FROM CLAMP TO CRAMP OF LOGIC: EXPLICATING THE CHAGRINS OFSYMBOLIC LOGIC. (pgs 30-47)


Anacletus Ogbunkwu
Department of Philosophy,
Godfrey Okoye University, Enugu.
franacletus@gmail.com.


Abstract
Though there is a general fanfare exalting symbolism as a basic appurtenance of logic and
characteristically fundamental to logic; this paper argues that symbolism rather cramps and
leads logic to a stampede. Using the methods of hermeneutics and logical syllogism, this
paper reviews the claims of symbolism as a clamp and fundamental appurtenance of logic.
The paper critically presents synopsis on understanding logic, symbolic logic and the tenets
of arguments surrounding symbolic logic. On appraisal, it is revealed that logic is an essential
need to humanity and the quintessence of humanness but symbolic logic denies humanity this
essential possession since it relativizes logic only to the initiates in symbolic logic. However,
this paper shows that even among the initiates, symbolic logic houses lots of inconsistencies,
and ambiguities as different symbols have been used by different logicians and systems of
logic in their claims that symbolic logic is needful for precision and elegance. Hence if
philosophy is only possible as logic following the claims of some philosophers and
symbolism is the fundamental appurtenance of logic, the implication is that it becomes an
impossibility for humanity everywhere to access logic and a difficulty for logic to fulfil its
objectives. It therefore holds that the fundamental objectives of logic towards sound
reasoning and argumentation are put on edge by symbolic logic and this is what the paper
refers as the chagrins of symbolic logic.
Key words: logic, symbolism, operators, chagrins, reason,
31
Introduction
There is a common assumption that the apogee of logicism is achieved in symbolic logic
while symbolic logic implies the science and method of representing logical expressions
through the use of symbols and variables, rather than in ordinary language. Hence symbolism
is a basic appurtenance of logic and characteristically fundamental to logic towards
facilitating logical ability1
. It is from this perspective that this paper refers to symbolic logic
as a clamp to logic that is, a strengthening brace to logic.
Nevertheless, this paper bears much concern on the inescapable reality before us that
symbolic logic mounts unbearable and devastating effects on the objectives of logic leading
to the cramp of logic. Thus these devastating effects inhibit and impede the actualization of
the ends of logic which is the quest for correct reasoning and sound argumentation. In
symbolic logic, these objectives are rather traded for precision and elegance.
Hence the task before us is the responsibility of explicating these inhibitions and
impediments which this paper refers to the chagrins of symbolic logic. Undoubtedly,
symbolism imposes heavy burden on logic in the process of transcription from ordinary
language to symbolic language. Unfortunately, during this process, logic is greeted with
ambiguity, inconsistency and lost of focus.
Our thesis in this essay is; in spite of the benefits of symbolic logic, its challenging effects are
burdensome minding the aims of logic. Hence this is a wakeup call on all logicians towards
finding ways to mitigate these afore mentioned challenges of symbolic logic.
Understanding Logic.
Logic is primarily an epistemological tool. Etymologically, logic is the anglicized form of the
Greek word, logikḗ {(λογική) meaning: “possessed of reason, intellectual, dialectical,
argumentative”} which has its root derived from logos {(λόγος) meaning: “word, discourse,
rule, thought, idea, argument, account, reason, or principle”}.2
The New Testament Gospel
recognizes logos as synonymous to God as used in the Gospel of John3
. This etymological
derivations put together appropriately defines logic as the principles of correct reasoning. It is
the method whoever wants to reason or argue correctly ought to follow but it is not within the
scope of logic to lure people into following these principles. Hence Frege states that it is the
task of logic to discover the laws of truth.4
Uduma adds, ‘these laws of thought must be laws
of correct reasoning’5
such that appraisal of reasoning becomes the subject matter of logic.
32
According to Hegel, logic is the form taken by the science of thinking in general6
.The subject
matter of logic is argument such that logic is indispensable to human existence. Quoting
Spencer, Uduma maintained that while birds can fly, only human beings can argue. Hence
argument for him, is the affirmation of our being. Like Spencer, he affirmed that human life
is directed by argumentation7
. It is the disposition to fundamental ordered action. Thus it is
the necessary condition for order and intelligibility in reality. Therefore, we boldly emphasize
here that, human thought process, actions and inactions are bye products of human reasoned
private arguments and judgements. Logic is indeed needful in life and existence. 8
Therefore, it is obvious that human life becomes wild and strange when one loses this
essential and distinguishing element of being human. Even the Christians recognize that in
creation, God made man different from animals just by the gift of reason. While he gave mere
instinct to the lower animals, he gave reason to man. Little wonder when one acts without
reason, he can be said to be inhuman or at best described as animal. Thus in such situation,
one may be said to have lost the quintessence of humanness. Uduigwomen sees logic as the
science which “helps us to weigh the merits and demerits of an action or decision before we
venture into it, and hence enables us to take a balanced action or decision. Instead of
engaging in endless controversies of trivial matters, it enables us to sift the evidence before
us”9
.
Logic can be said to be the pattern of thought found in everyday discourse of a people. It is in
this sense that one can conveniently talk about logic to have a cultural background. We agree
with Momoh as follows:
in everyday usage of natural language we talk of a person as being
logical if he is reasonable, sensible and intelligent; if he can
unemotionally and critically evaluate evidence or a situation; if he can
avoid contradiction, inconsistency and incoherence, or if he can hold a
point of view argue for and from it, summon counter-examples and
answer objections10
.
Logic is typically an element of culture. Whatever is judged, reasoned, thought or argued is
according to the categories of the judging, thinking or reasoning mind as given by the
environment. This mind is a product of a particular culture. Hence people’s background and
temperaments influence their logic and thought process. Just as the Westerners have their
ability to conduct their daily affairs following the givens of their environment, the Africans
33
too have the same ability as regulated by their immediate experience and world views. This
implies that the westerners, as well as the Africans are logical but their logic(s) is/are
products of their varied experiences.
Logic can be either formal or informal. The discussion on logic so far bothers on informal
logic whereby logic is interested in correct reasoning, right thinking and acts as agent of
meaningful living. It is the branch of logic whose task it is to develop non-formal standards,
criteria, procedures for the analysis, interpretation, evaluation, criticism and construction of
argumentation in everyday discourse.11 This form of logic is typical of African logic.
On the other hand, formal logic is concerned with specialties and specialization in logic. It
can also be referred as Aristotelian, mathematical, artificial or critical logic. This is typical of
western logic. The historical roots of logic go back to the work of Aristotle (384–322 BCE),
whose syllogism was the standard account of the validity of arguments. The ancient times till
19th century witnessed a wide acceptance of Aristotelian logic. Logic earlier this modern
period was championed by the Aristotelian method as contained in the Organon. Philosophers
and commentators after Aristotle grouped Aristotle’s six logical treatises into a manual they
called the Organon which is the Greek translation for “tool”. The Organon comprises the
following works of Aristotle: the Categories, On Interpretation, Prior Analytics, Posterior
Analytics, the Topics, and On Sophistical Refutations. These works give us a good
understanding of Aristotelian logic especially as it concerns; structure/rules of arguments and
syllogisms, logical structure of propositions, difference between induction and deduction, the
nature of scientific knowledge, basic fallacies, debate techniques, to mention but a few12
.
The basis of Aristotelian formal logic is anchored on his three fundamental laws of reason,
namely; the law of Identity, the law of Contradiction and the law of Excluded Middle. The
first law states that a thing is always equal to or identical with itself. The second law states
that a thing cannot be unequal or different from itself. Also, the third law continues the
former two laws; it states that if a thing is equal to itself, it cannot be unequal or different
from itself13. For example; if ‘y’ equals ‘z’, it cannot equal ‘non y’. Regarding the three laws
of thought, we think that Aristotle has made a giant stride in formal logic. Formal logic is
chiefly concerned with the processes of thinking and reasoning as well as the symbolic
expression of such process in verbal or written form.14
Understanding Symbolic Logic
34
Symbolic logic is the science and method of representing logical expressions through the use
of symbols and variables, rather than in ordinary language. This has the benefit of removing
the ambiguity that normally accompanies ordinary languages in order to give way to easier
operation of reason15. Symbolic logic can be referred as artificial language. This is the form
of logic which places precedence to logical forms than ordinary statements. There are many
systems of symbolic logic, such as classical propositional logic, first-order logic and modal
logic. Each may have separate symbols, or exclude the use of certain symbols.16
According to the analysis of C. I. Lewis, the three characteristics of symbolic logic are:
(1) The use of symbols to stand for concepts, rather than ordinary language
(2) The use of the deductive method.
(3) The use of variables.17
For the reason of the difficulties experienced in the use of natural language in logic, symbolic
logic claims to offer a better way forward. Hence Copi stated as follows;
the words used may be vague or equivocal, the construction of the argument
may be ambiguous, metaphors or idioms may confuse or mislead, emotional
appeals may distract … to avoid these difficulties, and thus move directly to
the logical heart of an argument, logicians construct an artificial symbolic
language, free of linguistic defects. With symbolic language we can
formulate an argument with precision18
.
According to Alfred Whitehead, symbols facilitate our logical ability19. Quite unlike
Aristotelian logic that had syllogism as its end in logical activity, modern logic has logical
connectives accounting for the internal structure of propositions and arguments.20 Hence
according to Uduma, ‘the development of symbolic logic is undoubtedly the most significant
in the two thousand years of logic, and arguably, one of the most important events in human
intellectual history.21
For Uduma, “symbolic reasoning is reasoning in a large scale with instruments appropriate to
such wholesale undertakings”22. Hence he argued that the employment of special symbols in
logic is both for practical convenience and logical necessity23
.
35
Gorge Boole is said to be the father of symbolic logic. The discipline of symbolic logic
exploded in complexity as techniques of algebra were applied to issues of logic in the work of
George Boole (1815–1864), Augustus de Morgan (1806–1871), Charles Sanders Peirce
(1839–1914), and Ernst Schroder (1841–1902) in the nineteenth century24. They applied the
techniques of mathematics to represent propositions in arguments hence treating the validity
of arguments like equations in applied mathematics25
.
Furthermore, connections between mathematics and logic developed into the twentieth
century with the work of Gottlob Frege (1848–1925) and Bertrand Russell (1872–1970), who
used techniques in logic to study mathematics. Their goals were to use the newfound
precision in logical vocabulary to give detailed accounts of the structure of mathematical
reasoning, in such a way as to clarify the definitions that are used, and to make fully explicit
the commitments of mathematical reasoning. This is manifest in the Principia Mathematica
(1912) of Russell and Alfred North Whitehead.
Logical Symbols
In logic, a set of symbols is commonly used to express logical representation.26 The following
table presents several logical symbols, their name and meaning, and some relevant notes. It
should be noted that different symbols have been used by different logicians and systems of
logic. Nevertheless, for the sake of clarity, the column on the left hand side shows the
symbols while the middle column shows the meanings or names of the symbols and the right
hand side of the column shows further notes on the symbols with alternative symbols or other
commonly-used symbols27
.
Symbol Meaning Notes
Operators (Connectives)
¬ negation (NOT) The tilde ( ˜ ) is used in alternative.
∧ conjunction (AND)
The ampersand ( & ) or dot ( · ) are also used
in alternative.
∨ disjunction (OR)
This is the inclusive disjunction, equivalent to
and/or/or both in English.
⊕ exclusive disjunction
(XOR)
⊕ means that only one of the connected
propositions is true, equivalent to either…or
36
but not both. Sometimes ⊻ is used.
|
alternative denial
(NAND)
Means “not both”. Sometimes written as
‘↑’
↓ joint denial (NOR) Means “neither/nor”.
→ conditional (if/then)
Many logicians use the symbol ⊃ instead. This
is also known as material implication.
↔ biconditional (iff)
(Triple bar)
Means “if and only if” ≡. This is also used for
material equivalence.
Quantifiers
∀ universal quantifier
Means “all or for all”, so ∀xPx means that Px
is true for every x.
∃
existential quantifier
Means “there exists or indicates that one or
two or more unspecified individuals have
property”. Hence ∃xPx means that Px is true
for at least one x.
Relations
⊨ implication α ⊨ β means that β follows from α
≡ equivalence
Also ⇔. Equivalence is two-way implication,
so α ≡ β means α β and β α.
⊢ probability
Shows provable inference. α β means that
from α we can prove that β.
∴ therefore
Used to signify the conclusion of an argument.
Usually taken to mean implication, but often
used to present arguments in which the
premises do not deductively imply the
conclusion.
⊩ forces
A relationship between possible worlds and
sentences in modal logic.
Truth-Values
⊤ tautology
May be used to replace any tautologous
(always true) formula.
⊥ contradiction May be used to replace any contradictory
37
(always false) formula. Sometimes “F” is
used.
Parentheses
( ) parentheses
Used to group expressions to show precedence
of operations. Square brackets [ ] are
sometimes used to clarify groupings.
Set Theory
∈ membership
Denotes membership in a set. If a ∈ Γ, then a
is a member (or an element) of set Γ.
∪ union
Used to join sets. If S and T are sets of
formula, S ∪ T is a set containing all members
of both.
∩ intersection
The overlap between sets. If S and T are sets
of formula, S ∩ T is a set containing those
elemenets that are members of both.
⊆ subset
A subset is a set containing some or all
elements of another set.
⊂ proper subset
A proper subset contains some, but not all,
elements of another set.
= set equality
Two sets are equal if they contain exactly the
same elements.
∁ absolute complement
∁(S) is the set of all things that are not in the
set S. Sometimes written as C(S), S or SC.

  1. Does everyone have the knowledge of symbolic logic? A no answer
    here implies that only the privileged initiates in symbolism who have
    knowledge of symbolic logic can think logically, communicate and act
    with logic. This cannot be true!
  2. Do the initiate symbolic logicians think with formal language or
    symbolic logic before they act? Do they run their daily affairs with
    symbolic logic? Definitely the answer ought to be in the negative!
    Natural language logic suffices here.
  3. Was the entire world non critical or non reflective before the advent of
    Aristotelian logic and the symbolic logic? Definitely the answer is in
    the negative. Even before Aristotle himself, man was critical and
    reflective. This was why he claimed that the distinguishing
    characteristic of man is his rationality.
    It is a bogus assumption and futile expectations to presume that my grandmother in the
    village cannot think or act logically because she is not a symbolic logician. I sit down here to
    imagine the confusion in her head when I go home and instead of telling her:
    mama this drug makes people powerful and beautiful
    you have taken the drug
    therefore you are powerful and beautiful’
    I tell her
    P ∧ B
    Or I tell her P . B
    Here it is obvious that symbolic logic is not for everyone but for the initiates in
    symbolism. Hence here, this paper is not denying the importance of symbolic logic but
    only proves that it is relative only to the initiates in symbolism.
    43
    Conclusion
    The paper has made dogged effort towards giving a good understanding of logic, symbolic
    logic and the tenets of arguments surrounding symbolic logic as a necessity in logic. The
    paper has emphasized logic as the principle of correct reasoning. Also, symbolic logic has
    been defined as the science and method of representing logical expressions through the use of
    symbols and variables, rather than in ordinary language.
    According to Uduma, ‘the development of symbolic logic is undoubtedly the most significant
    in the two thousand years of logic, and arguably, one of the most important events in human
    intellectual history.36 There is a general claim that the apogee and epicentre of logic lies in
    symbolic logic being one of the most recent developments in the field of logic.
    Nevertheless, our logic in this paper is that symbolism has not completely solved the
    problems it claims to solve in the field of logic. It rather creates more problems to logic than
    it solves. Symbolic logic claimes to be fundamental to logic but warrants lost of focus on the
    objectives of logic from correct reasoning and sound argumentation to precision and
    elegance.
    If the aims of logic as clarity of expression, avoidance of ambiguities and contradictions can
    be effectively conducted in natural languages, symbolic logic can therefore be said to be
    second hand logic. Thus the work of logic is already achieved before the application or
    translation to symbolism. More so even when this translation or transcription to symbolic
    logic is made, logic in most cases becomes ambiguous and vague following the fact that
    different symbols have different meanings and applications.
    If logic is the process of correct reasoning, it is not proven anywhere that anyone had ever
    reasoned with symbolic logic. We rather reason in ordinary language then re-write our reason
    to symbolism. Similarly, if symbolism is a necessary condition to logic, the non initiates in
    symbolism would be illogical in reasoning. This is not the case because those who do not
    know the formal or artificial language of logic still exercise good principles of correct
    reasoning and sound argumentation. Hence we claim that in spite of the benefits of symbolic
    logic, its challenging effects are burdensome minding the aims of logic.
    44
    End Notes
    1
    Alfred North Whitehead, An Introduction to Mathematics, (New York: H.
    Holt and Company publishers, 1861), p.1911.
    2
    Liddell & Scott, “Logic” http://www.iep.utm.edu/aris-log/ accessed on
    20/4/17
    3
    John 1.1ff
    4
    Liddell & Scott, “Logic” http://www.iep.utm.edu/aris-log/ accessed on
    20/4/17
    45
    5
    Uduma Orji Uduma, “Explicating the Formal Nature of Logic”, in The
    Nigerian Journal of Philosophy, Vol., 23, No. 2, (Lagos: University Press, 2009). P.
    127.
    6
    Georg Wilhelm Friedrich Hegel; The Science of Logic, (Cambridge:
    University Press, 2010), p. 24
    7
    Uduma O. Uduma, “Can There Be An African Logic”? in Uduigwomen A.,
    From Footmarks to Landmarks on African Philosophy, 2nd ed., (Lagos: Obaroh &
    Ogbinaka Pub., Ltd, 1997), p. 273
    8
    Copi, Irving. Introduction to Logic. 6th (ed). (New York: Macmillan, 1982).
    P.vi
    9
    Uduigwomen, A.F., How to Think: An Introductory Text on Logic,
    Argumentation and Evidence. (Aba: A.A.U. Vitalis Books Co., 2003), p.19
    10 Momoh, C.S. (ed) The Substance of African Philosophy. (Auchi: African
    Philosophy Projects Publications, 1989), P.174
    11 Ralph H. Johnson, The Rise of Informal Logic, (London: University Press,
    2014), p. 13.
    12James Fieser, Internet Encyclopadia of Philosophy,
    http://www.iep.utm.edu/eds/ accessed on May 15, 2017.
    13 John Venn, Symbolic Logic, (Cambridge: Cambridge University Press,
    1990), p.43
    14 Corpi, I.M. and Cohen, C. Introduction to Logic. (USA: Macmillan
    Publishers, 1994), p.23.
    15 http://study.com/academy/lesson/symbolic-logic-definition-examples.html
    accessed on 01/12/2017
    16 Franklin, C. L. (1889). “On some characteristics of symbolic logic”. In The
    American Journal of Psychology, 2(4), 543-567.
    17 https://proofwiki.org/wiki/Definition:Symbolic_Logic, 05/11/2017
    46
    18 Irving Copi and Carl Cohen, Introduction to Logic, 11th edi., (Singapore:
    Pearson Education, 2002), p. 299
    19 Alfred North Whitehead, An Introduction to Mathematics, 1911,
    20 Irving Copi and Carl Cohen, Introduction to Logic, 11th edi., (Singapore:
    Pearson Education, 2002), p.300
    21 Uduma Oji Uduma, Introduction to Logic and the Fundamentals of Formal
    Reasoning, 2nd ed., Accra Ghana: Africa Analytica Publishers, 2015, 237.
    22 Uduma Oji Uduma, Introduction to Logic and the Fundamentals of Formal
    Reasoning, 2nd ed., Accra Ghana: Africa Analytica Publishers, 2015, 243.
    23 Uduma Oji Uduma, Introduction to Logic and the Fundamentals of Formal
    Reasoning, 2nd ed., Accra Ghana: Africa Analytica Publishers, 2015, p.243.
    24 Ewald, William, ed. 1996. From Kant to Hilbert: A Source Book in the
    Foundations of Mathematics. (Oxford: Oxford University Press, 1996), p. 132.
    25 Alfred North Whitehead, An Introduction to Mathematics, 1911
    26 https://en.wikipedia.org/wiki/List_of_logic_symbols, 09/01/2018
    27 http://www.philosophy-index.com/logic/symbolic/, 09/01/2018
    28 Copi, Irving. Introduction to Logic. 6th (ed). (New York: Macmillan, 1982).
    P.vi
    29 http://study.com/academy/lesson/symbolic-logic-definition-examples.html,
    09/01/2018
    30 Alfred North Whitehead, An Introduction to Mathematics, 1911,
    31 Irving Copi and Carl Cohen, Introduction to Logic, 11th edi., Singapore:
    Pearson Education, 2002, p. 299
    32 Uduma Oji Uduma, Introduction to Logic and the Fundamentals of Formal
    Reasoning, 2nd ed., Accra Ghana: Africa Analytica Publishers, 2015, p.243.
    33 https://en.wikipedia.org/wiki/List_of_logic_symbols, 09/01/2018
    34 C. S. Momo, “The Logic Question in African Philosophy” in C. S. Momoh
    ed., The Substance of African Philosophy, (Auchi: African Philosophy Projects
    47
    Publications), p. 23 quoted in Uduma, O. Uduma, “Logic As An Element of Culture:
    In Defense of Logic in African Traditional Thought”, P. 187.
    35 Uduma, O. Uduma, “Logic As An Element of Culture: In Defense of Logic
    in African Traditional Thought”, p.184
    36 Uduma Oji Uduma, Introduction to Logic and the Fundamentals of Formal
    Reasoning, 2nd ed., (Accra Ghana: Africa Analytica Publishers, 2015), p. 237.

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