No-no paradox

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The No–no paradox is a distinctive paradox belonging to the family of the semantic paradoxes (like the Liar paradox). It derives its name from the fact that it consists of two sentences each simply denying what the other says.

Paradox statement that apparently contradicts itself and yet might be true

A paradox is a statement that, despite apparently valid reasoning from true premises, leads to an apparently-self-contradictory or logically unacceptable conclusion. A paradox involves contradictory-yet-interrelated elements that exist simultaneously and persist over time.

Semantics is the linguistic and philosophical study of meaning, in language, programming languages, formal logics, and semiotics. It is concerned with the relationship between signifiers—like words, phrases, signs, and symbols—and what they stand for in reality, their denotation.

In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that he or she is lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar is lying. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.

Contents

History

A variation on the paradox occurs already in Thomas Bradwardine’s Insolubilia. [1] The paradox itself appears as the eighth sophism of chapter 8 of John Buridan’s Sophismata. [2] Although the paradox has gone largely unnoticed even in the course of the 20th-century revival of the semantic paradoxes, it has recently been rediscovered (and dubbed with its current name) by the US philosopher Roy Sorensen, [3] and is now appreciated for the distinctive difficulties it presents. [4]

Thomas Bradwardine 14th-century Archbishop of Canterbury and theologian

Thomas Bradwardine was an English cleric, scholar, mathematician, physicist, courtier and, very briefly, Archbishop of Canterbury. As a celebrated scholastic philosopher and doctor of theology, he is often called Doctor Profundus.

Formulation

The notion of truth seems to be governed by the naive schema:

Truth philosophical concept

Truth is most often used to mean being in accord with fact or reality, or fidelity to an original or standard. Truth is also sometimes defined in modern contexts as an idea of "truth to self", or authenticity.

(T): The sentence ' P ' is true if and only if P

(where we use single quotes to refer to the linguistic expression inside the quotes). Consider however the two sentences:

(N1): (N2) is not true
(N2): (N1) is not true

Reasoning in classical logic, there are four possibilities concerning (N1) and (N2):

Classical logic is the intensively studied and most widely used class of logics. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world.

  1. Both (N1) and (N2) are true
  2. Both (N1) and (N2) are not true
  3. (N1) is true and (N2) is not true
  4. (N1) is not true and (N2) is true

Yet, possibilities 1. and 2. are ruled out by the instances of (T) for (N1) and (N2). To wit, possibility 1. is ruled out because, if (N1) is true, then, by (T), (N2) is not true; possibility 2. is ruled out because, if (N1) is not true, then, by (T), (N2) is true. It would then seem that either of possibilities 3. and 4. should obtain. Yet, both of those possibilities would also seem repugnant, as, on each of them, two perfectly symmetrical sentences would mysteriously diverge in truth value.

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.

Discussion

Generally speaking, the paradox instantiates the problem of determining the status of ungrounded sentences that are not inconsistent. [5] More in particular, the paradox presents the challenge of expanding one’s favourite theory of truth with further principles which either express the symmetry intuition against possibilities 3. and 4. [6] or make them acceptable in spite of their intuitive repugnancy. [7] Because (N1) and (N2) do not lead to inconsistency, a certain strand in the discussion of the paradox has been willing to assume both the relevant instances of (T) and classical logic, thereby deriving the conclusion that either possibility 3. or possibility 4. holds. [8] Such conclusion has in turn been taken to have momentous consequences for certain influential philosophical theses. Consider, for example, the thesis of truthmaker maximalism:

In classical deductive logic, a consistent theory is one that does not entail a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory is consistent if there is no formula such that both and its negation are elements of the set of consequences of . Let be a set of closed sentences and the set of closed sentences provable from under some formal deductive system. The set of axioms is consistent when for no formula .

(TM): If a sentence is true, there is something that makes it true [9]

If, as per possibilities 3. and 4., one of (N1) or (N2) is true and the other one is not true, then, given the symmetry between the two sentences, it might seem that there is nothing that makes true whichever of the two is in fact true. If so, (TM) would fail. [10] These and similar conclusions have however been contested by other philosophers on the grounds that, as evidenced by Curry's paradox, joint reliance on (T) and classical logic might be problematic even when it does not lead to inconsistency. [11]

Related Research Articles

In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic.

In philosophy, vagueness refers to an important problem in semantics, metaphysics and philosophical logic. Definitions of this problem vary. A predicate is sometimes said to be vague if the bound of its extension is indeterminate, or appears to be so. The predicate "is tall" is vague because there seems to be no particular height at which someone becomes tall. Alternately, a predicate is sometimes said to be vague if there are borderline cases of its application, such that in these cases competent speakers of the language may faultlessly disagree over whether the predicate applies. The disagreement over whether a hotdog is a sandwich suggests that “sandwich” is vague.

Contradiction logical incompatibility between two or more propositions

In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "One cannot say of something that it is and that it is not in the same respect and at the same time."

Moore's paradox concerns the apparent absurdity involved in asserting a first-person present-tense sentence such as, "It's raining, but I don't believe that it is raining" or "It's raining but I believe that it is not raining." The first author to note this apparent absurdity was G. E. Moore. These 'Moorean' sentences, as they have become known, are paradoxical in that while they appear absurd, they nevertheless:

  1. Can be true,
  2. Are (logically) consistent, and moreover
  3. Are not (obviously) contradictions.
Sorites paradox paradox that, if ① one million grains of sand is a heap of sand and ② a heap of sand minus one grain is still a heap, then it follows that one grain of sand is a heap

The sorites paradox is a paradox that arises from vague predicates. A typical formulation involves a heap of sand, from which grains are individually removed. Under the assumption that removing a single grain does not turn a heap into a non-heap, the paradox is to consider what happens when the process is repeated enough times: is a single remaining grain still a heap? If not, when did it change from a heap to a non-heap?

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.

Truthmaker theory is "the branch of metaphysics that explores the relationships between what is true and what exists."

Dialetheism is the view that there are statements which are both true and false. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called "true contradictions", dialetheia, or nondualisms.

A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences.

In the Middle Ages, variations on the liar paradox were studied under the name of insolubilia ("insolubles").

Fitch's paradox of knowability is one of the fundamental puzzles of epistemic logic. It provides a challenge to the knowability thesis, which states that every truth is, in principle, knowable. The paradox is that this assumption implies the omniscience principle, which asserts that every truth is known. Essentially, Fitch's paradox asserts that the existence of an unknown truth is unknowable. So if all truths were knowable, it would follow that all truths are in fact known.

The paradoxes of material implication are a group of formulae that are truths of classical logic but are intuitively problematic.

Sophismata in medieval philosophy are difficult or puzzling sentences presenting difficulties of logical analysis that must be solved. Sophismata-literature grew in importance during the thirteenth and fourteenth centuries, and many important developments in philosophy occurred as a result of investigation into their logical and semantic properties.

Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to be termed either philosophy of logic or philosophical logic if no longer simply logic.

Pinocchio paradox

The Pinocchio paradox arises when Pinocchio says "My nose grows now" and is a version of the liar paradox. The liar paradox is defined in philosophy and logic as the statement "This sentence is false." Any attempts to assign a classical binary truth value to this statement lead to a contradiction, or paradox. This occurs because if the statement "This sentence is false" is true, then it is false; this would mean that it is technically true, but also that it is false, and so on without end. Although the Pinocchio paradox belongs to the liar paradox tradition, it is a special case because it has no semantic predicates, as for example "My sentence is false" does.

The following is a list of works by philosopher Graham Priest.

The knower paradox is a paradox belonging to the family of the paradoxes of self-reference. Informally, it consists in considering a sentence saying of itself that it is not known, and apparently deriving the contradiction that such sentence is both not known and known.

References

  1. ↑ Bradwardine, T. (1970), Insolubilia, pp. 304–305, in Roure, M.-L. (1970). ‘La problématique des propositions insolubles au XIIIe siècle et au début du XIVe, suivie de l’édition des traités de W. Shyreswood, W. Burleigh et Th. Bradwardine’, Archives d’Histoire Doctrinale et Littéraire du Moyen Âge37, pp. 205–326.
  2. ↑ Buridan, J. (2001), Summulae de Dialectica, tr. G. Klima, New Haven: Yale University Press, p. 971.
  3. ↑ Sorensen, R. (2001), Vagueness and Contradiction, Oxford: Oxford University Press.
  4. ↑ Greenough, P. (2011), 'Truthmaker gaps and the no-no paradox', Philosophy and Phenomenological Research82, pp. 547–563.
  5. ↑ Herzberger, H. (1970), 'Paradoxes of grounding in semantics', The Journal of Philosophy67, pp. 145–167
  6. ↑ Priest, G. (2005), 'Words without knowledge', Philosophy and Phenomenological Research71, pp. 686–694.
  7. ↑ Sorensen, R. (2001), Vagueness and Contradiction, Oxford: Oxford University Press, pp. 165–184.
  8. ↑ Armour-Garb, B. and J. Woodbridge (2006), 'Dialetheism, semantic pathology, and the open pair', Australasian Journal of Philosophy84, pp. 395–416.
  9. ↑ Armstrong, D. (2004), Truth and Truth-Makers, Cambridge: Cambridge University Press.
  10. ↑ Sorensen, R. (2001), Vagueness and Contradiction, Oxford: Oxford University Press, p. 176.
  11. ↑ López de Sa, D. and E. Zardini (2007), 'Truthmakers, knowledge and paradox', Analysis67, pp. 242–250.
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