The idea underlying Tarski’s characterization is that no such thing can
happen to logical expressions; the logical expressions retain their
intended meaning in a model, no matter under which permutation of the
objects in the model’s domain we consider them.
One of the main questions about the notion of logical consequence is
how the precise, model-theoretic notion relates to the intuitive,
pre-theoretical notion of logical entailment with which we operate in
ordinary reasoning. The idea that the former can be extracted from
natural language, and in particular Glanzberg’s recent critique of this
idea, are discussed in Gil Sagi’s contribution to the special issue.
That there is an explanatory gap to be filled here has already been
pointed out by Tarski, who writes that
the concept of following is not distinguished from other concepts of
everyday language by a clearer content or more precisely delineated
denotation […] and one has to reconcile oneself in advance to the fact
that every precise definition of the concept […] will to a greater or
lesser degree bear the mark of arbitrariness. (2002, 176)
An influential contribution to the debate about logical consequence
which takes this question as its starting point is Etchemendy (1990). Roughly,
Etchemendy argues that Tarski’s model-theoretic definition of logical
consequence fails to capture the intuitive notion of logical
consequence, since it presupposes certain contingent, non-logical
assumptions about the cardinality of the universe, putting the notion
defined by Tarski at odds with the necessity of the intuitive notion.
There are different formal methods which one can apply to evaluate
the logical validity of an argument. One may for example rely on
semantic methods, such as those provided by a model theoretic semantics,
or on syntactical methods, such as the one provided by the natural
deduction calculus. In order to apply such formal
methods to systematically assess the quality of an argument, the
premises and conclusions of arguments have to be translated from the
natural language in which they are stated into a suitable formal
language. The process of translating a sentence of a natural language
into a formal language is the process of formalizing in the narrow
sense, as opposed to the wider sense which pertains to whole
arguments.
Besides this central technical reason, there are further reasons for
formalizing arguments. One important reason is that given a suitable
formal language, formalizing an argument forces us to clarify, in
different respects, its premises and conclusion. One respect of
clarification concerns the many ambiguities present in natural language.
Formal languages are often explicitly constructed to be unambiguous, so
that each sentence (or formula, if one prefers) of the language is
assigned a single, precise meaning. A well-worn example are ambiguous
natural language sentences involving quantifier phrases such as “Every
child gets a present.” Translating the sentence into the formal language
of first-order logic, we are forced to decide between two unambiguous
readings of the sentence (that every child gets its own present(s) or
that every child gets the same present(s)) by the variable-binding
structure of the quantifiers of the formal language. Dutilh-Novaes (2012,
ch. 4 and 7), furthermore argues that there is another respect in
which formalization helps us clarify the formalized parts of language,
namely that formal languages serve to eliminate certain cognitive
biases.
From the perspective of logic, formal languages are first and
foremost mathematical objects. More specifically,
they are identified with sets of formulas, where a formula is a sequence
of symbols which is generated from a set of symbols, the formal
language’s alphabet, based on a set of syntactic rules which give us a
recipe for generating all well-formed formulas of the respective
language. The resulting formal language is of course still devoid of
meaning, as it merely gives us an alphabet of symbols and rules for
constructing certain sequences of them. To interpret the language, a
semantics which defines meanings for all well-formed formulas of the
language is needed. The standard approach is to identify these meanings
with truth-values, reflecting the idea that semantics is about true or
false representation of an underlying structure which the sentences of a
language reflect or fail to reflect. But there is also an inferentialist
tradition which aims to characterize meaning in terms of the inferential
rules which govern the expressions of the language.
Formal languages and their semantic interpretations are legion, but
what constrains our choice of a formal language when formalizing an
argument? This section will focus on one rather important constraint,
namely the expressive strength of the formal language. General
philosophical constraints about the notions involved in an argument one
wants to formalize or pragmatic or sociological constraints tied to
certain context will hence not be discussed.
The notion of expressive strength is a semantic notion which concerns
not only an uninterpreted formal language, but rather a pairing of such
a language with a suitable semantics. It seems that, at least in some
cases, there is a notable asymmetry in the relation between the language
and the semantics when it comes to determining expressive strength: We
cannot extend the expressive strength of some language beyond a certain
threshold set by the expressions it contains by coupling it with a
different semantics. An example is the language of propositional logic
which simply lacks the syntactic expressions needed to capture the inner
logical structure of atomic formulas which grounds the felicity of
certain inferences which come out as valid in classical first-order
logic. One could try to compensate for the lack of syntactic structure
by adopting a particular translation scheme and by encoding the validity
of the logically invalid inferences in the semantics. E.g. if the
predicate “\(F\)” stands for “is a dog”
and “\(G\)” for “is an animal,” then
the valid first-order inference from “\(\forall x (Fx \rightarrow Gx)\)” and “\(\exists x Fx\)” to “\(\exists x Gx\)” could be simulated in the
language of propositional logic by assigning a propositional constant to
the English sentences “All dogs are animals,” “There is a dog,” and
“There is an animal” and by building it into one’s semantics of the
language of propositional logic that the two first entail the third. But
there are obvious limits to this strategy, since it e.g. makes the
semantics depend on a particular translation-schema from a natural into
the formal language and since it would make it a matter of stipulation
which propositional constants express logical truths or stand in
relations of logical entailment.
In order to allow us to adequately formalize an argument, the formal
language (together with a suitable semantic interpretation), has to be
able to capture enough of the logical structure of the argument as
stated in a natural language to make it an argument, i.e. a collection
of sentences one of which stands in a relation of inferential support to
the others. Intensional logic offers a wealth of examples which
highlight expressive limitations of certain formal languages. A
classical example from tense logic concerns the formalization of the
sentence (see e.g.
Cresswell 1990, 18):
- One day all persons now alive will be dead.
In the language of a simple tense logic which extends the language of
first-order logic with the sentential tense-operators \(\mathbf{P}\) (“It was the case that…”) and
\(\mathbf{F}\) (“It will be the case
that…”), if one uses the predicates \(A,D\) for “… is alive” and “… is dead”
respectively, the closest one can get to an adequate formalization of
(13) is:
- \(\mathbf{F} \forall x (Ax\rightarrow
Dx)\)
Since this formula says that it will be the case at a future time
that everyone alive at that time is dead at that time, this translation
is clearly inadequate. There are different ways to remedy this lack of
expressive strength. One is to add a sentential “now”-operator \(\mathbf{N}\) and to introduce a
double-indexed semantics for the language which allows one to evaluate
formulas relative to not one but two time indices, one of which is
specifies the time of evaluation. Figuratively speaking,
\(\mathbf{N}\)’s semantic contribution
to a formula is to force the evaluation of the formula in its scope at
the time of evaluation. So in
- \(\mathbf{F} \forall x
(\mathbf{N}Ax\rightarrow Dx)\)
\(\mathbf{N}\)’s job is to exempt
the atomic formula \(Ax\) from being
evaluated at the future time index introduced by \(\mathbf{F}\) and to force its evaluation at
the time index representing the time of evaluation, i.e. present time
from the perspective of someone evaluating the formula. The result is an
adequate formalization of (13) which could e.g. be used in the
formalization an argument involving (15) as a premise.
Interestingly, (13) can also be expressed without temporal operators,
if we instead allow the quantifiers of the language to range over times,
relativize predications to times, so that “\(Axt\)” and “\(Dxt\)” stand for “\(x\) is alive at time \(t\)” and “\(x\) is dead at time \(t\)” respectively, and take \(t_0\) to stand for the time of evaluation
(Cresswell 1990,
19):
- \(\exists t_1 (t_0 < t_1 \land \forall
x (A(xt_0)\rightarrow D(xt_1)))\)
This formula seems to adequately capture what (13) says relative to a
particular time of evaluation. Note that, as Cresswell (1990, 21) points out,
it might be argued to be objectionable that (16) produces an eternal
sentence for each value of \(t_0\). At
least it is, if we assume that the truth-value of (13) could change, if
e.g. technological advances would allow humans to attain
immortality.
The availability of (16) as a translation of (13) raises the question
of whether it wouldn’t be preferable to just work with the language of
first-order logic rather than with the extended language of first-order
tense logic which adds new operators. Considerations of parsimony
certainly seem to favour this strategy. Why introduce additional
operators if we can express the same things without them? Philosophical
reasons may be brought to bear on this question. Arthur Prior for
example argued that the tense logical formalization of (13) is
preferable, considerations of parsimony notwithstanding, since he took
tense, which is more naturally expressed using operators like \(\mathbf{F}\), \(\mathbf{P}\), and \(\mathbf{N}\), to be more fundamental than
time.
Questions about the choice of formal language are discussed in Hanoch
Ben-Yami and, with a historical focus on Frege’s
Begriffsschrift, in Jongool Kim’s contributions to the special
issue.
Translation Problems and a
Simple Quality Constraint
Assuming that a suitable formal language has been selected,
determining the logical form of a natural language sentence is still not
a straightforward matter. It seems clear that not every formula of such
a language can equally well be used to translate every natural language
sentence. But what then makes a formula or a set of formulas an adequate
or a correct formalization? Can we formulate general criteria for the
quality or admissibility for formalizations of a formal language?
A minimal constraint on the correctness of formalization of sentences
is that it should respect certain intuitively valid inferences involving
these sentences. In this subsection, the focus will be on two well-known
examples of problem cases for translations of natural language sentences
into the language of first-order logic which illustrate two different
attempts to ensure that this minimal constraint is met.
The first problem specifically concerns a particular type of
sentence, namely that of action sentences. Consider the following
sentence:
- Donald embraced Orman at noon.
The most-straightforward translation of this sentence into the
language of first-order logic is
- \(Edon\)
where \(Exyz\) is the three-place
predicate “\(x\) embraces \(y\) at time \(z\)” and \(d\), \(o\), \(n\)
are individual constants designating Donald, Orman, and the relevant
point in time respectively. The problem with this formalization of the
sentence is that it does not respect the inferential relation between
(17) and the following sentence:
- Donald embraced Orman.
Clearly, if Donald embraced Orman at noon, Donald embraced Orman.
Yet, if we translate (19) in the same straightforward manner as (17),
using a two-place predicate \(Fxy\)
which stands for a sentence of the form “\(x\) embraces \(y\),” we get the following formula:
- \(Fdo\)
But this formula is not logically entailed, in classical first-order
logic, by (18). A classic discussion of this problem is found in Davidson (1967).
Building on previous work by Reichenbach and Kenny, Davidson’s solution
to the problem is to propose an alternative formalization-pattern for
sequences describing events. According to his proposal, (17) should be
formalized as:
- \(\exists x (Gxdo \land
Hxn)\),
Here the predicate \(Gxyz\) stands
for “\(x\) is an embrace by \(y\) of \(z\),” the predicate \(Hxy\) for “\(x\) happened at time \(y\),” and the constants \(d\), \(o\), \(n\)
retain their earlier referents. This new formula directly entails the
formula
- \(\exists x Gxdo\)
which, following Davidson’s formalization pattern, is an adequate
formalization of (19). The problem is hence solved.
Davidson’s proposal gives us an example of a formalization pattern
which is sensitive to the content of the formalized sentence. As
Davidson put it: “Part of what we must learn when we learn the meaning
of any predicate is how many places it has, and what sorts of entities
the variables that hold these places range over. Some predicates have an
event-place, some do not” (1967, 93). Given the previous
discussion about the distinction between formal and material inferences,
one might think that Davidson’s proposal blurs the line between the two
kinds of inferences, if such a line can at all be drawn. One might
indeed think that both the example discussed by Davidson and the example
to be discussed next illustrate that it is, even in the case of
first-order logic, a genuinely open question to which extent formal
logic can account for the informal notion of entailment, including
ostensibly material entailments such as those from (7) to (8) and from
(9) to (10).
The second example illustrates a problem case of formalization which
arises even if one accepts external constraints on formalization. A
classical example discussed in the literature is De Morgan’s problem:
All horses are animals.
\(\therefore\) All heads of
horses are heads of animals.
There is a straightforward way to formalize (23) by simply
translating “is a horse” using the predicate-letter \(F\) and “is an animal” using the predicate
letter \(G\):
- \(\forall x (Fx \rightarrow
Gx)\)
If we formalize (24) in the same manner using the predicate-letter
\(H\) for “is a head of a horse” and
\(I\) for “is the head of an animal,”
we end up with:
- \(\forall x (Hx \rightarrow
Ix)\)
If we just consider (24) in isolation, this is may be a fine
formalization, but (26) is inadequate in the context of a formalization
of the argument from (23) to (24). The inference captured in this
argument is intuitively correct, but (25) does not logically entail (26)
.
There are different formalizations of (24) which solve the problem
(cf. Brun 2004,
193). One solution is to formalize (24) as follows, using the
binary predicate \(K\) to translate “is
the head of” in addition to \(F\) and
\(G\) which are still used to translate
“is a horse” and “is an animal” respectively:
- \(\forall x \forall y ((Fy \land Kxy )
\rightarrow (Gy \land Kxy ))\)
Alternatively, the following formula also does the trick:
- \(\forall x (\exists y(Fy \land Kxy)
\rightarrow \exists y(Gy \land Kxy ))\)
Both (27) and (28) are logical consequences of (25), so both (25) and
(27), as well as (25) and (28) give us formalizations of the argument
from (23) to (24) which can be said to meet the minimal requirement set
out earlier in this section. Interestingly however, (27) is logically
stronger than (28) in the sense that (28) is a logical consequence of
(27), but (27) not of (28). The fact that we can have two different, but
non-equivalent ways of formalizing the argument from (23) to (24) raises
several general questions about the formalization of arguments (cf. Brun 2004, 194).
We might for example ask whether the two variants can be compared
concerning their quality as formalizations of the natural language
argument they translate, and if so, which one of them offers us the
better formalization.
The discussion of the two classical formalization problems illustrate
two important general aspect of how we determine the correctness of a
formalization. The first and quite obvious point is that the intuitive
notion of inference we apply when reasoning using natural language gives
us a corrective for correct formalization. The correctness of a
formalization can never be a completely formal matter; i.e. logic alone
can never tell us whether a formula is a correct formalization of a
sentence. Second, whether a formula of a
formal language is an adequate formalization of a natural language
sentence cannot be determined by considering the sentence in isolation.
Correctness rather is a holistic notion which has to take relevant
inferential patterns in natural language into account. (Cf. Friedrich
Reinmuth’s contribution to this special issue.)
These two points give us constraints on adequate formalization, but
they obviously fall short of giving us general criteria for the
adequateness of formalizations which might, e.g. answer the mentioned
questions about the comparative quality of equally admissible
alternative formalizations.
General Quality Criteria
What shape could such a general criterion take? Brun distinguishes
two kinds of quality criteria, correctness criteria and
adequacy criteria (see 2004, 11). In his terminology, a
formalization is correct if its validity-relevant features are
just those of the sentence or of the argument which it formalizes. But
there is a fundamental problem for formalizing arguments which shows
that correctness alone is not enough to guarantee that a formalization
is a good formalization. Following Blau (1977), this problem has come to be
known as the problem of unscrupulous formalization.
To see the problem, consider the following example given in Brun (2004, 238):
Every prime number is odd or equal to 2.
There is no prime number which is not odd and not equal to
2.
These two sentences can arguably be recognized to say the same
without thinking much about their logical form, e.g. by pondering the
meanings of “every” and “there is no.” Let us, for the sake of the
argument, assume that we accept on an intuitive level that (29) and (30)
are equivalent. Using “\(P\)” for “is a
prime number” and “\(O\)” for “is an
odd number,” a scrupulous formalization of the two sentences would give
us the two following formulas:
\(\forall x (Px \rightarrow (Ox \lor
x=2))\)
\(\neg \exists x (Px \land (\neg Ox
\land \neg x=2))\)
Given these translations, we could now provide a formal explanation
of our informal judgement that (29) and (30) are equivalent by proving
that the two formulas are equivalent in first-order logic. An
unscrupulous formalization in contrast would for example be one which
translates both (29) and (30) as (31). The goal of our exercise in
formalization is to show that we can confirm our informal judgement that
(29) and (30) are equivalent and there is no easier equivalence proof
than one which demonstrates that a formula, trivially, but correctly, is
equivalent to itself. The point of the example is that if correctness is
all that matters, then there the unscrupulous formalization is as good
as the scrupulous one.
The example of unscrupulous formalizations shows that correctness
alone is not a guarantee of the quality of a formalization. This is
where adequacy enters the picture. Adequacy is a stricter
quality-criterion than correctness, that is, each adequate formalization
is a correct formalization, but not vice versa. The notion of adequacy
hence allows us to rule out correct, but still problematic
formalizations of the sort just discussed. Unscrupulous formalization
give us a clear adequacy-constraint: Adequate formalizations do not
trivialize non-trivial inferential connections between the resulting
formulas, ruling out e.g. a formalization which translates both (29) and
(30) as (31). Accordingly, adequacy criteria go beyond correctness
criteria in the sense that they ensure that the formalization not only
captures the validity-relevant features of the formalized sentences or
argument, but also does so in a non-trivial way.
There are, just as in case of the notion of logical entailment, two
different conceptions of correctness which are tied to two conceptions
of what validity-relevant features are. First, these features can be the
truth-conditions of the relevant sentences and formulas, giving us a
semantic conception of correctness. The idea then is that a
formalization is correct if the formalization has the same
truth-conditions as the sentence it formalizes relative to a logic and a
translation-schema (or correspondence schema in Brun’s terms) which
specifies the translations of all relevant expressions of natural
language into the relevant formal language.
The validity-relevant features can however also be inferential
features, giving us a syntactic conception of correctness. For
arguments, the formalization and the formalized argument as stated in
natural language have to have the same inferential structure, whereas
for the formalization of a single sentence, the formalization is correct
if the formally correct inferences in which it can occur are also valid
in an informal sense for the corresponding inferences made in natural
language.
The minimal constraint mentioned in the previous subsection hence
concerns the second, the inferential, notion of correctness. Sainsbury
discusses the following adequacy criterion for formalizations of English
sentences:
QC1.
A formalization is adequate only if each of its logical constants
is matched by a single English expression making the same contribution
to truth conditions. (Sainsbury 2001, 352)
This proposal is motivated by Sainsbury’s discussion of what he calls
the “Tractarian vision,” that every entailment is a logical entailment.
Friends of this idea might be tempted to ensure that material
entailments are really logical entailments by putting more structure
into the formalizations than the surface form of the sentences requires.
They might for example try to ensure that the argument from (7) (“The
ball is red”) to (8) (“The ball is coloured”) counts as logically valid
by formalizing its premise and conclusion as follows:
\(Rb \land Cb\)
\(Cb\)
A problem with this sort of translation and, more generally, with the
Tractarian vision is that it appears to conflate the two distinct
projects of analysing the meaning of a sentence and of isolating its
logical form. The motivation for formalizing (7)
as (33) has to draw on the semantic fact that to say that an object is
red is, implicitly, to say that it is coloured. To ensure that the
entailment is logical, the proposed formalization hence draws on a fact
about the meaning of the non-logical expressions involved in (7). So
while the formalization of the argument works on the formal level, it
indirectly violates the formality requirement: The formality of the
logical entailment between (33) and (34) is not mirrored by the premise
and conclusion of the argument as stated in English. Sainsbury’s
adequacy criterion QC1 systematically blocks ad hoc logicalizations of
arguments of this sort.
A drawback of QC1 is that it also threatens Davidson’s proposed
formalization schema for action sequences: There is arguably no single
English expression in “Donald embraced Orman at noon” which makes the
same contribution to the sentences’s truth conditions as the existential
quantifier in its formalization (21) does with respect to that formula
of first-order logic.
Purists who eschew the content sensitivity of Davidson’s
formalization pattern might see this as an advantage rather than a
drawback, but Brun argues that QC1 suffers from two further problems
which are less specific and more severe (see Brun 2004, 253f). First, it
presupposes an explanation of what it means for a natural language
expression to match or correspond to a logical constant in a formula of
the formal language into which one translates. Second, putting the first
problem aside, while QC1 rules out some problematic formalizations, such
as (33), it likewise rules out uncontroversial formalizations, including
in particular:
Müller is sad, Schmidt is happy.
\(Sm \land Hs\)
Crocodiles are green.
\(\forall x (Cx \rightarrow
Gx)\)
Hans owns a red bicycle.
\(\exists x (Bx \land Rx \land
Ohx)\)
The comma in (35) can hardly be said to make the same contribution to
its truth-conditions as the conjunction in (36) and the same can be said
about the quantifier and the material conditional in (38) and the
existential quantifier, as well as the two conjunctions in (40). QC1
helps rooting out some inadequate formalizations, but it throws the baby
out with the bathwater by classifying a range of standard formalizations
as inadequate.
There are however better adequacy criteria than QC1, such as the
following, (a simplified version of) Brun’s criterion of less precise
formalization which gives us a necessary condition for the adequacy of a
formalization:
QC2.
For a formula \(\phi\) to be a
correct formalization of a sentence \(A\), every formula \(\psi\) which is less precise than \(\phi\) has to be such that there is a
correct formalization of \(A\) which is
a notational variant of \(\psi\).
This principle needs a bit of unpacking.
First of all, “less precise” is here understood to be a relation which
holds between two formulas \(\phi\) and
\(\psi\) relative to a formalism
(i.e. a logic), which are formalizations of the same sentence and which
are such that \(\psi\) can be generated
from \(\phi\) by substituting a
logically more complex formula for a sub-formula of \(\phi\). Of two such formulas, one is less
precise than the other if the former gives us a less detailed picture of
the logical structure of the sentence. Consider for example the
following sentence:
- Paul Otto Alfred is an adopted son.
Letting the constant \(a\) stand for
the name “Paul Otto Alfred” and the predicate \(P\) for “is an adopted son,” we can
formalize (41) as:
- \(Pa\)
However, we could also use the two predicates \(Q\) and \(R\), standing for “is adopted” and “is a
son” to formalize (41) as:
- \(Qa \land Ra\)
Or we could still be more precise and formalize (41) as follows using
the predicate \(S\) to translate “is
male” and \(T\) to translate “is the
father of”:
- \(Qa \land Sa \land \exists x
(Txa)\)
(42)–(44) are all formalizations of the same sentence, namely (41);
furthermore, each of the three formulas can be generated by substitution
from the others; finally, the three formulas are
increasingly precise, revealing more and more of the formalized
sentence’s logical structure.
QC2 also involves the notion of a notational variant. This notion can
be understood in terms of substitution: A formula \(\phi\) is a notational variant of a formula
\(\psi\) if, and only if, \(\phi\) can be transformed into \(\psi\) by a one-to-one substitution of
non-logical predicates and vice versa (see Brun (2004), 301).
Now how does QC2 work? We can think of a logically complex
formalization as the result of a step-by-step procedure which starts
with an atomic formula and then begins capturing more of the formalized
sentence’s logical structure by analyzing it in terms of more complex
formulas which all are correct in the semantic sense of having the right
truth-conditions. What QC2 tells us is basically that to be an adequate
formalization is to only contain logical complexity which can be the
result of such a process of refinement. (44) for example counts as
adequate in this sense, since if we condense the second conjunction into
a single formula, we in any case get a formula which is a notational
variant of (43), and which is a semantically correct formalization of
the sentence.
With that said, let us return to De Morgan’s problem and the two
non-equivalent, but seemingly both admissible formalizations of (24),
(27) and (28):
\(\forall x \forall y ((Fy \land Kxy )
\rightarrow (Gy \land Kxy ))\)
\(\forall x (\exists y(Fy \land Kxy)
\rightarrow \exists y(Gy \land Kxy ))\)
Can QC2 help us decide whether one of the two is a more adequate
formalization of (24), the conclusion of De Morgan’s argument? Note
first that neither of these two formulas is more precise than the other
in the relevant sense, since the quantifiers and variables the two
formulas contain prevent us from generating one from the other by
substituting a logically more complex formula for a sub-formula in
either of the two. However, only one of the two formulas, namely (28)
stands in the “is more precise than”-relation to (26):
- \(\forall x (Hx \rightarrow
Ix)\)
We can generate (28) from (26) by substituting \(\exists y(Fy \land Kxy)\) and \(\exists y(Gy \land Kxy)\) for \(Hx\) and \(Ix\) respectively. (27) cannot be generated
in the same way, since the second universal quantifier in (27) cannot be
introduced by substituting logically more complex formulas for
sub-formulas of (26). The closest we can get to (26) is:
- \(\forall x \forall y (Mxy \rightarrow
Nxy)\)
However, it is not clear what the predicates \(M\) and \(N\) could stand for. Since both are
relational predicates, \(M\) would have
to correspond to something like “is a horse head of” and \(N\) to “is an animal head of.” Be that as
it may, since (45) is a less precise formula than (27), QC2 tells us
that (27) is an inadequate formalization of (24), unless there is a
notational variant of (45) which is an adequate formalization of (24)
(“All heads of horses are heads of animals”). If (45) turned out to be a
notational variant of (26), then this condition would be met. However,
this is not the case, since due to the presence of the second universal
quantifier in (27), we cannot generate it from (26) by one-for-one
substituting its non-logical predicates. So whether (27) is an adequate
formalization of (24) depends on whether (45) is an adequate
formalization of (24).
This opens up a way to informally argue that only (28) is an adequate
formalization of (24) by arguing that (45) is not a notational variant
of an adequate formalization of (24). Given QC2, the adequacy of (45)
cannot be justified by pointing out that it is a less precise formula
than the adequate formalization (27) since it is exactly the adequacy of
(27) which is at issue, so an independent justification is needed. One
might then for example argue that the additional logical complexity of
(45) gives us a reason to prefer (26) instead, or one might also target
the seemingly unnatural translation schema one would have to adopt to
make sense of (45).
Choice of Logic
Since our focus here is on deductive logic, the formalisms one has to
choose from when formalizing an argument are different logics. The one
logic which has the claim to being the default choice is classical
first-order logic. It has this status in virtue of some of its formal
properties—classical first-order logic is e.g. complete and sound—and
its expressive strength. First-order logic can be used to formalize a
range of mathematical theories, including e.g. some set theories and, as
we have seen, it can be used to express the same, or at least similar
claims, as intensional logics such as tense logic or modal logic (see Lewis 1968).
Still, there appear to be reasons to rely on alternative logics. One
reason is that one may be compelled to reject logical principles or
inference schemata which hold in e.g. classical first-order logic with
respect to certain contexts, or topics, or more generally for
philosophical reasons. Free logic provides an example of the latter
sort. As Karel Lambert describes it, free logic is “free of existence
assumptions with respect to its terms, general and singular” (1981, 123).
Classical first-order logic involves the assumption that every singular
term (e.g. each constant) refers to an object in the domain of
quantification. This, free logicians argue, is
problematic. Consider for example the sentence:
- Heimdallr exists.
In the language of first-order logic, this sentence can be formalized
as follows, using the constant \(h\)
for Heimdallr:
- \(\exists x (h = x)\)
Literally, this formula says that there exists something the same as
Heimdallr. Both this logico-literal restatement and (46) itself are, at
least insofar as common sense is concerned, false, since Heimdallr is an
object of fiction, i.e. an object which does not exist. Given the
mentioned assumption about the reference of singular terms, this formula
is however a logical truth of classical first-order logic. If we accept
first-order logic, we hence seem to be forced to accept an obvious
falsehood as true. Free logic offers a way out of this
problem, since it allows for the falsity of formulas like (47). This is
because unlike in classical logic, the rule of Existential
Generalization:
- \(A \vdash \exists x A(x/t)\)
fails in free logic. Here, \(A\) is
a formula of the language of first order logic and \(A(x/t)\) is the formula which results if we
replace any occurrence of the individual constant \(t\) by the variable \(x\) (if there are any). Existential
Generalization allows us to e.g. infer from (the formalization in the
language of first-order predicate logic of) “Heidallr owns Gjallarhorn”
to the existence of something which owns Gjallarhorn. In free logic,
this inference is not valid, since, briefly put, that a sentence is
satisfied by a particular individual constant does not entail the
existence of an object in the domain of discourse which satisfies the
formula. Other reasons for adopting
particular (non-classical) logics which have been given in the
philosophical literature include its adequacy for explaining vagueness
(cf. e.g. Machina
(1976) or Smith (2008)), or the need to move to
a non-classical logic in order to avoid semantic paradoxes such as the
liar paradox (cf. e.g.
Kripke 1975).
It is a fact that there are different logics, but which one should we
rely on in analyzing arguments? Carnap famously adopted a tolerant
stance towards logic. He assumed that any choice of logic is permissible
in principle and that which logic one relies on is ultimately a matter
of its usefulness for a particular purpose.
However, Carnap’s tolerant attitude is not shared by everyone and we may
ask whether, despite the fact that there are different logics, there is
one logic which is correct in the sense that it gives us the one correct
notion of logical consequence. This question is asked in the recent
discussion about logical pluralism, the view that there is more than one
correct logic and therefore also more than one correct notion of logical
consequence. A recently proposed methodology for
choosing between logics based on reflective equilibrium is criticized in
Bogdan Dicher’s contribution to the special issue. A question about the
independence of formalization and choice of logic is raised in Roy
Cook’s contribution.
Genesis of the Special Issue and
Acknowledgements
The initial idea for this special issue came about during the
workshop “Making it (too) precise” which I organized together with
Dominik Aeschbacher and Maria Scarpati in July 2017 at the University of
Geneva as part of the SNSF-funded research project “Indeterminacy and
Formal Concepts” (project nr. 156554) led by Prof. Kevin Mulligan. After
the editorial committee of Dialectica approved the proposal for
the special issue, an open call for papers was published online. 18
papers in total were submitted, including some of those presented at the
workshop in Geneva. All of these paper were subject to the same review
process which mirrored that passed by regular submissions to
dialectica, with the sole differences being that the guest
editor was both responsible for the organization of the review process
and for the initial internal review. The 13 papers which passed this
initial step were double-anonymously reviewed by two expert reviewers.
In a third and final step, the papers which were selected by the guest
editor based on the recommendations of the reviewers were presented to
the editorial committee and the editors who approved the guest editor’s
decision.
First and foremost, I would like to thank the authors for
contributing their papers and allowing them to be published in this
special issue. My second greatest debt is to all the reviewers whose
work made it possible for an interested bystander like myself to take
editorial decisions. I would also like to thank the editorial committee
of Dialectica, especially Matthias Egg for his helpful comments
and its managing editor Philipp Blum, for giving me the opportunity to
edit and for approving the special issue and the Swiss National Science
Foundation for financial support at the outset (“Indeterminacy and
Formal Concepts,” University of Geneva 2014–17, project number 156554,
PI: Kevin Mulligan). Finally, I would like to thank Philipp Blum and all
the people involved for the work they put into turning
Dialectica into an open access journal. It is a very happy
coincidence, one which only materialized after the reviewing process had
been well under way, that this special issue would be one of the first
issues of the journal to be freely and openly accessible to anyone over
the internet.
Overview of the Papers of the
Special Issue
In his paper “The Quantified Argument Calculus and Natural Logic,”
Hanoch Ben-Yami relates his Quantified Argument Calculus (acronym:
Quarc) to Larry Moss’s Natural Logic. The main selling point of
both of these logical systems is that they give us logics which are able
to account for the validity of certain intuitively correct argument
types, such as for example the argument from (7) to (8), which are
invalid in classical first-order logic. Ben-Yami shows that Quarc is
able to account for the same extended range of arguments which Moss’s
Natural Logic is designed to capture and furthermore argues that Quarc
has the advantage that it does not require to posit negative nouns to do
so.
In “Reflective Equilibrium on the Fringe: The Tragic Threefold Story
of a Failed Methodology for Logical Theorising,” Bogdan Dicher
critizises the idea due to Peregrin and Svoboda (2017) that
reflective equilibrium can serve as a method for choosing a logic. The
core idea of this approach is that the fact that the rules of inference
of a logic and the inferences in natural language which it is supposed
to formalize can be brought into a (virtuously circular) agreement with
each other provides us with a criterion for that logic’s adequacy.
Dicher’s argument against this idea is based on three case studies, one
focusing on the impact on harmony of moving from single- to
multiple-conclusion, another focusing on the question of how we may
distinguish between logics which deliver the same valid logical
entailments, focusing on classical first-order logic and strict-tolerant
logic (Cobreros
et al. 2012), and a third focusing on an application of the logic
of first-degree entailment (Anderson and Belnap 1975)
by Beall.
Jongool Kim’s paper “The Primacy of the Universal Quantifier in
Frege’s Concept-Script” focuses on the question of why Frege adopted the
universal, rather than the existential quantifier as a primitive of the
formal system developed in his Frege (1879). This question is not only
of historical interest, given that Frege’s book is one of the most
important contributions to the development of contemporary logic, but
also raises a general systematic question about factors motivating the
choice of a particular formal language. While Frege never explicitly
answered this question, Kim extracts, develops, and discusses three
arguments which support this choice from Frege’s works and singles out
one of them, a philosophical argument based on the idea that choosing
the existential quantifier as a primitive instead would have undermined
Frege’s logicist project of putting arithmetic on a purely logical
foundation, as the strongest.
Friedrich Reinmuth’s paper “Holistic Inferential Criteria of Adequate
Formalization” focuses on adequacy criteria for logical formalization.
Following e.g. Brun
(2004), Peregrin and Svoboda (2017)
and others, Reinmuth assumes that such criteria have to be holistic in
the sense that they have to take into account the consequences of the
choice one makes in formalizing a particular natural language sentence
not only for the target argument, but also for all other arguments
involving the same sentence as a premise or conclusion. He points out
shortcomings in existing proposals and motivates and develops criteria
which extend from arguments to more complex sequences of logical
reasoning and which e.g. allow one to distinguish between equivalent
formalizations of arguments which nonetheless lead to differences when
embedded in such sequences.
Gil Sagi’s paper “Considerations on Logical Consequence and Natural
Language” focuses on the relation between the notion of logical
consequence and ordinary language. Sagi in particular targets three
recent arguments due to Glanzberg (2015) to the conclusion
that the relation of logical consequence cannot be simply read off
natural language. Her paper rebuts these arguments and argues that one
of the two positive proposals made by Glanzberg for how one might go
beyond natural language in order to get at logical consequence is in
fact compatible with the view that this relation exists in natural
language.
In “‘Unless’ is ‘Or,’ Unless ‘\(\neg
A\) Unless \(A\)’ is Invalid,”
Roy T. Cook discusses the formalization of arguments involving the
expression “unless,” focussing in particular on the differences between
formalizations which rely on the same formal language, that of
propositional logic, but differ in that they assume classical or
intuitionistic logic as the background logic. One of Cook’s main points
is that his discussion questions the assumption that translations from
informal into formal language are logic neutral, in the sense that we
can settle for a logical formalization independently of first adopting a
particular logic.
Vladan Djordjevic’s paper “Assumptions, Hypotheses, and Antecedents”
focuses on an important distinction between three ways in which
deductive arguments can be cast both in formal languages and in natural
language. Djordjevic distinguishes “arguments from assumptions,” which
are arguments in which each premise is assumed to be logically true and
the logical truth of the conclusion is to be established, from
“arguments from hypotheses,” in which the validity of an inference from
the premises to the conclusion is at issue, and from assertions of
conditionals which contain the premises of an argument in their
antecedent and its conclusion in its consequent. The three categories
are often conflated and Djordjevic argues that certain philosophical
puzzles, including a standard argument for fatalism and McGee’s
counterexample to Modus Ponens can be resolved based on these
distinctions.
