An intuitionistic defence of Berkeley’s master argument

An intuitionistic defence of Berkeley’s master argument Abstract Berkeley’s ‘master argument’ for idealism has been the subject of extensive criticism. Two of his strongest critics, A.N. Prior and J.L. Mackie, argue that due to various logical confusions on the part of Berkeley, the master argument fails to establish his idealist conclusion. Prior (1976) argues that Berkeley’s argument ‘proves too little’ in its conclusion, while Mackie (1964) contends that Berkeley confuses two different kinds of self-refutation in his argument. This paper proposes a defence of the master argument based on intuitionistic argument. It begins by giving a brief exposition of the master argument and Prior's and Mackie's criticism. The following section explains why we might read the master argument along intuitionistic lines. The final section demonstrates that, according to intuitionistic logic, Berkeley's argument withstands the criticisms of Prior and Mackie. Berkeley’s ‘master argument’ for idealism has been the subject of extensive criticism. Two of his strongest critics, A.N. Prior and J.L. Mackie, argue that due to various logical confusions on the part of Berkeley, the master argument fails to establish his idealist conclusion. Prior (1976) argues that Berkeley’s argument ‘proves too little’ in its conclusion, while Mackie (1964) contends that Berkeley confuses two different kinds of self-refutation in his argument. In this paper, I put forward a defence of the master argument based on intuitionistic logic. I argue that, analysed along these lines, Prior’s and Mackie’s criticisms fail to undermine Berkeley’s argument. 1. Prior and Mackie on the master argument In arguing for idealism in the Dialogues (1965), Berkeley’s mouthpiece Philonous is ‘content to put the whole upon this issue. If you can conceive it possible for … any sensible object whatever, to exist without the mind, then I will grant it actually to be so’ (1965: 163). The challenge Philonous proposes is to conceive of a thing existing without the mind. In response, his interlocutor Hylas gives the example of a tree existing without anyone around to consider it. However, Philonous points out that by giving the example of a tree existing unconceived, Hylas himself is conceiving of it. This is a contradiction. Hylas cannot give an example of a thing that is unconceived, as any example he gives would be rendered false by the fact that he conceives it. Prior gives an account of what he thinks is right, and what is wrong, in the master argument. In his analysis of the argument he introduces the following formalization. ‘Person s thinks that p’ is represented by   T(s, p), and ‘person s thinks truly that p’ is   T(t)(s, p), which is to be understood as   T(s, p) & p. For ‘person s is thinking about/of a’ he substitutes ‘for some φ, s is thinking that Φa’, formalized as   ∃φ(T(s,φa)). Prior defines ‘a is thought-of (by someone)’ as   τa=∃x∃φ(T(x,φa)). Given these definitions, Prior shows what Berkeley gets right in the master argument: that if person s is thinking that a is unthought-of, then a is thought-of, and therefore person s cannot think truly that a is unthought-of   [T(s,∼τa)→τa] (1)  ∴¬T(t)(s, ∼τa). This can be generalized for any given object y to apply to any case where an individual says that y is unthought-of. It generates what Prior calls a ‘logical law’ (1976: 38), namely the necessary falsity of   ∃y(T(t)(s, ∼τy)). (2) Prior sees the master argument as establishing this law, and insofar as it does so the argument is formally ‘impeccable’ (1976: 37). However, for establishing Berkeley’s idealist conclusion that things cannot exist without the mind Prior sees the master argument as seriously flawed. In order to prove that things cannot exist without the mind, Berkeley has to show not only the falsity of (2), but also that   T(t)(s,∃y(∼τy)) (3) is necessarily false. In other words, Berkeley must prove that person s cannot even suppose there being some y that is unthought-of. The master argument fails to establish that (3) cannot be true, and therefore it fails to establish that objects cannot exist without the mind. Mackie takes the criticism of the master argument even further. In his criticism, Mackie distinguishes between three different kinds of self-refutation. First there is ‘absolute self-refutation’. Absolute self-refutation occurs when a proposition is internally contradictory, as in (p & ∼p). Next there is ‘pragmatic self-refutation’, where the mode in which something is presented conflicts with what is being presented, as if I say ‘I am not saying anything’, or write ‘I am not writing anything’. Between these two types of self-refutation Mackie identifies an intermediate type, called ‘operational self-refutation’. Operational self-refutation occurs when there is no way in which a statement may be coherently presented in any form. The assertion of an operationally self-refuting statement implies its own falsity, even though the statement may not express a contradiction as in absolute self-refutation. Examples of operational self-refutation include ‘I do not exist’ or ‘Nothing is being thought’. These statements cannot be asserted and true, but the sentences themselves do not express a contradiction (Levine 2013: 177). According to Mackie, Berkeley makes the mistake of confusing operational self-refutation with absolute self-refutation. The master argument relies for its force on the falsity of ‘x is unthought-of’ whenever it is entertained. This, as Mackie points out, is an instance of operational, not absolute self-refutation. The importance of this, according to Mackie, is that while proving that p is absolutely self-refuting establishes that p is necessarily true, establishing that p is merely operationally self-refuting does not establish that p is self-refuting. Mackie’s criticism is therefore that, even if Prior is wrong, and Berkeley can establish that   T(t)(s,∃y(∼τy)) (3) is false, the master argument will still fail to establish his idealist conclusion, because operational self-refutation does not establish necessary truths about the world. 2. Intuitionistic logic Intuitionistic logic is an alternative form of logic first formally proposed by L.E.J. Brouwer and Arend Heyting in the early 20th century. Its most distinctive difference from classical logic is that it doesn’t hold to the law of the excluded middle. Its defining feature is a focus on proof rather than truth when determining the meaning of propositions (Priest 2008: 104). This form of logic was proposed in answer to certain problems that arise when applying standard theories of truth and meaning to mathematics. A mathematical proposition can be shown to be true if we have a proof of it, or false if we can find a counterexample to it. For some propositions, however, we have neither proof nor counterexample. Consider the case of Goldbach’s Conjecture: that every even number greater than 2 is the sum of two primes. There is no proof for this conjecture, but neither have we found any instance that disconfirms it. Moreover, it is possible that the conjecture is unprovable, that we would never be in a position to verify it. The intuitionist’s response is to say that meaning is given by provability rather than by truth. The meaning of a proposition ‘is to be given, not by the conditions under which it is true, where truth is conceived as a relationship with some external reality, but by the conditions under which it is proved, its proof conditions – where proof is a (mental) construction of a certain kind’ (Priest 2008: 104). When we apply intuitionistic logic to natural language, the proof conditions that determine meaning are the conditions under which a sentence can be verified.1 For example, a proof of a disjunction must be a proof of one of its disjuncts. As a special case of this the proof of an existential statement ‘∃x(Fx)’ must include the construction of some object a, as well as the proof that a is F. Most important for the present discussion is the proof of a negation. According to intuitionistic logic, to prove a negation of a proposition is to prove that any proof of the proposition can be turned into a contradiction (Priest 2008: 104). For an existential statement, to prove a negation is therefore to prove that no object can be constructed that provably instantiates that statement. Truth itself can be characterized in different ways for the intuitionist. Michael Dummett (1973) proposes one possible characterization, which is simply that a statement is true if and only if it has been proven. Dummett also presents a different route intuitionistic logic from intuitionists such as Brouwer, a difference which is important in the context of defending the master argument. Brouwer’s intuitionism is motivated by the assumption that mathematical objects do not have an independent existence, a premiss which would render the master argument circular, as the impossibility of objects existing outside the mind is precisely what the master argument attempts to establish. Dummett, on the other hand, presents a case for intuitionistic logic that does not assume the unreality of objects outside the mind, but which uses the Wittgensteinian thesis that meaning is use to argue that the truth conditions of statements cannot transcend our use of them, which, he argues, requires us to be able to prove them. Dummett thus provides a way in which we can invoke intuitionistic logic in a non-circular way to defend the master argument. 3. An intuitionistic response to Prior and Mackie The rejection of the law of the excluded middle in intuitionistic logic and its focus on proof as the basis for truth suggest a way in which intuitionistic logic can be used to defend Berkeley’s argument from Mackie’s and Prior’s criticisms. Prior’s criticism of the master argument centres on the fact that, on the standard logical picture, Berkeley fails to show that the falsity of (2) implies that (3) is also false:   ∃y(T(t)(s, ∼τy)), (2)  T(t)(s,∃y(∼τy)). (3) Thus, the argument fails to show the falsity of ∃y(∼τy) and hence fails to establish his idealist conclusion. However, things change when we consider the argument on intuitionistic grounds, and truth is construed in terms of provability. According to intuitionistic logic, ∃y(¬τy) is true only if it can be proved. What would such a proof consist in? Proving ∃y(¬τy) would involve showing that there is something that y stands in place of, such that this thing is not thought-of. However, showing this would involve thinking truly that the thing is unthought-of. In other words it would require to be true:   ∃y(T(t)(s, ¬τy)). (2i) This has already been shown by Prior in classical logic to be necessarily false. Since any proof of ∃y(¬τy) involves showing that (2i) is true, there can be no proof of ∃y(¬τy). However, in intuitionistic logic, a proof that there is no proof of A is a proof of ¬A (Priest 2008: 104). This is because the truth of a sentence is conceived as the conditions under which it can be proved. A sentence with no proof cannot be true, and hence we can conclude, from the absence of a proof, the falsity of ∃y(¬τy). Intuitionistic logic thus allows Berkeley to go from the negation of (2i) to the negation of (3i):   ¬∃y(T(t)(s, ¬τy)), (¬ 2i)  ∴¬T(t)(s,∃y(¬τy)). (¬ 3i) Prior’s criticism, on this logical picture, fails to undermine the master argument. What about Mackie’s criticism of Berkeley? Recall that Mackie said that even if Berkeley did manage to establish the falsity of (3), his argument would still fail to establish his idealist conclusion, as the form of self-refutation involved is merely operational self-refutation. Operational self-refutation is not absolute self-refutation, and so it cannot establish necessary truths about the world. In responding to Mackie, we can first highlight what makes Mackie’s criticism convincing on the standard logical picture. It is convincing because standard logic allows for divergence between truth and coherent assertion. So while we may not be able to coherently assert that something exists unconceived, this tells us nothing at all about whether something does in fact exist unconceived or not. Thus operational self-refutation, which only shows that we are unable to coherently assert a certain proposition, tells us nothing about the truth or falsity of such propositions themselves; it does not: …enable us to establish as necessary truths such propositions as ‘I am (essentially) a thinking being’, ‘I know something’, or ‘Material objects do not exist unconceived’ … since the items opposed to these are at most operationally self-refuting, the detection of them does not lead to any such necessary truths. (Mackie 1964: 203) This divergence, between coherent assertion and truth, cannot take place in intuitionistic logic, however. Proof conditions are truth conditions, and hence the conditions under which a proposition can be asserted are the conditions under which it can be true. Intuitionistic logic gives meaning to sentences on the basis of assertability. The presence of operational self-refutation, however, means that there are no grounds on which we can assert a sentence. If a sentence is operationally self-refuting, there cannot be any way to verify it and hence know that it is true. Thus, for the intuitionist, there is no distinction between operational and absolute self-refutation, as to maintain this distinction the truth conditions of operationally self-refuting sentences must transcend our ability to verify them. On this view of logic, Mackie cannot appeal to this distinction, and hence he fails to undermine the master argument. 4. Conclusion On the standard logical picture, Prior’s and Mackie’s criticisms seriously undermine the master argument. According to intuitionistic logic, however, their criticisms lose their bite. Contrary to Prior’s criticism, in intuitionistic logic the conclusion Berkeley establishes in his argument does not ‘prove too little’. Operational self-refutation does not have the same status in intuitionistic logic as it does in standard logic, and so Mackie’s distinction between the two also does not show the master argument to be invalid. While the master argument may still fail on other grounds, on the intuitionistic picture it can withstand the criticisms of Prior and Mackie.2 Footnotes 1 While I will not discuss it here, it should be noted that giving an account of empirical verification is not unproblematic for the intuitionist. In the case of mathematics, we have a relatively clear idea of what verification is, namely proof. For empirical statements about the past, for example, it is much less clear exactly what can be said to count as verification. See Peacocke 2005 on this problem. 2 I would like to thank Arif Ahmed, Alex Oliver and James Levine for their excellent supervision and teaching and two anonymous referees for their very helpful comments. References Berkeley G. 1965. Three dialogues between Hylas and Philonous. In Berkeley’s Philosophical Writings , ed. Armstrong D.M., 129– 225. New York: Collier Books. Dummett M. 1973 The philosophical basis of intuitionistic logic. In his Truth and Other Enigmas , 215– 47. Cambridge, MA: Harvard University Press. Levine J. 2013. Logic and solipsism. In Wittgenstein’s Tractatus , eds. Sullivan P., Potter M., 170– 238. Oxford: Oxford University Press. Google Scholar CrossRef Search ADS   Mackie J.L. 1964. Self-refutation: a formal analysis. Philosophical Quarterly  14: 193– 203. Google Scholar CrossRef Search ADS   Peacocke C. 2005. Justification, realism and the past. Mind  114: 639– 70. Google Scholar CrossRef Search ADS   Priest G. 2008. An Introduction to Non-Classical Logic . Cambridge: Cambridge University Press. Google Scholar CrossRef Search ADS   Prior A.N. 1976. Berkeley in logical form. In his Papers in Logic and Ethics , 33– 8. London: Duckworth. © The Author(s) 2018. Published by Oxford University Press on behalf of The Analysis Trust. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis Oxford University Press

An intuitionistic defence of Berkeley’s master argument

Analysis , Volume Advance Article – May 19, 2018

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Abstract

Abstract Berkeley’s ‘master argument’ for idealism has been the subject of extensive criticism. Two of his strongest critics, A.N. Prior and J.L. Mackie, argue that due to various logical confusions on the part of Berkeley, the master argument fails to establish his idealist conclusion. Prior (1976) argues that Berkeley’s argument ‘proves too little’ in its conclusion, while Mackie (1964) contends that Berkeley confuses two different kinds of self-refutation in his argument. This paper proposes a defence of the master argument based on intuitionistic argument. It begins by giving a brief exposition of the master argument and Prior's and Mackie's criticism. The following section explains why we might read the master argument along intuitionistic lines. The final section demonstrates that, according to intuitionistic logic, Berkeley's argument withstands the criticisms of Prior and Mackie. Berkeley’s ‘master argument’ for idealism has been the subject of extensive criticism. Two of his strongest critics, A.N. Prior and J.L. Mackie, argue that due to various logical confusions on the part of Berkeley, the master argument fails to establish his idealist conclusion. Prior (1976) argues that Berkeley’s argument ‘proves too little’ in its conclusion, while Mackie (1964) contends that Berkeley confuses two different kinds of self-refutation in his argument. In this paper, I put forward a defence of the master argument based on intuitionistic logic. I argue that, analysed along these lines, Prior’s and Mackie’s criticisms fail to undermine Berkeley’s argument. 1. Prior and Mackie on the master argument In arguing for idealism in the Dialogues (1965), Berkeley’s mouthpiece Philonous is ‘content to put the whole upon this issue. If you can conceive it possible for … any sensible object whatever, to exist without the mind, then I will grant it actually to be so’ (1965: 163). The challenge Philonous proposes is to conceive of a thing existing without the mind. In response, his interlocutor Hylas gives the example of a tree existing without anyone around to consider it. However, Philonous points out that by giving the example of a tree existing unconceived, Hylas himself is conceiving of it. This is a contradiction. Hylas cannot give an example of a thing that is unconceived, as any example he gives would be rendered false by the fact that he conceives it. Prior gives an account of what he thinks is right, and what is wrong, in the master argument. In his analysis of the argument he introduces the following formalization. ‘Person s thinks that p’ is represented by   T(s, p), and ‘person s thinks truly that p’ is   T(t)(s, p), which is to be understood as   T(s, p) & p. For ‘person s is thinking about/of a’ he substitutes ‘for some φ, s is thinking that Φa’, formalized as   ∃φ(T(s,φa)). Prior defines ‘a is thought-of (by someone)’ as   τa=∃x∃φ(T(x,φa)). Given these definitions, Prior shows what Berkeley gets right in the master argument: that if person s is thinking that a is unthought-of, then a is thought-of, and therefore person s cannot think truly that a is unthought-of   [T(s,∼τa)→τa] (1)  ∴¬T(t)(s, ∼τa). This can be generalized for any given object y to apply to any case where an individual says that y is unthought-of. It generates what Prior calls a ‘logical law’ (1976: 38), namely the necessary falsity of   ∃y(T(t)(s, ∼τy)). (2) Prior sees the master argument as establishing this law, and insofar as it does so the argument is formally ‘impeccable’ (1976: 37). However, for establishing Berkeley’s idealist conclusion that things cannot exist without the mind Prior sees the master argument as seriously flawed. In order to prove that things cannot exist without the mind, Berkeley has to show not only the falsity of (2), but also that   T(t)(s,∃y(∼τy)) (3) is necessarily false. In other words, Berkeley must prove that person s cannot even suppose there being some y that is unthought-of. The master argument fails to establish that (3) cannot be true, and therefore it fails to establish that objects cannot exist without the mind. Mackie takes the criticism of the master argument even further. In his criticism, Mackie distinguishes between three different kinds of self-refutation. First there is ‘absolute self-refutation’. Absolute self-refutation occurs when a proposition is internally contradictory, as in (p & ∼p). Next there is ‘pragmatic self-refutation’, where the mode in which something is presented conflicts with what is being presented, as if I say ‘I am not saying anything’, or write ‘I am not writing anything’. Between these two types of self-refutation Mackie identifies an intermediate type, called ‘operational self-refutation’. Operational self-refutation occurs when there is no way in which a statement may be coherently presented in any form. The assertion of an operationally self-refuting statement implies its own falsity, even though the statement may not express a contradiction as in absolute self-refutation. Examples of operational self-refutation include ‘I do not exist’ or ‘Nothing is being thought’. These statements cannot be asserted and true, but the sentences themselves do not express a contradiction (Levine 2013: 177). According to Mackie, Berkeley makes the mistake of confusing operational self-refutation with absolute self-refutation. The master argument relies for its force on the falsity of ‘x is unthought-of’ whenever it is entertained. This, as Mackie points out, is an instance of operational, not absolute self-refutation. The importance of this, according to Mackie, is that while proving that p is absolutely self-refuting establishes that p is necessarily true, establishing that p is merely operationally self-refuting does not establish that p is self-refuting. Mackie’s criticism is therefore that, even if Prior is wrong, and Berkeley can establish that   T(t)(s,∃y(∼τy)) (3) is false, the master argument will still fail to establish his idealist conclusion, because operational self-refutation does not establish necessary truths about the world. 2. Intuitionistic logic Intuitionistic logic is an alternative form of logic first formally proposed by L.E.J. Brouwer and Arend Heyting in the early 20th century. Its most distinctive difference from classical logic is that it doesn’t hold to the law of the excluded middle. Its defining feature is a focus on proof rather than truth when determining the meaning of propositions (Priest 2008: 104). This form of logic was proposed in answer to certain problems that arise when applying standard theories of truth and meaning to mathematics. A mathematical proposition can be shown to be true if we have a proof of it, or false if we can find a counterexample to it. For some propositions, however, we have neither proof nor counterexample. Consider the case of Goldbach’s Conjecture: that every even number greater than 2 is the sum of two primes. There is no proof for this conjecture, but neither have we found any instance that disconfirms it. Moreover, it is possible that the conjecture is unprovable, that we would never be in a position to verify it. The intuitionist’s response is to say that meaning is given by provability rather than by truth. The meaning of a proposition ‘is to be given, not by the conditions under which it is true, where truth is conceived as a relationship with some external reality, but by the conditions under which it is proved, its proof conditions – where proof is a (mental) construction of a certain kind’ (Priest 2008: 104). When we apply intuitionistic logic to natural language, the proof conditions that determine meaning are the conditions under which a sentence can be verified.1 For example, a proof of a disjunction must be a proof of one of its disjuncts. As a special case of this the proof of an existential statement ‘∃x(Fx)’ must include the construction of some object a, as well as the proof that a is F. Most important for the present discussion is the proof of a negation. According to intuitionistic logic, to prove a negation of a proposition is to prove that any proof of the proposition can be turned into a contradiction (Priest 2008: 104). For an existential statement, to prove a negation is therefore to prove that no object can be constructed that provably instantiates that statement. Truth itself can be characterized in different ways for the intuitionist. Michael Dummett (1973) proposes one possible characterization, which is simply that a statement is true if and only if it has been proven. Dummett also presents a different route intuitionistic logic from intuitionists such as Brouwer, a difference which is important in the context of defending the master argument. Brouwer’s intuitionism is motivated by the assumption that mathematical objects do not have an independent existence, a premiss which would render the master argument circular, as the impossibility of objects existing outside the mind is precisely what the master argument attempts to establish. Dummett, on the other hand, presents a case for intuitionistic logic that does not assume the unreality of objects outside the mind, but which uses the Wittgensteinian thesis that meaning is use to argue that the truth conditions of statements cannot transcend our use of them, which, he argues, requires us to be able to prove them. Dummett thus provides a way in which we can invoke intuitionistic logic in a non-circular way to defend the master argument. 3. An intuitionistic response to Prior and Mackie The rejection of the law of the excluded middle in intuitionistic logic and its focus on proof as the basis for truth suggest a way in which intuitionistic logic can be used to defend Berkeley’s argument from Mackie’s and Prior’s criticisms. Prior’s criticism of the master argument centres on the fact that, on the standard logical picture, Berkeley fails to show that the falsity of (2) implies that (3) is also false:   ∃y(T(t)(s, ∼τy)), (2)  T(t)(s,∃y(∼τy)). (3) Thus, the argument fails to show the falsity of ∃y(∼τy) and hence fails to establish his idealist conclusion. However, things change when we consider the argument on intuitionistic grounds, and truth is construed in terms of provability. According to intuitionistic logic, ∃y(¬τy) is true only if it can be proved. What would such a proof consist in? Proving ∃y(¬τy) would involve showing that there is something that y stands in place of, such that this thing is not thought-of. However, showing this would involve thinking truly that the thing is unthought-of. In other words it would require to be true:   ∃y(T(t)(s, ¬τy)). (2i) This has already been shown by Prior in classical logic to be necessarily false. Since any proof of ∃y(¬τy) involves showing that (2i) is true, there can be no proof of ∃y(¬τy). However, in intuitionistic logic, a proof that there is no proof of A is a proof of ¬A (Priest 2008: 104). This is because the truth of a sentence is conceived as the conditions under which it can be proved. A sentence with no proof cannot be true, and hence we can conclude, from the absence of a proof, the falsity of ∃y(¬τy). Intuitionistic logic thus allows Berkeley to go from the negation of (2i) to the negation of (3i):   ¬∃y(T(t)(s, ¬τy)), (¬ 2i)  ∴¬T(t)(s,∃y(¬τy)). (¬ 3i) Prior’s criticism, on this logical picture, fails to undermine the master argument. What about Mackie’s criticism of Berkeley? Recall that Mackie said that even if Berkeley did manage to establish the falsity of (3), his argument would still fail to establish his idealist conclusion, as the form of self-refutation involved is merely operational self-refutation. Operational self-refutation is not absolute self-refutation, and so it cannot establish necessary truths about the world. In responding to Mackie, we can first highlight what makes Mackie’s criticism convincing on the standard logical picture. It is convincing because standard logic allows for divergence between truth and coherent assertion. So while we may not be able to coherently assert that something exists unconceived, this tells us nothing at all about whether something does in fact exist unconceived or not. Thus operational self-refutation, which only shows that we are unable to coherently assert a certain proposition, tells us nothing about the truth or falsity of such propositions themselves; it does not: …enable us to establish as necessary truths such propositions as ‘I am (essentially) a thinking being’, ‘I know something’, or ‘Material objects do not exist unconceived’ … since the items opposed to these are at most operationally self-refuting, the detection of them does not lead to any such necessary truths. (Mackie 1964: 203) This divergence, between coherent assertion and truth, cannot take place in intuitionistic logic, however. Proof conditions are truth conditions, and hence the conditions under which a proposition can be asserted are the conditions under which it can be true. Intuitionistic logic gives meaning to sentences on the basis of assertability. The presence of operational self-refutation, however, means that there are no grounds on which we can assert a sentence. If a sentence is operationally self-refuting, there cannot be any way to verify it and hence know that it is true. Thus, for the intuitionist, there is no distinction between operational and absolute self-refutation, as to maintain this distinction the truth conditions of operationally self-refuting sentences must transcend our ability to verify them. On this view of logic, Mackie cannot appeal to this distinction, and hence he fails to undermine the master argument. 4. Conclusion On the standard logical picture, Prior’s and Mackie’s criticisms seriously undermine the master argument. According to intuitionistic logic, however, their criticisms lose their bite. Contrary to Prior’s criticism, in intuitionistic logic the conclusion Berkeley establishes in his argument does not ‘prove too little’. Operational self-refutation does not have the same status in intuitionistic logic as it does in standard logic, and so Mackie’s distinction between the two also does not show the master argument to be invalid. While the master argument may still fail on other grounds, on the intuitionistic picture it can withstand the criticisms of Prior and Mackie.2 Footnotes 1 While I will not discuss it here, it should be noted that giving an account of empirical verification is not unproblematic for the intuitionist. In the case of mathematics, we have a relatively clear idea of what verification is, namely proof. For empirical statements about the past, for example, it is much less clear exactly what can be said to count as verification. See Peacocke 2005 on this problem. 2 I would like to thank Arif Ahmed, Alex Oliver and James Levine for their excellent supervision and teaching and two anonymous referees for their very helpful comments. References Berkeley G. 1965. Three dialogues between Hylas and Philonous. In Berkeley’s Philosophical Writings , ed. Armstrong D.M., 129– 225. New York: Collier Books. Dummett M. 1973 The philosophical basis of intuitionistic logic. In his Truth and Other Enigmas , 215– 47. Cambridge, MA: Harvard University Press. Levine J. 2013. Logic and solipsism. In Wittgenstein’s Tractatus , eds. Sullivan P., Potter M., 170– 238. Oxford: Oxford University Press. Google Scholar CrossRef Search ADS   Mackie J.L. 1964. Self-refutation: a formal analysis. Philosophical Quarterly  14: 193– 203. Google Scholar CrossRef Search ADS   Peacocke C. 2005. Justification, realism and the past. Mind  114: 639– 70. Google Scholar CrossRef Search ADS   Priest G. 2008. An Introduction to Non-Classical Logic . Cambridge: Cambridge University Press. Google Scholar CrossRef Search ADS   Prior A.N. 1976. Berkeley in logical form. In his Papers in Logic and Ethics , 33– 8. London: Duckworth. © The Author(s) 2018. Published by Oxford University Press on behalf of The Analysis Trust. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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AnalysisOxford University Press

Published: May 19, 2018

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