Rules of Inference: A Study in Early Analytic Philosophy
A short Abstract
Inference is a key notion in the philosophy of logic of the early analytic philosophers. Both Frege and Russell hold that inferential justification involves an appeal to logical rules, but they acknowledge that the rules that justify inference cannot be expressed in the language that they govern, on pain of regress. Nonetheless, these rules are taken to encode substantive logical content — the same content that they ascribe to the axioms and laws of their systems. Wittgenstein argues, by contrast, that this account of rules of inference reflects a deep confusion about the nature of logic and its relation to thought: logic cannot be both formal (in the strong sense of informing all thought) and substantive (in the sense that there is genuine knowledge of logic). Rules of inference, according to Wittgenstein, are “senseless and superfluous”. The aim of my dissertation is to reveal the continuing relevance of Frege, Russell and Wittgenstein to contemporary debates on the nature of reasoning and the epistemology of logic. By drawing on these contemporary debates I articulate reasons for accepting Wittgenstein’s claim that inferential justification does not depend on the mediation of logical rules.
A Longer Abstract
The aim of my dissertation is to reveal the continuing relevance of Frege, Russell and Wittgenstein to contemporary debates on the epistemology of logic and the nature of reasoning. My understanding of these contemporary debates allows me to articulate what reasons there might be for us to accept Wittgenstein’s claim that inferential justification does not depend on the mediation of logical rules.
Inference is a key notion in the philosophy of logic of early analytic philosophers. Both Frege and Russell hold that inferential justification involves an appeal to logical rules, but they acknowledge that the rules that justify inference cannot be expressed in the language that they govern, on pain of regress. Nonetheless, these rules are taken to encode substantive logical content — the same content that they ascribe to the axioms and laws of their systems. Wittgenstein argues, by contrast, that this account of rules of inference reflects a deep confusion about the nature of logic and its relation to thought: logic cannot be both formal (in the strong sense of informing all thought) and substantive (in the sense that there is genuine knowledge of logic).
In Tractatus 5.132 Wittgenstein claims that rules of inference are “senseless” and “superfluous”. Given the central role played by rules of inference in modern accounts of proof and inference, such a rejection of rules can only make sense on the background of a sweeping revision of our understanding of the relation between logic, language and thought. For Wittgenstein, the axiomatic systems of proof offered by Frege and Russell present an inadequate model of the shape of human mindedness. Logic has no content of its own and it must not be assimilated to a contentful science; logic is the form of the activity through which we come to know our world and ourselves. It is not just the interpretation of the logical axioms as substantive truths that Wittgenstein rejects, but also the idea that the justification of the simplest inferential steps depends on the application of rules.
From Wittgenstein’s point of view, Frege and Russell separate the understanding of propositions from the inferential activity which these propositions are involved in. By contrast to their atomistic conception of the understanding of propositions, Wittgenstein proposes a holistic approach. Since in understanding each proposition we are already assessing its inferential relations to other propositions, no appeal to logical rules is needed to justify our inferences.
The first three chapters of the dissertation inquire into Frege’s and Russell’s accounts of inference. Both Frege and Russell introduce rules of inference in a way that responds to the threat of regress. They realize that if the justification of any inference requires further inferential reasoning, and this further reasoning itself stands in need of justification, which again involves inference, no inference would ever be justified. They conclude that to avoid the regress, the logical rules to which we appeal in order to justify inferences must be inexpressible within the system that they govern.
In Chapters 1 and 2 I inquire into Frege’s account of inference. Frege the rules of inference, together with the axioms of a logical system, capture the entire content from which all of the theorems are to be derived. To understand Frege’s claim that the rule of inference must nonetheless be inexpressible, I compare his discussion of inference to his discussions of the special status of the judgment-stroke and of the truth-predicate. In all these cases, Frege seeks to preserve the intuition that the logical, normative status of logical act cannot be part of the content of the proposition. This intuition stands in tension with the way he conceives of the relation between the role of rules of inference with the role of the axioms in his system of proofs. Rules of inference can be converted into axioms, and vice versa; and since the axioms are taken to convey universal truths, the rules to which they can be converted count as equally contentful.
In Chapter 3 I discuss Russell’s often neglected account of inference. Russell argues that a regress in the justification of inference can only be avoided if we acknowledge that we have a primitive, non-discursive capacity to appreciate the applicability of rules.
In Chapter 4 I contrast my reading of Wittgenstein’s criticism of Frege and Russell with two alternative readings: traditional readings (e.g. Mounce, Baker) that take Wittgenstein to argue that Frege’s and Russell’s rules of inference do not manage to put a stop to the regress of justifications, and new readings (e.g. Ricketts, Proops) that take Frege and Russell to be immune to the threat of regress, and therefore take Wittgenstein to target a different aspect of Frege’s and Russell’s conception of proof, rather than their rules of inference. I show that Wittgenstein does target the rules of inference, although this is not because he takes them to be vulnerable to the regress of justifications. Despite the fact that Frege and Russell distinguish rules of inference from the expressible propositions of their systems, they continue to think of rules as encoding part of the content of the logical system — the logical content from which mathematical theorems are to be derived. The idea that such rules are inexpressible but contentful is for Wittgenstein a sign of deep confusion.
In Chapter 5 I locate the ground of Wittgenstein’s critique of rules in the distinction he draws between what can be said and what only shows itself through our use of language, but cannot be said in it. Frege and Russell are already committed to a variant of this distinction. Rules of inference, on their account, are just such things that show themselves and cannot be directly expressed. However, they continue to think of rules as contentful, and as grounding the propositions that can be expressed. Several prominent readers of Wittgenstein, including Anscombe and Hacker, think this is also Wittgenstein’s way of drawing the distinction between what can be said and what only shows itself — that certain things that show themselves could still be thought of contentful. By attributing this variant of the distinction to Wittgenstein himself, they mistake his target for his view. They hold that Wittgenstein appeals to the distinction, thus understood, in order to convey metaphysical truths that cannot be directly communicated. By contrast, I hold that the purpose of the distinction is to clarify the sense in which Frege’s and Russell’s approach is untenable. What can only be shown has to be radically different from what can be said — for otherwise, it could be said. More generally, the fact that logical principles cannot be made explicit in a non-circular way indicates that we should not assimilate them to ordinary propositions in any way — in particular, we should not think of them as in any way contentful. Logic is the form, not the content of thought.
In Chapter 6 I develop the Wittgensteinian account of reasoning as a logically informed activity. To fully understand a proposition presupposes the appreciation of many of the inferential relations it stands in. It is then incoherent to claim that we can understand propositions, premises of an inference, and yet are not able to appreciate their inferential relations before we apply a rule of inference to them. A rich, holistic account of understanding renders the rules of inference senseless and superfluous.
I take Wittgenstein’s philosophical endeavor to be aimed at counteracting the philosophical confusions that lead us to form distorted conceptions of reasoning — of the various activities through which we make meaningful contact with our world. We misconstrue the nature of our own mind by making it seem that the activity of justification itself stands in need of justification. But any attempt to think of logic as providing justification — including Frege’s and Russell’s attempt to capture the logical form of reasoning in terms of contentful rules of inference — only deepens our puzzlement about the very possibility of rationality.
A short Abstract
Inference is a key notion in the philosophy of logic of the early analytic philosophers. Both Frege and Russell hold that inferential justification involves an appeal to logical rules, but they acknowledge that the rules that justify inference cannot be expressed in the language that they govern, on pain of regress. Nonetheless, these rules are taken to encode substantive logical content — the same content that they ascribe to the axioms and laws of their systems. Wittgenstein argues, by contrast, that this account of rules of inference reflects a deep confusion about the nature of logic and its relation to thought: logic cannot be both formal (in the strong sense of informing all thought) and substantive (in the sense that there is genuine knowledge of logic). Rules of inference, according to Wittgenstein, are “senseless and superfluous”. The aim of my dissertation is to reveal the continuing relevance of Frege, Russell and Wittgenstein to contemporary debates on the nature of reasoning and the epistemology of logic. By drawing on these contemporary debates I articulate reasons for accepting Wittgenstein’s claim that inferential justification does not depend on the mediation of logical rules.
A Longer Abstract
The aim of my dissertation is to reveal the continuing relevance of Frege, Russell and Wittgenstein to contemporary debates on the epistemology of logic and the nature of reasoning. My understanding of these contemporary debates allows me to articulate what reasons there might be for us to accept Wittgenstein’s claim that inferential justification does not depend on the mediation of logical rules.
Inference is a key notion in the philosophy of logic of early analytic philosophers. Both Frege and Russell hold that inferential justification involves an appeal to logical rules, but they acknowledge that the rules that justify inference cannot be expressed in the language that they govern, on pain of regress. Nonetheless, these rules are taken to encode substantive logical content — the same content that they ascribe to the axioms and laws of their systems. Wittgenstein argues, by contrast, that this account of rules of inference reflects a deep confusion about the nature of logic and its relation to thought: logic cannot be both formal (in the strong sense of informing all thought) and substantive (in the sense that there is genuine knowledge of logic).
In Tractatus 5.132 Wittgenstein claims that rules of inference are “senseless” and “superfluous”. Given the central role played by rules of inference in modern accounts of proof and inference, such a rejection of rules can only make sense on the background of a sweeping revision of our understanding of the relation between logic, language and thought. For Wittgenstein, the axiomatic systems of proof offered by Frege and Russell present an inadequate model of the shape of human mindedness. Logic has no content of its own and it must not be assimilated to a contentful science; logic is the form of the activity through which we come to know our world and ourselves. It is not just the interpretation of the logical axioms as substantive truths that Wittgenstein rejects, but also the idea that the justification of the simplest inferential steps depends on the application of rules.
From Wittgenstein’s point of view, Frege and Russell separate the understanding of propositions from the inferential activity which these propositions are involved in. By contrast to their atomistic conception of the understanding of propositions, Wittgenstein proposes a holistic approach. Since in understanding each proposition we are already assessing its inferential relations to other propositions, no appeal to logical rules is needed to justify our inferences.
The first three chapters of the dissertation inquire into Frege’s and Russell’s accounts of inference. Both Frege and Russell introduce rules of inference in a way that responds to the threat of regress. They realize that if the justification of any inference requires further inferential reasoning, and this further reasoning itself stands in need of justification, which again involves inference, no inference would ever be justified. They conclude that to avoid the regress, the logical rules to which we appeal in order to justify inferences must be inexpressible within the system that they govern.
In Chapters 1 and 2 I inquire into Frege’s account of inference. Frege the rules of inference, together with the axioms of a logical system, capture the entire content from which all of the theorems are to be derived. To understand Frege’s claim that the rule of inference must nonetheless be inexpressible, I compare his discussion of inference to his discussions of the special status of the judgment-stroke and of the truth-predicate. In all these cases, Frege seeks to preserve the intuition that the logical, normative status of logical act cannot be part of the content of the proposition. This intuition stands in tension with the way he conceives of the relation between the role of rules of inference with the role of the axioms in his system of proofs. Rules of inference can be converted into axioms, and vice versa; and since the axioms are taken to convey universal truths, the rules to which they can be converted count as equally contentful.
In Chapter 3 I discuss Russell’s often neglected account of inference. Russell argues that a regress in the justification of inference can only be avoided if we acknowledge that we have a primitive, non-discursive capacity to appreciate the applicability of rules.
In Chapter 4 I contrast my reading of Wittgenstein’s criticism of Frege and Russell with two alternative readings: traditional readings (e.g. Mounce, Baker) that take Wittgenstein to argue that Frege’s and Russell’s rules of inference do not manage to put a stop to the regress of justifications, and new readings (e.g. Ricketts, Proops) that take Frege and Russell to be immune to the threat of regress, and therefore take Wittgenstein to target a different aspect of Frege’s and Russell’s conception of proof, rather than their rules of inference. I show that Wittgenstein does target the rules of inference, although this is not because he takes them to be vulnerable to the regress of justifications. Despite the fact that Frege and Russell distinguish rules of inference from the expressible propositions of their systems, they continue to think of rules as encoding part of the content of the logical system — the logical content from which mathematical theorems are to be derived. The idea that such rules are inexpressible but contentful is for Wittgenstein a sign of deep confusion.
In Chapter 5 I locate the ground of Wittgenstein’s critique of rules in the distinction he draws between what can be said and what only shows itself through our use of language, but cannot be said in it. Frege and Russell are already committed to a variant of this distinction. Rules of inference, on their account, are just such things that show themselves and cannot be directly expressed. However, they continue to think of rules as contentful, and as grounding the propositions that can be expressed. Several prominent readers of Wittgenstein, including Anscombe and Hacker, think this is also Wittgenstein’s way of drawing the distinction between what can be said and what only shows itself — that certain things that show themselves could still be thought of contentful. By attributing this variant of the distinction to Wittgenstein himself, they mistake his target for his view. They hold that Wittgenstein appeals to the distinction, thus understood, in order to convey metaphysical truths that cannot be directly communicated. By contrast, I hold that the purpose of the distinction is to clarify the sense in which Frege’s and Russell’s approach is untenable. What can only be shown has to be radically different from what can be said — for otherwise, it could be said. More generally, the fact that logical principles cannot be made explicit in a non-circular way indicates that we should not assimilate them to ordinary propositions in any way — in particular, we should not think of them as in any way contentful. Logic is the form, not the content of thought.
In Chapter 6 I develop the Wittgensteinian account of reasoning as a logically informed activity. To fully understand a proposition presupposes the appreciation of many of the inferential relations it stands in. It is then incoherent to claim that we can understand propositions, premises of an inference, and yet are not able to appreciate their inferential relations before we apply a rule of inference to them. A rich, holistic account of understanding renders the rules of inference senseless and superfluous.
I take Wittgenstein’s philosophical endeavor to be aimed at counteracting the philosophical confusions that lead us to form distorted conceptions of reasoning — of the various activities through which we make meaningful contact with our world. We misconstrue the nature of our own mind by making it seem that the activity of justification itself stands in need of justification. But any attempt to think of logic as providing justification — including Frege’s and Russell’s attempt to capture the logical form of reasoning in terms of contentful rules of inference — only deepens our puzzlement about the very possibility of rationality.
I have recently published a paper which brings together some of the results of chapters 4-6 of the dissertation:
- “Are Rules of Inference Superfluous? Wittgenstein vs. Frege and Russell,” Teorema, Vol. XL/2, 2021, pp. 45-61. Published version available here, open-access.
For my other publications, see here.
