Universal Instantiation Research Articles

The Adoption Problem and the Epistemology of Logic

Abstract After introducing the adoption problem (AP) as the claim that certain basic logical principles cannot be adopted, I offer a characterization of this notion as a two-phase process consisting in (1) the acceptance of a basic logical principle, and (2) the development, in virtue of Phase 1, of a practice of inferring in accordance with that principle. The case of a subject who does not infer in accordance with universal instantiation is considered in detail. I argue that the AP has deep and wide implications for the epistemology of logic, extending well beyond Kripke’s original target, viz. Putnam’s proposal for the empirical revision of logic and its background Quinean epistemology. In particular, the AP questions whether basic logical principles could have a fundamental role in our inferential practices, drawing our attention to the nature of basic inferences and the need to have a clearer conception of them before taking a stand on the matter of the epistemic justification of the logical principles.

Trying to adjunct without knowing how: adjunction and the adoption problem

Abstract The adoption question asks whether there are logical rules that cannot be adopted if one does not already infer in accordance with them. Several philosophers, most famously Saul Kripke and Romina Padró, agree that there are such rules. Accordingly, they agree that there is an adoption problem. However, there is disagreement over which rules are unadoptable. In particular, while most agree that if there is an adoption problem, modus ponens and universal instantiation are in its scope, many would exclude adjunction from the list. In this paper, I argue that adjunction is in the scope of the adoption problem. Then I show that the most straightforward counterargument against adjunction being in the scope of the problem, which tries to reformulate the adjunction rule into a more palatable one, does not work.

What the Adoption Problem Does Not Show

Saul Kripke proposed a skeptical challenge that Romina Padró defended and popularized by the name of the Adoption Problem. The challenge is that, given a certain definition of adoption, there are some logical principles that cannot be adopted—paradigmatic cases being Universal Instantiation and Modus Ponens. Kripke has used the Adoption Problem to argue that there is an important sense in which logic is not revisable. In this essay, I defend two independent claims. First, that the Adoption Problem does not entail that logic is never revisable in the sense that Kripke addresses. Second, that, to assess whether an agent can revise their logic in the sense that Kripke addresses, it is best to consider a different definition of adoption, according to which Universal Instantiation and Modus Ponens are sometimes adoptable.

On the Adoption Problem and Meta-Logical Monism

According to the Adoption Problem (AP) certain basic logical principles cannot be adopted. Drawing on the AP, Suki Finn presents an argument against logical pluralism: Modus Ponens (MP) and Universal Instantiation (UI) both govern a general structure shared by every logical rule. As such, analogues of these two rules must be present in every meta-logic for any logical system L, effectively imposing a restriction to logical pluralism at the meta-level through their presence constituting a “meta-logical monism”. We find a tension in the dual role that the “unadoptable rules” must play in Finn’s “meta-logical monism” rendering it ineffective to restrict logical theories and systems. Consequently, we argue they cannot be both analogues of MP and UI and inferentially productive. We conclude with a series of suggestions regarding where a more satisfying and robust interpretation of the AP could lie.

Solving the Conjunction Problem of Russell's Principles of Mathematics

The quantification theory of propositions in Russell’s Principles of Mathematics has been the subject of an intensive study and in reconstruction has been found to be complete with respect to analogs of the truths of modern quantification theory. A difficulty arises in the reconstruction, however, because it presents universally quantified exportations of five of Russell’s axioms. This paper investigates whether a formal system can be found that is more faithful to Russell’s original prose. Russell offers axioms that are universally quantified implications that have antecedent clauses that are conjunctions. The presence of conjunctions as antecedent clauses seems to doom the theory from the onset, it will be found that there is no way to prove conjunctions so that, after universal instantiation, one can detach the needed antecedent clauses. Amalgamating two of Russell’s axioms, this paper overcomes the difficulty.

The Russell-Prawitz embedding and the atomization of universal instantiation

Abstract Given the recent interest in the fragment of system $\mathbf{F}$ where universal instantiation is restricted to atomic formulas, a fragment nowadays named system ${\mathbf{F}}_{\textbf{at}}$, we study directly in system $\mathbf{F}$ new conversions whose purpose is to enforce that restriction. We show some benefits of these new atomization conversions: (i) they help achieving strict simulation of proof reduction by means of the Russell–Prawitz embedding of $\textbf{IPC}$ into system $\mathbf{F}$, (ii) they are not stronger than a certain ‘dinaturality’ conversion known to generate a consistent equality of proofs, (iii) they provide the bridge between the Russell–Prawitz embedding and another translation, due to the authors, of $\textbf{IPC}$ directly into system ${\mathbf{F}}_{\textbf{at}}$ and (iv) they give means for explaining why the Russell–Prawitz translation achieves strict simulation whereas the translation into ${\mathbf{F}}_{\textbf{at}}$ does not.

Semantics for first-order superposition logic

Abstract We investigate how the sentence choice semantics (SCS) for propositional superposition logic (PLS) developed in Tzouvaras (2018, Logic Journal of the IGPL, 26, 149–190) could be extended so as to successfully apply to first-order superposition logic (FOLS). There are two options for such an extension. The apparently more natural one is the formula choice semantics (FCS) based on choice functions for pairs of arbitrary formulas of the basis language. It is proved however that the universal instantiation scheme of first-order logic, $(\forall v)\varphi (v)\rightarrow \varphi (t)$, is false, as a scheme of tautologies, with respect to FCS. This causes the total failure of FCS as a candidate semantics. Then we turn to the other option, which is a variant of SCS, since it uses again choice functions for pairs of sentences only. This semantics however presupposes that the applicability of the connective | is restricted to quantifier-free sentences, and thus the class of well-formed formulas and sentences of the language is restricted too. Granted these syntactic restrictions, the usual axiomatizations of FOLS turn out to be sound and conditionally complete with respect to this second semantics, just like the corresponding systems of PLS.

Limiting logical pluralism

In this paper I argue that pluralism at the level of logical systems requires a certain monism at the meta-logical level, and so, in a sense, there cannot be pluralism all the way down. The adequate alternative logical systems bottom out in a shared basic meta-logic, and as such, logical pluralism is limited. I argue that the content of this basic meta-logic must include the analogue of logical rules Modus Ponens (MP) and Universal Instantiation (UI). I show this through a detailed analysis of the ‘adoption problem’, which manifests something special about MP and UI. It appears that MP and UI underwrite the very nature of a logical rule of inference, due to all rules of inference being conditional and universal in their structure. As such, all logical rules presuppose MP and UI, making MP and UI self-governing, basic, unadoptable, and (most relevantly to logical pluralism) required in the meta-logic for the adequacy of any logical system.

The Liar Hypodox: A Truth-Teller’s Guide to Defusing Proofs of the Liar Paradox

It seems that the Truth-teller is either true or false, but there is no accepted principle determining which it is. From this point of view, the Truth-teller is a hypodox. A hypodox is a conundrum like a paradox, but consistent. Sometimes, accepting an additional principle will convert a hypodox into a paradox. Conversely, in some cases, retracting or restricting a principle will convert a paradox to a hypodox. This last point suggests a new method of avoiding inconsistency. This article provides a significant example. The Liar paradox can be defused to a hypodox by relatively minimally restricting three principles: the T-schema, substitution of identicals and universal instantiation. These restrictions are not arbitrary. For each, I identify the source of a contradiction given some presumptions. Then I propose each restriction as a reasonable way to deal with that source of contradiction.

LOGICAL ANALYSES OF VIPASSANĀ MEDITATION

ABSTRACTIn this paper we propose logical analyses of Vipassanā Meditation as modus ponens or universal instantiation, as based upon identification, analysis, and interpretation of the meditation as a (spiritual) thought experiment (STE like e.g. huàtóus and kōans in Chan and Zen).*The analyses consist in completing the unfinished enthymematic argument of the Sutta-nipāta by explication of hidden major. We speculate the thought experiment inference might be brought about by a faculty like mindfulness. The thought experiment analyses link the meditation to global cross-cultural theories and arguments in philosophy as Ibn Sīnā’s Flying Man and, notably, René Descartes’s Cogito. Last, we will give some indications how Vipassanā Meditation may be explained with help of western theories of consciousness.

Paradoxes and Restricted Quantification

Andrew Bacon, John Hawthorne, and Gabriel Uzquiano (Bacon, Hawthorne, and Uzquiano 2016) have recently argued that free logics—logics that reject or restrict Universal Instantiation—are ultimately not promising approaches to resolving a family of intensional paradoxes due to Arthur Prior (Prior 1961). These logics encompass ramified and contextualist approaches to paradoxes, and broadly speaking, there are two kinds of criticism they face. First, they fail to address every version of the Priorean paradoxes. Second, the theoretical considerations behind the logics make absolutely general statements about all propositions, properties of propositions, etc., but because this sort of intensional quantification is always restricted in the logics, they cannot even express those considerations. I present a novel sort of free logic, which rejects the standard Universal Instantiation but validates a restricted form of the rule, and which addresses both kinds of criticism by allowing some propositions to be unrestricted in their quantification.

Tree Trimming: Four Non-Branching Rules for Priest’s Introduction to Non-Classical Logic

In An Introduction to Non-Classical Logic: From If to Is Graham Priest (2008) presents branching rules in Free Logic, Variable Domain Modal Logic, and Intuitionist Logic. I propose a simpler, non-branching rule to replace Priest’s rule for universal instantiation in Free Logic, a second, slightly modified version of this rule to replace Priest’s rule for universal instantiation in Variable Domain Modal Logic, and third and fourth rules, further modifying the second rule, to replace Priest’s branching universal and particular instantiation rules in Intuitionist Logic. In each of these logics the proposed rule leads to tableaux with fewer branches. In Intuitionist logic, the proposed rules allow for the resolution of a particular problem Priest grapples with throughout the chapter. In this paper, I demonstrate that the proposed rules can greatly simplify tableaux and argue that they should be used in place of the rules given by Priest.

Students\u2019 understanding of the structure of deductive proof

While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students’ understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Pre-structural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.

Thought Experiment Analyses of René Descartes' Cogito

ABSTRACT: René Descartes' Cogito is an example of a paradigmatic thought experiment, herald of both subjectivism and new science in Europe's Modern Age, that seems to have escaped the attention of thought experiment philosophers. On deep analysis, the Cogito appears as universal instantiation (or modus ponens with implicit major 'whatever has the property of thinking, exists'). The Cogito has strong rhetorical effects for it narratively generalizes from I to all human kind, and its historical and philosophical success can be explained from its concise enthymematic structure that rings true in many possible senses. We consider it a preeminent example of a thought experiment as it states the power of thinking as its very contents. From Descartes' methodology of doubt we can conclude that, e.g., on a Wittgensteinian interpretation, the Cogito is a logical thought experiment rather than a psychological one.

Higher-order free logic and the Prior-Kaplan paradox

AbstractThe principle of universal instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior's paradox and Kaplan's paradox and the debate between necessitism and contingentism. We outline a distinctively free logical approach to the intensional paradoxes and note how the free logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramificationist approaches to higher-order logic. Our assessment of both approaches is largely pessimistic, and we remain reluctantly inclined to take Prior's and Kaplan's derivations at face value.

Some clearer reactions: Gauker on the validity of universal instantiation

It is part of the logical orthodoxy that quantifiers are interdefinable and that the rules of Universal Instantiation (UI) and Existential Generalization (EG) hold or fail together. Christopher Gauker has presented some cases which seemingly undermine the validity of UI but nonetheless leave EG untouched, and has developed a very sophisticated theory to explain why this is so. In the process, he has rejected several attempts to explain the asymmetry, especially those aiming at saving the logical orthodoxy by showing what is wrong with the counterexamples to UI. In this paper I argue that some of those proposals are better grounded than Gauker thinks and that ultimately they should be preferred over his since they satisfactorily explain the apparent counterexamples.

Space and time in perceptual causality.

Inferring causality is a fundamental feature of human cognition that allows us to theorize about and predict future states of the world. Michotte suggested that humans automatically perceive causality based on certain perceptual features of events. However, individual differences in judgments of perceptual causality cast doubt on Michotte's view. To gain insights in the neural basis of individual difference in the perception of causality, our participants judged causal relationships in animations of a blue ball colliding with a red ball (a launching event) while fMRI-data were acquired. Spatial continuity and temporal contiguity were varied parametrically in these stimuli. We did not find consistent brain activation differences between trials judged as caused and those judged as non-caused, making it unlikely that humans have universal instantiation of perceptual causality in the brain. However, participants were slower to respond to and showed greater neural activity for violations of causality, suggesting that humans are biased to expect causal relationships when moving objects appear to interact. Our participants demonstrated considerable individual differences in their sensitivity to spatial and temporal characteristics in perceiving causality. These qualitative differences in sensitivity to time or space in perceiving causality were instantiated in individual differences in activation of the left basal ganglia or right parietal lobe, respectively. Thus, the perception that the movement of one object causes the movement of another is triggered by elemental spatial and temporal sensitivities, which themselves are instantiated in specific distinct neural networks.

The three-form reasoning of new Hetu-vidya in Indian logic from the perspective of modern logic

Comparing the three-form reasoning of new Hetu-vidya with Western logic, scholars have put forward four perspectives. Combining their strengths and shortcomings, and the examples of Hetu-vidya reasoning, we can conclude that the three-form reasoning should have four forms: (1) the affirmative expression of formal implication; (2) the modus ponens of hypothetical reasoning concerning sufficient conditions after universal instantiation; (3) the negative expression of a formal implication; and (4) the modus tollens of hypothetical reasoning concerning sufficient conditions after universal instantiation.

Words Without Meaning (review)

Reviewed by: Words without meaning by Christopher Gauker Xuelin He Words without meaning. By Christopher Gauker. Cambridge, MA: MIT Press, 2003. Pp. xi, 299. ISBN 0262571625. $28. Basing his own approach to linguistic communication on the rejection of two significant notions of speaker meaning and word meaning, Gauker casts new light on some of the most controversial issues in the philosophy of language, ranging from logic puzzles to belief report. This short but impressive monograph consists of twelve chapters, each one providing its own focus on the discussion, but each reinforcing the others to form a well-argued presentation of G’s theory of mind. Ch. 1, ‘The received view’, gives a critical review of the propositional approach to meaning in linguistic communication, one that claims the speaker reveals her belief through the right choice of words and sentences whose meanings are also shared by the hearer. Ch. 2, ‘Mental representation’, criticizes the views that beliefs are mental structures bearing propositional content. Ch. 3, ‘Elements of an alternative’, introduces G’s context-logical approach that features the replacement of truth conditions for sentences with assertibility conditions that are formulated in terms of G’s concept of the context. G assumes that any action in pursuit of a conversational goal may or may not accord with a set of sentences, or the context in G’s definition. The idea of assertibility in a context means that a sentence is assertible in a context if and only if it is a member of the set. The rest of the book concerns the application of this context-logical approach to formulating the assertibility conditions for complicated sentences (i.e. sentences that include logical devices, such as quantifiers, conditionals, truth predicates, the predicate ‘believes’, etc.). Ch. 4, ‘Domain of discourse’, reveals the inadequacy of the view that the speaker’s states of mind function as the sole determinant for the hearer to identify the referents of demonstrative phrases and the domains for quantifiers in the speaker’s sentences. Ch. 5, ‘Presupposition’, discusses problems for the pragmatic theory of presupposition, in particular, informative presupposition and presupposition coordination. Ch. 6, ‘Implicature’, questions the core idea in Gricean theory that the hearer can retrieve the meaning only by contemplating what is in the speaker’s mind. Ch. 7, ‘Quantification’, may be viewed as a nice solution to the validity asymmetry between universal instantiation and existential generalization. Ch. 8, ‘Conditionals’, shows how the context-logical approach can assure the validity of the inference in conditionals in a context in which it is uttered; Ch. 9, ‘Truth’, describes how the new approach avoids some famous semantic paradoxes and shows that no new paradoxes will arise within this approach. Ch. 10, ‘The communication conception’, argues that the account of belief attributions is the account of the nature of beliefs and desires. Ch. 11, ‘Explanation and prediction’, refutes the idea that human behaviors must be attributed to folk psychology. For example, one can also explain or predict the other’s behavior by straight induction or by his/her language competence. Ch. 12, ‘Semantics and ontology’, provides a semantic theory of belief attributions and ontological views on the nature of beliefs. This book will benefit scholars who are familiar and yet not satisfied with the standard solutions to the quandaries in the philosophy of language, but will still be accessible to readers with only elementary knowledge of first-order logic (thanks to the author’s effort to avoid philosophical verbosity and too many ‘isms’ in his narration). G’s brief introduction in the preface and his comparison of his ideas with current philosophical theories in the epilogue also help to facilitate readers’ understanding of the book. Xuelin He Guangdong Foreign Studies University Copyright © 2006 Linguistic Society of America

Open Questions and the Manifest Image

The essay argues that, on their usual metalinguistic reconstructions, the open question argument and Frege's puzzle are variants of the same argument. Each are arguments to a conclusion about a difference in meaning; each deploy compositionality as a premise; and each deploy a premise linking epistemic features of sentences with their meaning (which, given certain meaning‐platonist assumptions, can be interpreted as a universal instantiation of Leibniz's law). Given these parallels, each is sound just in case the other is. They are, in fact, unsound. The essay first argues that reformulations of these arguments directly in terms of Leibniz's law are unsound and then that subarguments of the metalinguistic versions are unsound for structurally similar reasons. Finally, given how the theory/observation distinction is deployed in linguistic practice, the meaning‐platonist assumptions are shown to be optional.Shouldn't philosophers be permitted to rise above faith in grammar? All due respect for governesses–but hasn't the time come for philosophy to renounce the faith of governesses?FRIEDRICH NIETZSCHE, Beyond Good and Evil, 1886