=Paper=
{{Paper
|id=Vol-1994/Bridging2017_paper4
|storemode=property
|title=Informalizing Formal Logic
|pdfUrl=https://ceur-ws.org/Vol-1994/Bridging2017_paper4.pdf
|volume=Vol-1994
|authors=Antonis Kakas
|dblpUrl=https://dblp.org/rec/conf/cogsci/Kakas17
}}
==Informalizing Formal Logic==
https://ceur-ws.org/Vol-1994/Bridging2017_paper4.pdf
Informalizing Formal Logic?
Antonis Kakas
Department of Computer Science, University of Cyprus, Cyprus
antonis@ucy.ac.cy
Abstract. This paper discusses how the basic notions of formal logic
can be expressed in terms of argumentation and how formal classical
(or deductive) reasoning can be captured as a dialectic argumentation
process. Classical propositional logical entailment of a formula is under-
stood via the wining arguments between those supporting the formula
and arguments supporting its contradictory or negated formula. Hence
both informal and formal logic are captured uniformly in terms of an
argumentation and its dialectic process.
1 Introduction
Informal Logic is usually equated with argumentation as used in real-life every-
day situations. On the other hand, formal logic is concerned with the strict and
precise reasoning in mathematics and science. There are several works aiming to
capture informal logic in a precise formal setting such as that found in the arti-
cle “Formalizing informal logic” [10] where informal logic is placed in the formal
argumentation framework setting of the Carneades Argumentation System [3].
This paper is concerned with the other direction of linking formal logic to
informal logic - taking informal logic as synonymous to argumentation. The aim
is to reconstruct formal logic entirely in terms of argumentation enabling us
to view formal deductive reasoning of classical logic as a process of dialectic
argumentation.
The paper rests on the technical work of Argumentation Logic [5, 6] where
this reformulation of classical Propositional Logic in terms of argumentation is
carried out in formally precise terms. This work is based on notions coming from
the fairly recent development of argumentation theory in Artificial Intelligence.
The purpose of this paper is to unravel the technical results and present them in
a generally accessible way thus providing a uniform argumentation view of both
informal and formal logic.
Informalizing formal logic will be possible as a limiting case of “strict di-
alectic argumentation” where the arena of arguments together with the notions
of counter-argument and defending argument are tightly fixed. This rigidity of
the argumentation framework is to be expected since our task is to recover
strict formal reasoning. The importance though of this reformulation of formal
logical reasoning is that the strictness in the argumentation framework can be
?
A full version of this paper is in preparation.
subsequently relaxed in cases where this is appropriate, as for example in com-
monsense reasoning. As a result we have a uniform way of capturing both formal
and informal reasoning, smoothly moving from one to the other.
2 Logical Arguments
The construction of arguments in informal logic typically follows some accepted
argument schemas that would link premises to a conclusion or a position of
the argument. Logical arguments are arguments whose link between premises
and supporting position rests on a precise logical proof in some formal logical
system such as Classical Logic1 . Hence to informalize formal logic one starts by
considering the set of proof rules in a logical proof system as argument schemes.
Arguments can then be identified with sets of logical formulae that under
some of the proof rule argument schemes derive and thus support a conclusion
or position of the argument. The chosen proof rule argument schemes are called
direct argument schemes. The support of a conclusion φ by an argument
A is given through a direct derivation of φ from A. This will be denoted by
A `DD φ where DD denotes the chosen set of proof rules.
There are two important conditions that need to be applied to this choice of
proof rule argument schemes. The first is that these argument schemes of proof
rules need to be considered as strict schemes, i.e. that arguments constructed
under these can not be defeated by questioning the validity of the chosen proof
rules. The other condition has a more technical nature and requires that the
proof rules of Reduction ad Absurdum (RA) are excluded from this initial
choice of core argument schemes. This rests on the observation that the RA rules
contain an element of evaluation of arguments, as they rest on first recognizing
that their posited hypothesis (or argument) is inconsistent (invalid), and hence
cannot be considered as a primary scheme of construction of arguments.
The main technical task then in the re-formulation of formal logic in argu-
mentation terms is to recover at the semantic level of argumentation the RA
proofs of formal logic.
Let us illustrate these ideas with a simple example. Suppose that the premises
of propositional logic theory, T , are given by:
q → ¬p (1)
r → ¬p (2)
Given additional premises about q and r we can construct arguments for and
against p. For example, if in addition we are given q in T then we can construct
an argument A1 with premises the sentences (1) and q supporting the conclusion
1
We will confine ourselves to the case of Classical Logic and more specifically to clas-
sical Propositional Logic although the ideas presented would apply more generally
to other formal logics.
¬p, as there is a direct derivation (using the proof rule of Modus Ponens) of this
conclusion from these premises.
Given this theory T of premises to construct arguments that support p we
would need to base these on formulae that are outside T . We will call such
premises hypotheses and arguments that are build on them hypothetical ar-
guments. For example, we can simply build an argument A2 supporting p based
on the hypothesis of itself. We will see below the significance of this difference in
the type of premises used when we consider the argumentation process between
arguments. As expected, arguments whose premises are entirely drawn from the
given theory will be stronger or preferred to hypothetical arguments allowing for
example A1 to win over A2 and as a result the theory T to logically conclude
¬p.
3 Logical Reasoning as Dialectic Argumentation
In an argumentation framework, given a position of interest we can distinguish
pro arguments and con arguments, i.e. arguments that support the position
and arguments that oppose the position. Arguments from these different classes
attack each other or are counter-arguments of each other based on some
form of conflict between them2 . In formal logic contradiction is capture via the
conflict between formulae and their negation. Generally, this (symmetric) attack
relation for formal propositional logic can be captured through the joint direct
derivation of an inconsistency, namely of any formula and its negation, normally
denoted by ⊥. So two arguments A1 and A2 attack each other if and only if
A1 ∪ A2 `DD ⊥.
We will then view formal logical reasoning as a dialectic argumentation pro-
cess for and against formulae and their negation. Arguments will be evaluated
with respect to the other arguments that can be constructed and in particular
evaluated against their counter-arguments. Arguments are acceptable when
they exhibit a good dialectic quality, namely that they can defend against all
attacking arguments. Analogously, an argument is non-acceptable if there
is at least one counter-argument that it cannot be defend against.
To turn this into a precise definition that would then capture the strict logi-
cal reasoning of propositional logic we notice that the defence argument against
any counter-argument must also be required to be acceptable and importantly
to be acceptable within the context of the original argument that we want to
be acceptable. Thus the central notion of acceptability of arguments is a rela-
tive notion that is recursively specified by (here S and S0 are sets of arguments):
2
In Artificial Intelligence the attacking relation in an argumentation framework [4, 2]
often contains more information than simply this symmetric incompatibility of the
arguments involved. This extra information, as we will see below, pertains to the
relative strength or preference of the arguments involved in the attack.
“S is acceptable w.r.t. S0 if for any attacking argument, A against S there
exits a defending argument D that is acceptable w.r.t. S0 extended by S.”
Analogously, for the non-acceptability of S w.r.t. S0 we need to have an
attacking argument whose all possible defences are recursively non-acceptable
w.r.t. S0 extended by S.
The defence relation between arguments normally captures the relative
strength or preference between arguments. An argument can defend against
a counter-argument if it is preferred over the attacking argument or they are
non-comparable in preference. The preference and its defence relation in many
domains of argumentation comes from domain specific information. Nevertheless
for the quite general framework of logical reasoning as captured by Argumenta-
tion Logic [5] the preference and ensuing defence relation is minimal. It consists
of two elements:
– Arguments which are entirely made out of premises in the given theory T
are strictly preferred over arguments that contain hypothetical sentences
and thus can be defended against only by other arguments that also consist
entirely from premises in T .
– A hypothetical formula φ and its complement φc are equally preferred.
We are free to choose equally between the two (provided that one is not also
a direct consequence of the given theory T ) with this choice allowing us to
take the side appropriately needed to defend against attacks.
Note that when the given premises T are consistent the first element of
defence means that attacking arguments that are made entirely from T can not
be defended against. Hence an argument that is attacked by an argument made
entirely of premises in T cannot be acceptable. Similarly, an argument S made
entirely from T is attacked only by arguments containing hypothetical formulae
and so can always be defended against by S. Hence such arguments are always
acceptable.
In the simple example given above, we can then see that p is acceptably
supported, given that the premises T contains the sentences (1), (2) and q.
To illustrate a more complex case of the dialectic argumentation process and
how this captures formal logical conclusions of PL, let us consider that instead
of q we have the premise:
¬q → r (3)
Hence we are now considering the theory T consisting of sentences (1), (2),
and (3). The position of p can only be supported by arguments that contain
this as a hypothesis or directly derive this from a set of formulae that contains
hypotheses. Then the non-acceptability of such arguments can be determined
by considering the counter-argument consisting of the premises 1 from T and
hypothesis {q}. The dialectical process of argumentation that shows that this
attack cannot be defended against is depicted in figure 1 where for simplicity we
only show the hypothesis part of the arguments involved3 .
{p} O
attack : T ∪{p}∪{q}`DD ⊥
{q} KS
defence by opposing the hypothesis
{¬q}
O
attack : T ∪{¬q}∪{p}`DD ⊥
{p}
Fig. 1. Dialectic process of argumentation for determining the non-acceptability of
p, with respect to the empty set of arguments, given T = {(1), (2), (3)}, in order to
determine the classical entailment of ¬p from T . Arguments are shown only by their
hypotheses as indicated in brackets.
The informal reading of this figure is as follows: The argument of supposing
p is attacked by the hypothesis q. The canonical objection or defence to this
counter-argument is to assume ¬q. But this defence is in conflict with the argu-
ment p that it is meant to be defending as together they directly derive through
(2) and (3) an inconsistency. Hence although in general there is a defence against
the objection (by taking the opposite view) this is not possible in the context of
the particular argument that we wish to be acceptable.
This example illustrates how defences against attacks to an argument must
“hold well together” in the sense that they need to be conflict free or in other
words they should not contain an internal attack or counter-argument relation
between them. None of the defences should form a con argument against any
of the other defences and of course against the original argument of interest. In
general, for any acceptable argument there must exist a set of defences against
all its attacks that is attack free, i.e. directly consistent under `DD .
This rationality property of the set of defences points to the connection be-
tween acceptability of arguments and the formal logical notion of satisfiability
of the formulae composing the argument. In fact, in the case where the given
premises T are classically consistent and we have a model for the set of formu-
lae in an argument A then we can choose the defences for A from the set of
formulae that are made true in this model and hence A would be acceptable.
In other words, satisfiability implies acceptability and vice versa and thus for
classically consistent premises T formal logical entailment coincides with scep-
tical argumentation reasoning given by: a formula φ is sceptically concluded
by argumentation if and only if φ is dialectically acceptable (w.r.t. the empty
set of initially accepted formulae) and ¬φ is dialectically non-acceptable. This
3
For simplicity we also assume here that `DD contains only the Modus Ponens proof
rule argument schema.
then gives the logical equivalence of formal classical (propositional) logic and
argumentation logic thus informalizing formal logic.
4 Beyond Classical Reasoning
Propositional Logic or full Classical Predicate Logic are not equipped or de-
signed to deal with contradictory information. When the given premises T are
inconsistent formal classical reasoning collapses where every formula is trivially
entailed. In contrast, argumentation is concerned exactly with how to deal with
conflicting information and positions. Hence the argumentation based reformu-
lation of formal classical reasoning that we have described above, mainly for the
case of consistent premises T , would or should carry through when T becomes
classically inconsistent.
Consider in our example that we are given p as an additional premise to those
of {(1), (2), (3)} that we already have. This turns the set of premises classically
inconsistent. But the argumentation-based formulation of logic that we have
described above will not trivialize. For example, it would sceptically conclude
that p holds, without also concluding that ¬p holds as PL does.
The dialectic argumentation proof in figure 1, that gave us earlier the conclu-
sion that p is non-acceptable (and hence that ¬p holds since also it is acceptable),
now changes. Indeed, we now have another way to defend against the attack(s)
containing the hypothesis q. We can now defend using the premise p that, as we
have explained above, is preferred over arguments that contain hypotheses as
it is made purely of sentences in the given premises T . On the other hand, any
argument supporting ¬p will be attacked by the argument made purely from
the premise p. But this cannot be defended against since there is no direct or
explicit information to its contrary in the premises. Hence p is acceptable and
any argument supporting ¬p is not acceptable, thus p is sceptically concluded.
Another example of escaping from the trivialization of formal classical reasoning
would be the case where we have in the premises T both p and ¬p. Then each
of these premises would defend against each other and so both arguments would
be acceptable and therefore there will be no sceptical conclusion for whether p
holds or not.
The strict conditions on the argumentation framework that we have imposed
so that we can match the strict reasoning of formal logic can be further relaxed by
allowing the premises themselves to take a defeasible nature, e.g. implications to
represent only a “normally or mostly” nature of associating their condition with
their conclusion. Then relative preferences amongst this defeasible knowledge
enriches the defence relation by rendering some arguments stronger and hence
able to defend against (some of) their counter-arguments but not vice-versa.
This is particularly appropriate when we consider the informal logic of com-
mon sense human reasoning [9, 8, 1] where people normally reason within a
context and where the common sense knowledge is in this form of loose asso-
ciations. General or individual human biases give preference to some of these
statements and we can then understand common sense reasoning in argumenta-
tion terms in the same way we have expressed formal logical reasoning. Argumen-
tation thus gives a uniform umbrella framework covering the whole spectrum of
reasoning from the very strict formal reasoning to the flexible informal reasoning.
5 Conclusions
We have described how formal classical reasoning can be captured through the
same process of dialectic argumentation that is normally associated with informal
logic. This reformulation of logic in terms of argumentation has been shown [6]
to be complete for propositional logic. The same approach can be applied more
generally to first order predicate logic. An interesting example of this is that of
Aristotelian syllogisms when these are seen as canonical forms of strict classical
reasoning of predicate logic. Furthermore, syllogisms have been studied as an
example of human cognitive reasoning, see e.g. [7], where it is observed that
humans do not reason according to formal logic but that their interpretation
of syllogism is indeed a case of informal logic. In a recent challenge4 to model
the cognitive syllogistic reasoning of humans, argumentation was shown as a
promising approach towards this goal.
By varying the degree of flexibility within the argumentation framework and
its dialectic process we can move from formal logic to informal logic and back.
Argumentation thus provides a way to unify these two worlds of logic, normally
considered as very different, under the same conceptual framework. It provides a
uniform umbrella framework covering the whole spectrum of reasoning from the
very strict formal reasoning to the extremely flexible informal human reasoning.
Acknowledgement
I would like to thank Loizos Michael and Francesca Toni for their continued
collaboration on argumentation. This has been very useful in writing this paper.
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