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...he real point is simply that the speaker is the one to judge whether the substitution of S’ for S in the present context will forward his present or evolving program of activity to his satisfaction» (=-=[56]-=-, p. 160; emphasis added). Differences among different supporters of a logic would not then concern as to how a certain informal operator should be captured. Indeed, each different paraphrase cannot b...

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...a compatibility in a third - integrative - way. To this aim, he developed the idea of a pragmatic analysis of sentences that was originally suggested by Frege [27, 28] and then refined by Reichenbach =-=[65]-=-. According to this conception, every sentence can be analyzed in terms of two distinct linguistic components (or features), each of them having different semiotic roles: the pragmatic mood and the ra...

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... just a change of demarcation, either, between what to call logical truth and what to call extra-logical truth. It is a question rather of outright rejection of part of our logic as not true at all» (=-=[58]-=-, pp. 80-81; emphasis added). 2.2 Quine on alternative logics 2.2.1 Two kinds of conflicts As one would expect, the first attempts to argue for a revision of some classical principle or, in Carnap’s w...

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...is a] mere imaginary hook, from which the occurrences are supposed to hang [...] But when we take away the properties, and try to imagine the substance by itself, we find that there is nothing left» (=-=[69]-=-, p. 193). A suggestive attempt to individuate the essence of an operator, due to Priest, is worth mentioning: «Someone who rejects classical logic, say a paraconsistent logician, need not deny that t...

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... Philo there were disputes over the nature of conditionals, which were so long-lasting that «Callimachus wrote an epigram saying, ’Even the crows on the roofs crow about the nature of conditionals’» (=-=[35]-=-, p. 128). On the other hand, however, there are reasons to be sceptic about the relevance that Priest’s distinction has for Quine’s challenge. First of all, it seems deeply unclear how atheoryoflogic...

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...ereas this was considered in disagreement with the view according to which mathematical results are products of a construction-based activity [12]; Later, intuitionistic logic was defended by Dummett =-=[22, 23, 24]-=-, on the basis that it is the only logic in agreement with what he considered the necessary constraints for any theory of meaning, namely the so-called “principle of manifestability”; -Quantumlogic,as...

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...dely proposed in conflict with CL, viz., as rivals. They are known as deviant (or alternative) logics and constitute a non-conservative critical logical response to orthodox logic. According to Haack =-=[31]-=-, a deviant logic can in first instance be described as a system such that the class of its wffs coincides with the class of the wffs of CL, but such that the class of its theorems/valid inferences di...

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...tic assignment function -throughtherecursiveclauses-isolatesthepropertiesthatanynotionoftruth must satisfy. 14The rules are only analogous to the rules given in Heyting’s definition of “truth” for IL =-=[33]-=-. In Dalla Pozza’s words, they are intuitionistic-like rules (notice that they indeed do not mention the constraint that the radical must be atomic).CHAPTER 4. LOGICAL PLURALISM 63 his methods clearl...

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...ereas this was considered in disagreement with the view according to which mathematical results are products of a construction-based activity [12]; Later, intuitionistic logic was defended by Dummett =-=[22, 23, 24]-=-, on the basis that it is the only logic in agreement with what he considered the necessary constraints for any theory of meaning, namely the so-called “principle of manifestability”; -Quantumlogic,as...

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...clear only to the extent that we know how to divide up such firmly held principles into those that are “meaning constitutive" or “analytic" and those which aren’t, and this is notoriously difficult» (=-=[25]-=-, p. 176). Quine has anyway an advantage over the others, for he can tolerate the absence of a clear idea of “meaning” better than the supporter of the weak notion. Quine’s reply to Field could be ind...

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...which contain only atomic radicals, endowed with one of the usual sets of axioms for IL in assertive disguise (e.g., the axiom ((s ) ), where and only contain atomic radicals). A suitable 8See, e.g., =-=[68]-=-,[16],[45],[11].CHAPTER 4. LOGICAL PLURALISM 57 rule for the pragmatic renderings of modus-ponens is of course included in both calculi ( ` , `( ! ) ` for ACPC; , for AIPC, where and only contain ato...

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...t trivial (thus preventing to derive every sentence). 2 For example, Bohr’s model of the atom was inconsistent with Maxwell’s equations, but no trivialization was conceded on the whole; - Fuzzy logic =-=[77]-=- rejected the Principle of Bivalence on the basis that it does not allow for the treatment of sentences containing vague expressions. The peculiar character of deviant systems is a bit clearer now. Th...

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...-cluster concepts are constituted not by a bundle of properties [...] but by a cluster of laws [and any of them] can be abandoned without destroying the identity of the law-cluster concept involved» (=-=[53]-=-, pp. 50-52). Such an identity would be enough to allow for a Gc-scenario and for the refinement of a logic, which would be thus able to individuate a core meaning of the operators, within a minimalis...

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...ion of logical consequence» ([4], p. 491, emphasis added). The idea that the intuitive notion of logical consequence is in fact not really welldetermined is not new in logical studies. Already Tarski =-=[74]-=- was aware that the clarity of this notion is not superior to the clarity of other concepts of everyday-language, so 43CHAPTER 4. LOGICAL PLURALISM 44 that every attempt to define it would inevitably...

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...tronger evidence of a change in usage than the repudiation of what had been obvious, and no stronger evidence of bad translation than that it translates earnest affirmations into obvious falsehoods» (=-=[57]-=- pp. 351, 355; emphasis added). 7 Finally, from the reasoning in (G), one concludes that the only possible scenario for a conflict is the one where the deviant logician challenges CL not as inadequate...

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...t chapter, we will introduce two recent approaches in this sense, in the perspective of some logical pluralism.Chapter 4 Logical Pluralism 4.1 Beall and Restall’s logical pluralism Beall and Restall =-=[4, 5, 6]-=- proposed a view about plurality in logic based on the tenet that the central notion in logical theorizing is the notion of “logical consequence” - or “follow from”. They hold that this notion is esse...

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...nciple of Bivalence, as Lukasiewicz maintains). Not unexpectedly then, Graham Priest considers «the analogy between non-standard logics and non-Euclidean geometries an important and interesting one» (=-=[51]-=-, p. 156). Euclid’s fifth postulate was indeed considered correct (better, true) forcenturies,eveniftherehadalwaysbeen doubts about its self-evidence. 3 [14], p. xv.CHAPTER 2. QUINE’S CHALLENGE 11 Ho...

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...hat it is the only logic in agreement with what he considered the necessary constraints for any theory of meaning, namely the so-called “principle of manifestability”; -Quantumlogic,asdefendedbyPutnam=-=[54]-=-,rejectedtheLawofDistributivityon the basis of its being “empirically refuted” by the experiments; - Three-valued logic rejected the Principle of Bivalence and was proposed by Lukasiewicz [36] on the ...

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.... Let us notice however that dpGP has been already applied also to other logics (e.g., Quantum logic [29] and Substructural logics [7]) and that there is a lot of work in progress on the subject (see =-=[8, 59, 60]-=-). From what we said throughout, it seems that, when one wants to claim that two systems of logics are compatible, one can only show either that a complete inter-translation is possible which save the...

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...nly atomic radicals, endowed with one of the usual sets of axioms for IL in assertive disguise (e.g., the axiom ((s ) ), where and only contain atomic radicals). A suitable 8See, e.g., [68],[16],[45],=-=[11]-=-.CHAPTER 4. LOGICAL PLURALISM 57 rule for the pragmatic renderings of modus-ponens is of course included in both calculi ( ` , `( ! ) ` for ACPC; , for AIPC, where and only contain atomic radicals). ...

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... the resolution of each and every mathematical problem, whereas this was considered in disagreement with the view according to which mathematical results are products of a construction-based activity =-=[12]-=-; Later, intuitionistic logic was defended by Dummett [22, 23, 24], on the basis that it is the only logic in agreement with what he considered the necessary constraints for any theory of meaning, nam...

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... contain only atomic radicals, endowed with one of the usual sets of axioms for IL in assertive disguise (e.g., the axiom ((s ) ), where and only contain atomic radicals). A suitable 8See, e.g., [68],=-=[16]-=-,[45],[11].CHAPTER 4. LOGICAL PLURALISM 57 rule for the pragmatic renderings of modus-ponens is of course included in both calculi ( ` , `( ! ) ` for ACPC; , for AIPC, where and only contain atomic r...

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.... Let us notice however that dpGP has been already applied also to other logics (e.g., Quantum logic [29] and Substructural logics [7]) and that there is a lot of work in progress on the subject (see =-=[8, 59, 60]-=-). From what we said throughout, it seems that, when one wants to claim that two systems of logics are compatible, one can only show either that a complete inter-translation is possible which save the...

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...hanging the laws is changing the subject. [Yet] such a person needs only be suggesting a revision of a theory of logic, not logic itself. One cannot simply assume that classical logic gets it right» (=-=[48]-=-, p. 102; italics in the text). 3 A deviant logician could then genuinely disagree with CL at the level of the theory, namely about which logic best describes a certain underlying phenomenon (the info...

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...o external data to “get right”» ([42], p. 542). Hence, his meaningchange thesis could be so expanded: “change of logic, change of subject, change of 10 Or “rational reconstruction”, in Carnap’s words =-=[13]-=-. Carnap introduced this notion to explain how formal methods can be applied to philosophical problems involving informal concepts of ordinary language.CHAPTER 2. QUINE’S CHALLENGE 21 language”. With...

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...e has in an imaginary pragmatic scenario. In this work, we will then focus mainly on clues about the first argument, since touching upon «the thorny issues related with indeterminacy of translation» (=-=[42]-=-, p. 542) would lead us too far from our aims. CHAPTER 2. QUINE’S CHALLENGE 14 G1 (first argument): -(a)AGc-scenarioinvolvingadeviantandaclassicallogicianwouldbepossible only if they both wanted t...

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...-rules for an operator. The second one is the “holistic meaning” -meanttobe«themeaningof[theoperator]asencodedintherelationshipsbetween[theoperator] itself and the other [operators] in the language» (=-=[43]-=-, p. 556). According to Paoli, in the framework of a deductive system, namely a sequent calculi S, one can express both these two aspects. He calls the first one operational meaning, definedbytheopera...

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...uctural distinction for a sequent calculi S is not firm, but dependent on the specific axiomatization of S. For a more detailed critique to Paoli, see [34]. Other minimalist attempts are developed in =-=[64]-=- and [67]. 9One of the classical problems here is connected indeed to the notion of meaning defined by the introduction-rules and elimination-rules for an operator. This notion must face the tradition...

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.... Let us notice however that dpGP has been already applied also to other logics (e.g., Quantum logic [29] and Substructural logics [7]) and that there is a lot of work in progress on the subject (see =-=[8, 59, 60]-=-). From what we said throughout, it seems that, when one wants to claim that two systems of logics are compatible, one can only show either that a complete inter-translation is possible which save the...

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... to ban the choice of this or that logic, but to explore the consequences of such choices. [For Carnap], all there is to logic is language adoption, and language adoption is radically unconstrained» (=-=[67]-=-, pp. 429, 430). The unique accepted notion of validity from this perspective is an internal one - a systemrelative validity - so that an argument cannot be merely said to be valid/invalid, but valid/...

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...ee-valued logic rejected the Principle of Bivalence and was proposed by Lukasiewicz [36] on the grounds that CL was unable to govern future-tense sentences without leading to fatalism; -Relevancelogic=-=[63]-=-wasdefendedbyarguingagainsttheclassicalparadoxesof implication and on the basis that any deduction should be such that the premises are effectively and materially involved in the inference toward the ...

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...tenet that (classical) logic deals with the recognition of the truth-value of a sentence, rather than with the truth-value itself. This was the original conception of logic as developed by Frege (see =-=[72]-=-). Recently, it has been endorsed by Martin-Löf [37].CHAPTER 4. LOGICAL PLURALISM 60 They are different things tout court, but equally employable by users, giventhatthey belong to the same language i...

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...pplied to natural argumentation is still in its earliest stages, but much recent work toward the provision of a general account of logical systems may lend itself to the advancement of this program» (=-=[1]-=-, p. 698. See, e.g., Sambin’s Basic Logic [70]). This minimalism about logic - not to be confused with the minimalism about the meaning of logical operators indicated above - is developed nevertheless...

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...t chapter, we will introduce two recent approaches in this sense, in the perspective of some logical pluralism.Chapter 4 Logical Pluralism 4.1 Beall and Restall’s logical pluralism Beall and Restall =-=[4, 5, 6]-=- proposed a view about plurality in logic based on the tenet that the central notion in logical theorizing is the notion of “logical consequence” - or “follow from”. They hold that this notion is esse...

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...bove, let us just notice that the operational/structural distinction for a sequent calculi S is not firm, but dependent on the specific axiomatization of S. For a more detailed critique to Paoli, see =-=[34]-=-. Other minimalist attempts are developed in [64] and [67]. 9One of the classical problems here is connected indeed to the notion of meaning defined by the introduction-rules and elimination-rules for...

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...ptable proofs. Have I revised my logic, while continuing to mean the same by “not” and “or” or have I decided to use those words with a different meaning? I don’t perceive a fact of the matter here» (=-=[66]-=-, p. 180). From the reasoning in the conclusion of the last section, it follows that a relevant critique to Quine’s meaning-change thesis should be addressed to G1, more than to G2. Now, G1 is mainly ...

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...pic of logical revision do not appear stable throughout his texts. Usually, this passage of Quine in Two Dogmas of Empiricism (1960) is quoted to express the ideas of the “radical Quine” - as Shapiro =-=[2]-=- calls it: «[No sentence] is immune to revision. Revision even of the logical law of the excluded middle has been proposed as a mean of simplifying quantum mechanics; and what difference is there in p...

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...om B&R’s approach); (i) - It should constitute a general way to understand the relations between different logics and to resolve possible conflicts. We hold that Dalla Pozza’s global pluralism (dpGP) =-=[19, 29, 7, 20]-=- meets these desiderata. The starting point of dpGP is the following tenet: two logics cannot both deal with the notion of truth or with the same logical operators. In order to better appreciate Dalla...

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...metries an important and interesting one» ([51], p. 156). Euclid’s fifth postulate was indeed considered correct (better, true) forcenturies,eveniftherehadalwaysbeen doubts about its self-evidence. 3 =-=[14]-=-, p. xv.CHAPTER 2. QUINE’S CHALLENGE 11 However, there is something about logic that one would say is “special”: «If sheer logic is not conclusive, what is? What higher tribunal could abrogate the lo...

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...calobjects,and the Bohr construction is a model of an atom. In other words, a formal language displays certain features of natural languages, or idealizations [...] while simplifying other features» (=-=[17]-=-, p. 500). In this way, the behaviour and the properties of the logical operators in the system constitute a model of the behaviour and the properties of the logical expressions in the natural languag...

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...culus -extended logics: modal, tense, deontic, epistemic, preference, erotetic logics -deviant logics: many-valued, intuitionistic, quantum, free logics -inductive logics -Classification2(Dalla Pozza =-=[21]-=-): -standard logic: classical logic -non-standard logics: supplementary logics: extended: modal,epistemic,deontic,tenselogics integrative: pragmaticlogic deviant logics: many-valued: 3-valued,fuzzylog...

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...ereas this was considered in disagreement with the view according to which mathematical results are products of a construction-based activity [12]; Later, intuitionistic logic was defended by Dummett =-=[22, 23, 24]-=-, on the basis that it is the only logic in agreement with what he considered the necessary constraints for any theory of meaning, namely the so-called “principle of manifestability”; -Quantumlogic,as...

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...inean meaning-change thesis and to reach a compatibility in a third - integrative - way. To this aim, he developed the idea of a pragmatic analysis of sentences that was originally suggested by Frege =-=[27, 28]-=- and then refined by Reichenbach [65]. According to this conception, every sentence can be analyzed in terms of two distinct linguistic components (or features), each of them having different semiotic...

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...om B&R’s approach); (i) - It should constitute a general way to understand the relations between different logics and to resolve possible conflicts. We hold that Dalla Pozza’s global pluralism (dpGP) =-=[19, 29, 7, 20]-=- meets these desiderata. The starting point of dpGP is the following tenet: two logics cannot both deal with the notion of truth or with the same logical operators. In order to better appreciate Dalla...

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...ical) logic in principle; he is instead conservative about logic in practice (through the application of his maxim of minimum mutilation). 2.3.2 Immanent vs. transcendent operators According to Haack =-=[32]-=-, one can say that a formal system is a logic when an isomorphism is shown to hold between the system and a “structured part of reality”. For example, she considers many-valued calculi as logics in th...

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...byPutnam[54],rejectedtheLawofDistributivityon the basis of its being “empirically refuted” by the experiments; - Three-valued logic rejected the Principle of Bivalence and was proposed by Lukasiewicz =-=[36]-=- on the grounds that CL was unable to govern future-tense sentences without leading to fatalism; -Relevancelogic[63]wasdefendedbyarguingagainsttheclassicalparadoxesof implication and on the basis that...

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...ion of the truth-value of a sentence, rather than with the truth-value itself. This was the original conception of logic as developed by Frege (see [72]). Recently, it has been endorsed by Martin-Löf =-=[37]-=-.CHAPTER 4. LOGICAL PLURALISM 60 They are different things tout court, but equally employable by users, giventhatthey belong to the same language in a deep sense. Choosing one of them is not a metaph...

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...matic theory of understanding (the meaning) - and between a semantic notion (truth) and a pragmatic criterion to establish the truth of a sentence ([32], pp. 86-91; for details on such critiques, see =-=[38, 44, 19]-=-).CHAPTER 2. QUINE’S CHALLENGE 25 mechanics. They concluded from this that the theories are incommensurable, i.e. that no direct comparison of content is possible between them. But they did not infer...

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...sentences), combining within one and the same language both the descriptive aspect and the pragmatic/illocutionary aspect of a sentence - in a logical-semiotic perspective already suggested by Morris =-=[40]-=-. Let us describe in more details the pragmatic language introduced by Dalla Pozza to treat the supposed conflict between CL and IL. We will see that Dalla Pozza’s crucial philosophical idea - that no...

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...many-valued: 3-valued,fuzzylogics non-classical 2-valued: intuitionistic,quantum, paraconsistent logics substructural logics: non-monotonic, relevance, linear, basic logics -Classification3(Palladino =-=[41]-=-): -classical logic -extended logics: Kripke’s, minimal modal, alethic, deontic, epistemic, tense logics -alternative logics: many-valued, relevance, conditional, intuitionistic, paraconsistent logics...

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...matic theory of understanding (the meaning) - and between a semantic notion (truth) and a pragmatic criterion to establish the truth of a sentence ([32], pp. 86-91; for details on such critiques, see =-=[38, 44, 19]-=-).CHAPTER 2. QUINE’S CHALLENGE 25 mechanics. They concluded from this that the theories are incommensurable, i.e. that no direct comparison of content is possible between them. But they did not infer...

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...logic, that is to a clear incompatibility-scenario; and one cannot really see how mere a conflict can be whose outcome is an incompatibility. Indeed, philosophers such as Dummett [23, 22] and Prawitz =-=[47]-=- have argued - contra Quine, in their view - that even a terminological disagreement can lead to a real conflict if two logics disagree on general meaning-theoretical principles. The typical case is t...

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...a deep sense). In this way, one would show that in fact an equal dignity is guaranteed to each logic. But such a result is anyway not easy to reach in B&R’s framework, because - as observed by Priest =-=[49]-=- - different types of cases endorse as a matter of fact different theories of vernacular connectives, which are based on different truth-conditions (so, different meanings) and cannot all be right, if...

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... as a mean of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristotle?» (=-=[55]-=-, p. 43). This passage is then put in tension with the rationale in “Philosophy of Logic” (1970) [55] that we have sketched above, which is supposed to imply that (classical) logic cannot be reformed....

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...ts earliest stages, but much recent work toward the provision of a general account of logical systems may lend itself to the advancement of this program» ([1], p. 698. See, e.g., Sambin’s Basic Logic =-=[70]-=-). This minimalism about logic - not to be confused with the minimalism about the meaning of logical operators indicated above - is developed nevertheless in a perspective that we do not consider part...

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...are accepted in the metalanguage for PL - shows that non-classical procedures of proof are not required in order to express intuitionistic logic. An analogous point of view is defended by Sundholm in =-=[73]-=-, where he maintains that constructivism and intuitionistic logic must not necessarily go hand in hand. Besides, these results strengthen Quine’s observation that one can practice «a very considerable...

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...ly there would not be an interesting pluralism at all, for the universality of logic would be impossible and our rationality 2The latter is meant to be exhaustively described by Tarski’s truth theory =-=[75]-=- and adequately captured only by CL.CHAPTER 4. LOGICAL PLURALISM 53 should be considered defeated. We hold that dpGP is a promising extensive approach able to satisfy both constraints. In order to ob...