Abstract
To think about inconsistencies involves reflecting on several basic notions widely used in order to talk about human knowledge and actions, such as negation, opposition, denial, assertion, truth, falsity, contradiction and incompatibility, just to name the more perspicuous ones. All of them are regularly used in natural language and for each one several definitions or conceptions have been proposed throughout the history of Western thought. That being so we tend to think that we have a good enough intuitive understanding of them. Yet a closer examination shows many ways in which “contradiction” and related words can be understood. Thus, a more precise definition would help to clarify their meaning and assist us to use them in a more appropriate manner. In this paper I will try to clarify these notions and thus make a terminological proposal. The general background will be the reflexion on paraconsistency. A main purpose will be to show that the confusion between contraries and contradictories—although they were clearly distinguished in the original square of opposition—is very common and it paves the way to the rejection of all forms of “inconsistencies” without making distinctions, and also to the wrong assumption that regarding all the main aspects the effects of contrary opposition are equivalent to the ones of contradictory opposition.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For example the glossary in Sainsbury says:
“contradictories: two propositions are contradictories iff it is logically impossible for both to be true and logically impossible for both to be false.” [13, p. 394]
- 2.
“What is logically impossible involves some kind of contradiction, […]” [13, p. 15].
- 3.
That is the denomination in Church [3].
- 4.
This is used sometime by Strawson [15, pp. 164 ff].
- 5.
Strawson [15, pp. 164 ff.], examining the question about the interpretation of the system of traditional logic, especially by means of modern predicate logic, addresses the problem of existential import and says that in the case of the particular statements (I and O) the only reasonable solution is that they do carry existential commitment, but for the case of the universal (A and E) he present it as a dilemma: “Either the A and E forms have existential import or they do not. If they do, one set of laws has to be sacrificed as invalid; if the do not, another set has to go. Therefore no consistent interpretation of the system as a whole, within the prescribed limits is possible.” (Strawson [15, p. 165]) He develops the dilemma in a more formal way maintaining that one option is to take the A and O statements just in terms of their standard formalization in the predicate logic, respectively ‘∼(∃x)(fx. ∼ gx)’ and ‘∼(∃x)(fx.gx)’, and the other is to conjoin these formalizations with “an assertion of the existence as far as the first ‘term’ is concerned (‘(∃x(fx)’)” [15, p. 165]. Then he shows that in the first option the rule for contradiction still holds, but the rules for contraries and subcontraries do not hold, while in the second option the rule for contraries is preserved but not for subcontraries and contradictories. Yet, to me what I think is more interesting is his conclusion about the later: “A and O are no longer contradictories, but only contraries; since, while both cannot be true for a given example, both may be false, in the case where the positively existential component of A is false. Similarly, E and I are only contraries.” [15, p. 166]
- 6.
For example in Spanish, ‘contradecir’—the equivalent to contradict—is defined in the «canonical dictionary», this way:
“contradecir (Del. lat. contradicere, -onis) Decir uno lo contrario de lo que el otro afirma, o negar lo que da por cierto.” (Real Academia Española: Diccionario de la Lengua Española. Madrid: Epasa-Calpe, 1992, p. 556).
It is almost the same as in NSOED definition 2, (translated it would be: “to say the contrary to what someone else asserts or to deny what he takes as true”). So both cases are clearly stated there.
- 7.
In the context of the square of opposition ‘kontradiktorishenGegensatz’ and ‘konträrenGegensatz’ are use for ‘contradictory opposition’ and ‘contrary opposition’, respectively [cf. Die Philosophie (Mannheim: DudenVerlag, 1985) p. 151]. Meanwhile, the verb ‘widersprechen’ is the term that normally will be used to translate ‘contradict’ and the noun ‘Widerspruch’ for ‘contradiction’, but also the term ‘Gegensatz’ is commonly used to express the confrontation between two statements, so it can be translated as ‘opposition’ but also as ‘contrariety’ as well as ‘contradiction’ (cf. Waibl, E./Herdina, P.: German Dictionary of Philosophical Terms, vol. 1 German-English. München: K. G. Saur/London: Routledge, 1997, p. 99). Similarly with related words as the verb ‘entgegengesetzten’, the adjective ‘entgegengesetzt’, and the noun ‘Entgegensetzung’. In fact, Hegel used ‘Gegensatz’ for what is known as his ‘antithesis’ (cf. ibid.).
- 8.
Following definitions like the one in Honderich 1995:
“term. A word or phrase denoting an individual or class, or the propositional component it expresses. Thus ‘John is a man’ contains two terms ‘John’ and ‘man’ (or ‘is a man’), denoting John and the set of men respectively. More generally, any word or phrase that determines the proposition expressed. In this sense, the above sentences contains the syncategorematic term ‘is’, which does not denote and individual or class.” [5, p. 869 (Entry written by Wayne A. Davis)]
- 9.
Horn [6, p. 268] presents a different proposal: what I have called ‘incompatible terms’, there are called “contraries (incompatibles)”; my ‘contrary terms’ are called “mediate (weak) contraries” (which are subdivided into “simple (reductive) contraries” and ‘”polar (absolute) contraries”, subdivision that I do not have; my ‘contradictory terms’ are called “immediate (strong, logical) contraries”. The main reason for such differences is that he thinks, following Aristotle, that terms cannot be contradictories, only propositions (cf. Ibid. p. 39). I will come back to this in the next chapter, but for the moment let me point out that, for the purposes of his chap. 5, he also uses contradictories for terms (cf. [6, p. 269]).
- 10.
It is interesting to see how it is presented: “Law of Trichotomy. Also called the law of comparability. In general, a division of entities into three sets that are pairwise disjoint (that is, non-overlapping) and exhaustive.” [4, p. 61] That is followed by specific definitions for the theory of real numbers and for set theory.
- 11.
The most relevant definitions in the NSOED are:
“Incompatible […] adj. 1 Incapable of existing together in the same person, opposed in character, discordant. (Foll. by with, [obsolete] to.) lME. […] c Of an item of equipment: unable to be used in conjunction with some other item. m20. 2 Unable to agree or be in harmony together, at variance. m16. […]” (NSOED [10, p. 1339]).
References
R. Audi (ed.), The Cambridge Dictionary of Philosophy, 1st edn. (Cambridge University Press, Cambridge, 1995) [2nd edn (1999)]
B.A. Brady, in Glossary of Logical Terms, vol. 5, ed. by Edwards (The Macmillan Company & The Free Press/Collier-Macmillan Limited, New York/London, 1967), pp. 58–77
A. Church, The history of the question of existential import of categorical propositions, in Logic, Methodology and Philosophy of Science, ed. by Y. Bar-Hillel (Nort-Holland, Amsterdam, 1965), pp. 417–424
M. Detlefsen, D.C. McCarty, J.B. Bacon, Logic from A to Z (Routledge, London/New York, 1999) [Consist of the separated publication of the “Glossary of logical and mathematical terms” of the Routledge Encyclopedia of Philosophy (Craig/Floridi, 1998)]
T. Honderich (ed.), Oxford Companion to Philosophy (Oxford University Press, Oxford, 1995)
L.R. Horn, A Natural History of Negation (The University of Chicago Press, Chicago, 1989)
J.J. Katz, The Philosophy of Language (Harper and Row, New York/London, 1966)
W. Marciszewski (ed.), Dictionary of Logic as Applied in the Study of Language (Martinus Nijhoff Publishers, The Hague, 1981)
T. Mautner (ed.), Dictionary of Philosophy (Penguin, London, 1997) [1st edn, Blackwell, Oxford, 1996]
NSOED, in New Shorter Oxford English Dictionary, 1st edn, ed. by L. Brown (editor-in-chief) (Oxford University Press, Oxford, 1933) [4th edn, 1993]
M. Sainsbury, Paradoxes, 1st edn. (Cambridge University Press, Cambridge, 1987) [2nd edn (1995), repr (1997)]
M. Sainsbury, Can rational dialetheism be refuted by considerations about negation and denial? Proc. Natl. Acad. Sci. U.S.A. 10, 215–228 (1997)
M. Sainsbury, Logical Forms. An introduction to Philosophical Logic, 1st edn. (Blackwell, Oxford, 1991) [2nd edn (2001)]
B.H. Slater, Paraconsistent logic? J. Philos. Logic 24, 451–454 (1995)
P.F. Strawson, Introduction to Logical Theory, 1st edn. (Methuen, London, 1952) [paperback edn (1963), repr (1967)]
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Andrés, B.M. (2017). The Many Faces of Inconsistency. In: Béziau, JY., Basti, G. (eds) The Square of Opposition: A Cornerstone of Thought. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-45062-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-45062-9_9
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-45061-2
Online ISBN: 978-3-319-45062-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)