1
Basic Logical Knowledge1
Concha Martínez-Vidal
Department of Logic and Moral Philosophy
University of Santiago de Compostela (Spain)
mconcepcion.martinez@usc.es
Abstract.My purpose in this paper is to appraise the prospect of counting on a justification for
basic logical inference rules and for basic logical laws.
The offer at the philosophical menu is as wide as usual in our field. I am not going to cover all
possibilities available. On the contrary, I will present a widely discussed proposal by Paul
Boghossian to reject it and put forward a view that shares a lot with the one advanced by Gila
Sher (Sher 2011), and that takes some elements from Corcoran’s view of logic as model.
The kind of justification for logic different authors suggest is not usually independent of their
philosophical orientation and motivation to start thinking about the issue. My worry has to do
with trying to understand the role of logic in our intellectual lives, and my starting point is our
contemporary understanding of logic.
Nowadays, one frequent way of explaining what is logic about is by saying logic is the discipline
that formulates criteria for the evaluation of arguments, thus given our best logical theory—
the model-theoretical account of logical truth and logical consequence—, logic is understood
as “(L2) the study of formally valid inferences and logical consequence”(Hofweber 2012).2 On
the other hand, traditionally logic has been understood as “(L4) the study of general features,
or forms, of judgments” (Hofweber, 2012). Now, when the focus of interest is the need and
possibility of the justification of logical knowledge, clarifying the relation between both ways
of understanding logic will prove crucial in order to clarify how our current logical knowledge is
to illuminate or determine our conception of the role logic plays in our intellectual lives. After
all, to assay whether our (basic) logical inferences are justifiable, amounts to judging the
possibility of counting on a justification for the rules we use (maybe we should use) when we
judge.
The problem
In the case of logic, any inferential justification for a given rule is going to be circular. Thus,
either some kind of non-inferential justification is available, or something else is to be done in
order to show that a justification is possible, even if it is circular. Otherwise, no justification for
logic will be available.
1
This research was supported by the FFI2009-08828 of the Spanish Ministry of Science and Innovation
and FEDER (Cognitive Attitudes and Knowledge Justification) and by the “The Bounds of Judgment” a
project of the MLAG Group of the Institute of Philosophy (Porto University) financed by the Portuguese
FCT.
2
In the following exposition of the relation between logic as the study of valid inferences and logic as
the form of judgment I am paraphrasing Hofweber in his 2012.
2
Let us illustrate the problem by supplying a reconstruction of an inferential justification for
modus ponens:3
If we want to justify MPP inferentially, then we want to justify that p and p  q, imply q.
So, we assume
1p
we assume
2 pq
3q
(it follows by MPP from 1 and 2)
4 (pq)q
obtains by conditional introduction
5 p [ (pq) q] obtains by conditional introduction
Clearly, Modus Ponendo Ponens, the rule whose validity we intend to prove, is used in the
argumentation. A rule-circular justification obtains. Once more, claiming that this kind of
argumentation provides a justification for our belief in MPP at all involves explaining why we
should not care about its circularity or proving that they are not circular after all.
Boghossian (Boghossian 2000, 2003) contends that this kind of justification, rule-circular
justification, is, nevertheless, non-trivial, though it might not satisfy the skeptic. He argues4
that there is an independent (independent from our knowledge of the validity of Modus
Ponens) entitlement for our use of Modus Ponens in our going through the inferential
justification of the validity of Modus Ponens above, step 3. Note that ‘entitlement’ is a species
of warrant; the kind of warrant on the basis of which human beings are granted human rights,
or the rights that the Spanish constitution bestows on us because we are Spanish citizens. This
is the kind of warrant dispositions have, according to Boghossian, while ‘justification’ amounts
to giving reasons, and it is the kind of warrant beliefs have (Boghossian 2000, 230). Our
entitlement to use Modus Ponens—so Boghossian claims—comes from the fact that Modus
Ponens is meaning-constituting for the conditional. Hence, assenting to the rule is necessary to
have the concept of conditional. As a result, we are entitled to use the rule even if we lack an
explicit justification for it. The reason is we count on a blind, but blameless justification to use
Modus Ponens. Blind because, let us say, we cannot give non-circular reasons, but blameless
because we cannot give reasons because such reasons are not available: assenting to the rule
is a pre-requisite for our using it. 5
If this kind of entitlement were really attainable, then it would allow us to overcome the
circularity problems that lurk logical justification; it would grant us independent warrant for
our use of the logical rule in the inferential justification of the validity of that very rule.
3
See Boghossian 2000 for a detailed presentation of the issue.
See Boghossian’s papers on the issue: 2000, 2003a, 2003b, etc.
5
It should be apparent that we are analyzing the problem of the justification of logic from an internalist
viewpoint. There are different internalist views about justification:
•
Access internalism maintains that we enjoy a special kind of access to justification factors: they
are always recognizable on reflection.
•
Mentalism considers justification is internal because justification factors are always mental
states; what ultimately justifies a belief is a certain mental state.
•
And deontological internalism claims that one is justified if the kind of justification one has
allows her to comply with her epistemological duties.
Boghossian’s proposal might be interpreted as a kind of deontological internalism, though some
contend it would rather be a kind of externalism.
4
3
Moreover, it would allow us to claim that basic logical knowledge is epistemically analytic in
Boghossian’s sense of the term. 6
Unfortunately, there are problems. One of them is that we need to distinguish between rules
that are actually meaning-constituting versus rules that are only apparently so. Boghossian is
well aware of it, and considers several cases. Among the latter a well-known example is that of
‘tonk’.7 The introduction and elimination rules for ‘tonk’ cannot be meaning-constituting,
Boghossian contends (Boghossian 2000; 2003); the reason is that ‘tonk’ does not refer to a
concept at all.
A
A tonk B
A tonk B
B
He also examines other cases such as that of ‘Boche’. The introduction and elimination rules
for ‘boche’ are:
x is German
---------------x is Boche
x is Boche
---------------x is Cruel
‘Boche’ would refer to a concept, according to Boghossian, but to a defective concept since its
introduction and elimination rules are not truth-preserving (Boghossian 2003).
The other problem was pointed out by Timothy Williamson (Williamson 2003) who supplied a
counterexample to Boghossian’s contention that assenting to the rule is necessary to have the
concept of conditional. Williamson illustrates his objection by presenting Vann McGee’s case,
an expert logician, who is well known for rejecting certain uses of Modus Ponens, yet no one
doubts,—even if McGee might be wrong in his verdict about those uses of the conditional as
Williamson himself thinks—, he has the concept of conditional.
Moreover, and in the line of Williamson’s objection, I would like to bring up a remark Wright
and Resnik made both independently of each other and independently of our worries in this
paper. Both Wright (Wright 1986) and Resnisk (Resnik 1997) agree that if a sentence  follows
from  according to logic L, then the conditional, ‘if  then  (according to logic L)’ is
objectively true. It is clear then, that in order to see whether a given logic complies to certain
standards it is not necessary to take a compromise with those principles, that is to say, in order
to see whether a given proof is intuitionistically acceptable one does not have to be an
intuitionist, but only to agree that if those rules are in place then a given theorem follows. On
the other hand, such an agreement is far from being available when the discussion has to do
with which logic is the right logic. (See Martínez Vidal 2005).
In other words, logicians discuss logical principles, and they agree that a certain theorem
obtains if certain meaning is associated to logical constants, even if they do not accept the
6
See Boghossian 1996, 2003b for a chareacterization of the notion of epistemic analyticity in opposition
to Carnap’s metaphysical analyticity.
7
Prior first mentioned the example (Prior 1960). The introduction and elimination rules for tonk are:
(p tonk q)hence q; and p hence (p tonk q).
4
logic in question. Analogously, scientists do discuss scientific hypothesis even if they do not
accept them. Boghossian acknowledges that, in the case of empirical concepts
“…the concept should not be designated in such a way that only those who believe a certain
creed are allowed to possess it” (Boghossian 2003, 245)
But he claims things work differently in the case of logical concepts. Now, given the
abovementioned facts about logicians practice, Boghossian’s claim that only unconditionalized
versions (more on what is meant by this below) are available for logical concepts seems clearly
unteanable.
Boghossian (Boghossian 2003) also contends that some concepts are defective concepts
because they do not allow for disagreement. Hence, he sustains that those concepts that are
formulated in their unconditinalized version when a conditionalized one is available are
defective. Non-defective concepts should allow for disagreement. Boghossian introduces the
distinction between conditionalized and unconditionalized versions of a concept in order to
provide one a criterion to distinguish defective from non-defective concepts. To that purpose,
Boghossian, sustains that:
“If we follow Russell, Carnap and Lewis, and represent neutrino theory
T(neutrino)
as the conjunction of the two propositions
(S) (x) Tx
and
(M) (x) TxT(neutrino)
then the point is that we think of possession of the concept neutrino as requiring someone to
affirm only M and not S as well.” (Boghossian 2003, 245)
Hence, in order to allow for discussion, a conditionalized version is to be preferred if available.
When a conditionalized version is available but not used, then the resulting concept is a
defective one.
Now, for Boghossian there is no conditionalized version in the case of logical concepts, hence
the unconditionalized version applies. The reason why there is no conditionalized version for
the conditional, is that it would have to be used in order to give a conditionalized version of
the concept of conditional. But then again, in the light of the objections signaled above, it is
clear that logicians discuss about the appropriateness of logical rules and they do use logic in
the process. It seems Boghossian’s proposal has carried us through a long detour but the
circularity problem is still with us.
Towards an Alternative Account
Two arguments have been waved against Boghossian’s proposal:
1) An expert logician who is widely acknowledged to have the concept of conditional, rejects
certain uses of Modus Ponens;
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2) And we have also signaled that logicians discuss basic logical principles just as scientist
discuss scientific hypothesis.
Hence, it seems that, following Boghossian’s route to a solution to the problem, some kind of
conditionalized version for logical notions should be available, but then we go circular again.
So, whether basic logical rules are justifiable and if so how, is still our problem.
If we examine Wright and Resnik’s remark above, it seems clear that logicians agree that if a
certain meaning is assigned to logical expressions (by providing introduction and elimination
rules for them), then a set of theorems follows. McGee is surely not contesting that; rather, he
is contesting that Modus Ponens can appropriately explain certain of our uses of the
conditional. In the same line, intuitionists doubt that LEM (Law of excluded Middle) is in place,
or Field rejects that it applies generally if problematic sentences such as the liar are allowed in
a given language.
One thing is to establish the validity of a rule, or the soundness of a logical system, or that 
follows from  according to logic L, and quite another is to establish that a given logic is
appropriate to account for the underlying logic in a given domain or argumentative practice, or
that a given logic that complies with a given set of philosophical constraints (the ones
proposed by Dummett or by Field, for instance) has to be in force.
Boghossian’s characterization of meaning-constituting rules as being such that accepting them
is a precondition for having the corresponding concept, seems not to take into consideration
that one thing is to have the concept, and another thing is to agree it applies.8
Purely deductive methodology has always consisted in making sense of structures that might
not be realized empirically:
“The distinction is drawn and discussed between a conception of a priori truth, which is first
found in Plato and is found in Leibniz under the heading of “a priori truth,” according to which
it is truth about a species of structure and can be understood and studied independently of
whether or not this kind of structure is exemplified in the natural world, …” (Tait 1992, 18)
“In the exact sciences concepts are employed which do not literally apply to the phenomena
and propositions are assumed which are not literally true of them. Thus, in the case of
geometry, we do not perceive point, lines, surfaces, etc. and the assumption that, for example,
any two magnitudes are comparable is not supported by our actual experience in measuring,
in which the results of measuring are inexact and unstable under repeated measurements.10
[…] Geometry was the study, not of empirical shapes and magnitudes directly, but rather of
the kind or form of structure that they imperfectly exemplify.” (Tait 1992, 22-23)
As Resnik puts it:
“The development of non-Euclidean geometries and abstract algebra further promoted the
purely deductive methodology of the axiomatic method through showing mathematicians how
to make sense of structures that might not be realized physically. It also promoted a shift from
8
According to Williamson (Williamson 2003), for Boghossian to have a concept is to understand the
word that expresses it.
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viewing mathematical sentences as unqualifiedly true or false to regarding them as true or
false of structures of various types. These two developments have further insulated
mathematics against empirical refutation.” (Resnik 1997, 130)
When we develop logical systems, we make sense of structures that allow us to characterize
an appropriate semantics for them. When we claim that a given logic (language and deduction
system plus the corresponding semantics) accounts for the logic underlying a given domain or
practice, we are contending that the use of logical expressions underlying the practice,—or
allowing us to establish the corresponding truths in that domain— is properly accounted for by
the corresponding logic, the logical concepts involved are the ones characterized by the logic.
Logical, like mathematical models, are developed independently of their being useful for
accounting for underlying logical practices. This peculiarity of them, probably explains (at least
in part) that logic and mathematics are frequently considered as paradigmatic cases of a priori
knowledge. For instance, logicians construct mathematical models, not only for the actual
underlying logics of the mathematician but also for putative logics—i.e., for logics which are
envisaged by various theories of logic but which do not necessarily correspond in any very
faithful way to actual underlying logics. When logicians are involved in this task, their
methodology is exclusively that of deduction and the resulting systems provide norms for
correct reasoning. In those cases, being consistent or sound is all that is required; no attention
is paid to their matching a certain practice, or any possibilities of application. Being consistent
or sound is enough for it to be interesting, though not enough for it to be adequate to account
for a given practice:
“…internal questions are perfectly definite mathematical questions concerning the logic as a
mathematical object. And if they are answered, then they are answered by the same means
used to answer any mathematical question—viz. by logical reasoning from the definitions of
the systems together with the relevant mathematical laws” (Corcoran, 1974, p. 119).
In many other cases logical theories are established from above; then we get prescriptive
views of logic. As a general rule, these approaches rely on considerations that are basic to
their philosophical projects and determine—on their basis—which logic should be used. They
are also cases in which logical models are developed independently of any practice.
Some examples of LOGICAL theories that are developed prescriptively:
a) Those for which the norm is founded in considerations in the theory of meaning: one good
example would be Dummett’s.
b) Considerations (Brouwer) about the ontological status of numbers.
c) Considerations about how the liar paradox should be solved (Dialethists like Priest or
paracomplete logicians like Field)
Both mathematical intuitionists and Dummett, an antirealist, defended that classical
mathematical practice should be changed: classical logic should be abandoned and replaced
with intuitionistic logic. Brouwer, the father of intuitionistic mathematics, developed
intuitionistic logic. According to both these projects, classical logic is wrong and should be
abandoned. In the case of intuitionistic mathematicians, classical logic should be replaced by
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intuitionistic logic in mathematical practice. Many of them do not claim that it is to be
abandoned in other areas. Dummett’s proposal is more radical. Classical logic is wrong. It
should not be used in any field except for finite domains. This difference has to do with the
dissimilarities between both projects: while classical intuitionists contend classical logic is
wrong because it presupposes the existence of a mathematical realm of objects that exist
independently of us, Dummett claims that classical logic is wrong because it presupposes a
transcendent notion of truth.
According to Corcoran, providing evidence that a given mathematical model is adequate to
account for a given practice is an external question;9 so would be establishing that a given set
of prescriptions is to be applied and hence the logic underlying to such set of prescriptions is to
be adopted. Deciding whether a given logical logic is appropriate is decided empirically;
deciding whether a set of prescriptions is to be in force might be a philosophical decision, and
whence a priori.
Now, the philosopher worried about the role logic plays in our intellectual lives might object
that logical pluralism and mathematical logic have to do with logic as a discipline, not with
logic as the study of the general form of thought.
But it seems to be the case that Boghossian’s view is at least compatible with the existence of
a direct relation between logic as the study of the relation between of logical consequence,
and logic as the study of the laws of thought. Given the two senses of logic we have just
mentioned,10 —that is to say logic understood as (L2) the study of formally valid inferences and
logical consequence, and logic understood as (L4) the study of general features, or forms, of
judgments – there is a way of understanding what judging or thinking is that allows for a close
relationship between the two abovementioned ways of understanding the term:
“If thoughts, and thus judgments, are realized by minds having a certain relation to mental
representations, and if these representations are themselves structured like a language, with a
“syntax” and a “semantics” (properly understood), then the form of a judgment could be
understood just like the form of a sentence. Such a view of thoughts is commonly called the
Language of Thought hypothesis, and if it is correct then in the language of thought there
might be logical and non-logical vocabulary. The form of a judgment could be understood
along the lines we understood the form of a linguistic representation when we talked about
formally valid inferences. Thus the relationship between (L2) and (L4) is rather direct. On both
conceptions of logic we deal with logical constants, the difference is that one deals with a
system of mental representations, the other with a system of linguistic representations. Both,
presumably, would deal with corresponding sets of logical constants. Even though mental and
linguistic representations form different sets of representations, since they are closely
connected with each other, for every logical constant in one of these sets of representations
there will be another one of the corresponding syntactic type and with the same content, or at
least a corresponding inferential role.” (Hofweber 2012)
9
Corcoran 1974.
I follow Hofweber presentation of the issue in Hofweber 2012.
10
8
Boghossian (Bogossian 2003) sounds quite happy with assuming we think in a language. Hence
I think we can easily establish the relation in his case between logic as the study of formally
valid inferences and logical consequences, and logic as the study of the laws of thought:
“Before proceeding, it is worth emphasizing that although I follow tradition in construing
analyticity as a property of linguistic items, that feature of the presentation of the view is
entirely optional. For example, I will talk of grasp of the meaning of a sentence as sufficing for
justified belief in the proposition it expresses; but I could equally well simply have talked about
grasp of a proposition p as sufficing for justified belief in p. Thus, too, I will talk about words
being synonymous with each other; but I could equally well have talked about concepts being
identical to one another. Finally, I will talk of holding some sentences true, as a condition of
meaning some specific proposition by them; but I could equally well have talked of believing
some propositions as a condition of having some of their ingredient concepts. Although I find
it convenient for expository purposes to assume that we think in a language (which I will
pretend is in fact English), no central issue hangs on that assumption.” (Boghossian 2003, 1516)
It seems to me that, given we can establish a link between both views, it should be clear that
the abovementioned objections based on what logicians do, apply.
An Alternative Account
Any justification for logic is going to entail the use of logic, and hence some circularity.
Nevertheless we manage to reason about which logic is the correct logic, and we do manage to
make a difference between logical systems that are ‘good’ and those that are not. My claim is
that we manage to do so because we check that logical rules are valid in a given set of abstract
structures (our contemporary way of ‘imagining’ possible models, in the sense of model
theoretic semantics) or we manage to check that a given logical model (in the sense in which
scientists develop models) is appropriate to account for a given practice. In the first case, we
develop logical models (logical systems plus the corresponding semantics), in the second case,
we apply pre-existent models in order to account for the logic underlying a given
argumentative practice or domain. When we check whether a logical model is appropriate to
account for a given practice we test whether the set of truths that obtains according to a given
practice is the set of truths our logical model predicts that obtain. Analogously, we check
whether a given logical model allows us to obtain truths in a given domain when we start with
truths in that domain. Either externally or internally, the notion of truth is playing a role, be it
the notion of truth in a model or truth simpliciter.
Where do we stand? Are there basic logical truths? Is Logic A Priori?
One thing is clear: Boghossian’s proposal, as it stands today, does not allow for us to contend
that we can count on a justification for basic logical inference rules that were not grossly
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circular. Boghossian’s main purpose is not to try to explain the kind of propositional warrant11
available for Modus Ponens; he is trying to elucidate whether it is possible for us, human
subjects, to establish that modus ponens is valid. Boghossian is worried about doxastic
justification, and he intends to proof that we can know a priori that a basic logical rule is truthpreserving in the real world.
We have argued that we cannot know a priori that a basic logical rule is truth-preserving in the
real world. We can only know a priori that a given basic logical rule is truth-preserving
according to a given logic.
Mathematical logic has developed mathematical tools in order to establish whether a given
logical system is truth-preserving: proofs of consistency, soundness, completeness. When
mathematical logicians work at this level, I take it they establish the ‘goodness’ of the different
systems independently of experience. Proof is the only working methodology.
We can also establish a priori which logic is the one that complies with set of constraints. We
might even establish a priori, whether a given set of constraints is the one that is to guide a
given inferential practice, the issue might be a philosophical one. But when we check whether
a logical model is appropriate to account for a given practice, we are not applying an a priori
methodology. At this point what happens with logical systems is analogous to what happened
with Euclidean Geometry. This is a problem of adequacy of the model, and the solution has to
be established empirically.
All these considerations should be sufficient to understand that demanding a justification for
logic is not to be identified with demanding a justification for logic that does not use logic, a
justification for logic that is non-circular (Sher 2011; Shapiro 1991). In other words, it is
possible to count on a foundation for logic, even if the structure of justification is not going to
be a foundationalist one. We use logic in order to evaluate our logical systems, but that does
not convey that the justification we achieve is a vicious one. This kind of justification might be
circular but not vicious because we count on independent evidence; we check whether the
logic is or is not truth-preserving in a given structure or in the empirical world.
References
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Philosophical Studies (Minneapolis), 73:2/3 (1994:Mar.) p.109
11
“Propositional warrants are those warrants a subject has, in a given situation, for believing a
particular proposition P, independently of whether he appreciates them or not and even if he does not
form the belief that P.” (Coliva 2010) Or, “A subject may have a propositional warrant to believe P while
not actually believing P and, indeed, while having no inclination to believe P. We might highlight this by
speaking of an abstract space of warrants just as we speak of a logical space of entailment relations.”
(Davies 2008, 2) It seems to me to correspond or involve what Frege called “grounding”. In contrast, “If
a subject’s believing P is to be doxastically warranted, then there should be the right relation between
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10
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