The law of non-contradiction

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Pre-Socratics and Sophists

While there is no clear demarcation of a time period of the pre-Socratics but approximately 6th and 5th century BCE (Curd 2016) and the Sophists of the 5th and 4th century BCE (Struck 2009) , the importance for this thesis is that they both predate Aristotle at 384 BCE (Shields 2016). As Aristotle, in Metaphysics Gamma (1993) responded strongly to at least two philosophers from this prior period, they will form the focus of this initial historical section. However, “historical” might be too strong. What this section and others do is analyse perceptions of the philosophers and consider what their original intents were or might be on a charitable interpretation. Part of the methodological problematic is a lack of original source texts coupled with the degree of interpretation apparent in the paraphrases. Heraclitus (2013:1238) is an obvious target given the controversial nature of his claim about stepping in the same river twice. Some characterise his views into three separate principles: the theory of flux, the theory of the unity of opposites, and his monist ontology (Graham 2011). Our concerns are the first two.

The law of non-contradiction

In Metaphysics Gamma (1993), Aristotle covers a number of logical and metaphysical topics. One of these topics has a unique character. In particular, he addresses “… the firmest of all principles …” and a “… principle about which it is impossible to be in error …” which is the law of non-contradiction (LNC) (Aristotle 1993:1421). In Metaphysics Gamma, Aristotle states that this principle can be demonstrated through the method of refutation but not directly proven (1993:1006a: 1–10). There are three different formulations of the LNC found in Metaphysics Gamma: ontological law of non-contradiction, logical law of noncontradiction, and psychological law of non-contradiction, each is defined in turn:6 From 1005b, the ontological formulation of the LNC (OLNC): “For the same thing to hold good and not to hold good simultaneously of the same thing and in the same respect is impossible” (1993:19–21).


Kierkegaard’s religious background created intellectual and emotional conflicts that he sought to resolve (McDonald 2017). He labelled many of these “paradoxes”, for instance in the Philosophical Fragments (1936:46) he writes about the supreme paradox of all thought that humankind engages in: “The supreme paradox of all thought is the attempt to discover something that thought cannot think. This passion is at bottom present in all thinking, even in the thinking of the individual, in so far as in thinking he participates in something transcending himself. But habit dulls our sensibilities, and prevents us from perceiving it.” In his famous work Fear and Trembling (1986), Kierkegaard thinks through the Abraham and Isaac story from Genesis 22, where God calls upon Abraham to sacrifice his son, Isaac. The story Kierkegaard tells in the Problemata is one of anguish (1986). How can God ask him to do this when he has waited so long for a son? How can his love for God and his son be tested at the same time? For Abraham, he is being asked to sacrifice his son to God, yet, society will see him as a murderer. How can this one act be looked upon in so many conflicting ways? Simply, it is a tale of irrationality and one of a man’s anguish.

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Strawson was an influential analytic philosopher in the last century. His work included an interpretation of The Critique of Pure Reason by Immanuel Kant which he called The Bounds of Sense (1975). He also wrote about the role of free will in Freedom and Resentment (1974), personhood in Individuals (1964), and extensively on the philosophy of logic (1952). In particular, his Introduction to Logical Theory lays out a formal system along with relevant thoughts about the relationship between ordinary language and formal logic (1952). He begins this work by writing about inconsistency in detail in both respects. Strawson (1952:15) uses the linguistic distinction of first-order and secondorder languages. Ordinary language is first-order; formal logic is second-order. His point is that we construct the ideal of a formal language from an ordinary language (1952:15). In one sense, formal logic is a caricature of logic in ordinary language because it simplifies and minimises the relationships between ideas without the subtleties of ordinary language, particularly in context and use.

Table of Contents :

  • Declaration
  • Abstract (English)
  • Abstract (isiZulu)
  • Abstract (seSotho)
  • Curriculum Vitae
  • Acknowledgements
  • Table of Contents
  • Introduction
  • Background
  • Objectives
  • Research questions
  • Aims
  • Unique contributions of the research
  • Methodology
  • Structural outline
  • Chapter 1: A brief history of inconsistency
    • 1.1 Introduction
    • 1.2 Pre-Socratics and Sophists
    • 1.3 Aristotle
      • 1.3.1 The law of non-contradiction
      • 1.3.2 Aristotle on the Pre-Socratics
      • 1.3.3 Contemporary criticisms of Aristotle on the LNC
      • Cohen
      • Łukasiewicz
      • Dancy
      • Aristotle’s critics: some reflexions
    • 1.4 Historical paradoxes
      • 1.4.1 The sorites paradox
      • 1.4.2 The liar paradox
      • 1.4.3 The heterological paradox
      • 1.4.4 Linguistic paradox of material implication
    • 1.5 Other influential ideas and philosophers
      • 1.5.1 Arnauld and Nicole
      • 1.5.2 Descartes
      • 1.5.3 Hegel
      • 1.5.4 The existentialists
      • Kierkegaard
      • Camus
      • 1.5.5 Analytic philosophy
      • Strawson
      • Paraconsistency
    • 1.6 Synthesis and analysis
    • 1.7 Concluding thoughts
  • Chapter 2: Propositional deductive logic
    • 2.1 Introduction
    • 2.2 Formal systems, logical systems, and languages
    • 2.3 Propositional deductive logic (semantic)
      • 2.3.1 System considerations, well-formed formulas, and translations
      • 2.3.2 Truth tables for the connectives
      • 2.3.3 WFF evaluation: tautology, contingency, and contradiction
      • 2.3.4 Argument evaluation: validity and invalidity
    • 2.4 Propositional deductive logic (syntactic)
      • 2.4.1 Propositional proofs
      • 2.4.2 Simplification rules
      • 2.4.3 Implication rules
      • 2.4.4 Proofs
    • 2.5 Inconsistency and contradiction in propositional logic
      • 2.5.1 Negation
      • 2.5.2 Inconsistency in arguments
        • Semantic: ex falso quodlibet
        • Syntactic: ex contradictione quodlibet
  • 2.6 Concluding thoughts
  • Chapter 3: Critiquing propositional deductive logic as a logical system
    • 3.1 Introduction
    • 3.2 Classical and non-classical logics
    • 3.3 Contra Quine
    • 3.4 Propositional logic
      • 3.4.1 Principles of inference versus implicative transformation rules
      • Problems with modus ponens and modus tollens
      • 3.4.2 Contra EFQ
      • 3.4.3 Contra ECQ
      • 3.4.4 Monotonicity, non-monotonicity, and constrained conclusions
      • 3.4.5 Contra-translations as representative of natural language arguments
    • 3.5 Negation, translation, and WFFs
    • 3.6 Concluding thoughts
  • Chapter 4: The theory and justification of the semantic conception of inconsistency
  • Chapter 5: Critiquing the pedagogy of propositional logic as critical reasoning
  • Chapter 6: Applying the semantic conception of inconsistency to the pedagogy of critical reasoning
  • Chapter 7: Conclusions and implications

The theory and pedagogy of semantic inconsistency in critical reasoning

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