Negation, Absurdity and Contrariety

Tennant, Neil

doi:10.1007/978-94-015-9309-0_10

Cited by 101 publications

(24 citation statements)

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“…Notice that this is a much stronger point than that made byTennant (1999), discussed above. Tennant argues that '0 = 1' cannot be used in a philosophically adequate definition of negation because derivations of negations in non-arithmetical discourses would violate relevance considerations.…”

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confidence: 73%

What Negation is not: Intuitionism and '0=1'

Cook¹,

Cogburn

2000

Analysis

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It is evident that the differences between intuitionistic logic (and mathematics) and its classical counterpart(s) can be traced back at least in part to their differing treatments of negation. The intuitionistic failures of double negation elimination and classical reductio ad absurdum are the most obvious facts supporting this. As a result, the definition and/or rules formulated for negation are crucial in comparing these competing logics. One quite common method of handling intuitionistic negation, however, turns out to be untenable.The method in question is the definition of intuitionistic negation in terms of '0 = 1' and the conditional. In Elements of Intuitionism Michael Dummett writes that 'surprisingly, negation is definable in intuitionistic arithmetic ' (1977: 35), giving essentially 1 the following definition:This definition of negation is common among intuitionists, and its origin can be traced to Heyting's first formulations (1930) of intuitionistic reasoning.Before giving our objections, it should be pointed out that others have argued against this same definition. For example, Neil Tennant objects to defining negation in terms of '0 = 1' because this definition makes negation dependent on arithmetic, raising the question of 'how one would justify inferential passage from patent absurdities in a non-arithmetical discourse to the supposedly primal absurdity 0 = 1' (1999) in accounting for the behaviour of negation. On Tennant's account, if P and Q are two contrary sentences of some non-arithmetic discourse, say 'The ball is red' and 'The ball is green', then we would need to be able to justify (possibly non-logical) inferences from P and Q to '0 = 1', otherwise we cannot conclude ¬Q (i.e. Q → 0 = 1) from P. It is far from clear how the justification of this inference would proceed, and Tennant observes that for a relevance logician (such as himself) no justification seems possible. This is a subtle philosophical argument 2 for rejecting this manner of 1 Dummett actually writes this as a biconditional instead of using = df .2 A related philosophical criticism has to do with the use of '0 = 1' in the definition. One wonders why we should not just define ¬A as (A → 0 = 3) or (A → 0 = 247) or (A → 47 = 396). None of these seem any less arbitrary than using '0 = 1', yet one may worry that they impose different meanings on the negation operator. The objections raised in this paper, however, demonstrate that all definitions of this sort are inadequate, and thus that these quibbles are irrelevant.

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confidence: 73%

What Negation is not: Intuitionism and '0=1'

Cook¹,

Cogburn

2000

Analysis

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“…Tennant proposes “a rule based account of negation” (Tennant, , 199). Its meaning is specified relative to “metaphysico‐semantical fact[s] of absurdity” (Tennant, , 202), or incompatibility, such as “ a is red and a is green”, “ a is here and a is over there simultaneously”, “ e is both earlier and later than f ”, “You and I are the same person”. Tennant generalises the notion of incompatibility to allow not only two, but any finite number of propositions to stand in that relation.…”

Section: How To Eliminate Negation?mentioning

confidence: 99%

“…Tennant generalises the notion of incompatibility to allow not only two, but any finite number of propositions to stand in that relation. Having arrived at incompatible propositions p 1 … p n in a deduction, we write ⊥ to mark the event:

(⊥^{T}) \frac{p_{1} \cdot \cdot \cdot p_{n}}{⊥}

The incompatibility of p 1 … p n “arises by virtue of what the sentences mean and various ways that we understand the world simply cannot be ” (Tennant, , 217). Speakers of a language grasp primitively that certain atomic propositions are incompatible with each other.…”

Section: How To Eliminate Negation?mentioning

confidence: 99%

An Argument for Minimal Logic

Kürbis

2019

Dialectica

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The problem of negative truth is the problem of how, if everything in the world is positive, we can speak truly about the world using negative propositions. A prominent solution is to explain negation in terms of a primitive notion of metaphysical incompatibility. I argue that if this account is correct, then minimal logic is the correct logic. The negation of a proposition A is characterised as the minimal incompatible of A composed of it and the logical constant ¬. A rule-based account of the meanings of logical constants that appeals to the notion of incompatibility in the introduction rule for negation ensures the existence and uniqueness of the negation of every proposition. But it endows the negation operator with no more formal properties than those it has in minimal logic.There is implanted in the human breast an almost unquenchable desire to find some way of avoiding the admission that negative facts are as ultimate as those that are positive. Russell

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“…A more significant reason why Rumfitt's choice of Smiley over NonContradiction and Reductio is surprising is that he devotes some discussion to the status of ⊥, which does not occur at all in the formalisation that uses Smiley. Rumfitt follows Neil Tennant's proposal (Tennant (1999)) and sees ⊥ not as a proposition or a speech act, but as a 'punctuation mark in the deduction' indicating a 'dead end' (Rumfitt (2000): 794).…”

Section: Two Formalisations Of Bilateral Logicmentioning

confidence: 99%

Some Comments on Ian Rumfitt’s Bilateralism

Kürbis

2016

J Philos Logic

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Ian Rumfitt has proposed systems of bilateral logic for primitive speech acts of assertion and denial, with the purpose of 'exploring the possibility of specifying the classically intended senses for the connectives in terms of their deductive use' (Rumfitt (2000): 810f). Rumfitt formalises two systems of bilateral logic and gives two arguments for their classical nature. I assess both arguments and conclude that only one system satisfies the meaning-theoretical requirements Rumfitt imposes in his arguments. I then formalise an intuitionist system of bilateral logic which also meets those requirements. Thus Rumfitt cannot claim that only classical bilateral rules of inference succeed in imparting a coherent sense onto the connectives. My system can be extended to classical logic by adding the intuitionistically unacceptable half of a structural rule Rumfitt uses to codify the relation between assertion and denial. Thus there is a clear sense in which, in the bilateral framework, the difference between classicism and intuitionism is not one of the rules of inference governing negation, but rather one of the relation between assertion and denial.

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Negation, Absurdity and Contrariety

Cited by 101 publications

References 0 publications

What Negation is not: Intuitionism and '0=1'

What Negation is not: Intuitionism and '0=1'

An Argument for Minimal Logic

Some Comments on Ian Rumfitt’s Bilateralism

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