Foundations of Logical Consequence (2009-2012)

The Foundations of Logical Consequence Project is funded by the AHRC.

Previous Events:

January 2009: a Workshop on Proof-Theoretical and Model-Theoretical Semantics

October 2009: a Workshop on The Logic of Denial

March 2010: a Workshop on Propositions, Context and Consequence

May 2010: OSU/Maribor/Rijeka joint Workshop on The Philosophy of Logical Consequence

June 2010: the first Conference on Foundations of Logical Consequence

September 2010: a Mini-Course and Workshop on ‘Logic or Logics?’

April 2011: a Workshop on Paradox and Logical Revision

October 2011:a Workshop on the Epistemology of Logic

December 2011: a Workshop on Indefinite Extensibility and Logical Paradoxes jointly with the Plurals, Paradox and Predicates project

Forthcoming Events:

June 2012: the second Conference on Foundations of Logical Consequence

Weekly Events:

The FLC research seminar runs every week on Mondays from 16.30-18.30, and the Arché Logic Group runs every week on Mondays from 9.00-11.00. A schedule of planned talks and reading material for these meetings is available on the Arché Calendar.

Multi-media broadcasts of some of the sessions are now available from the Arché Projects website

The Project Team

Principal Investigator: Stephen Read.

Co-Investigators: Graham Priest and Stewart Shapiro.

Independent Auditor: Hartry Field (NYU).

Post-Doctoral Fellows: Colin Caret and Toby Meadows

Other Members: Patrick Greenough, Ole Hjortland, Michael De, Roy Dyckhoff and Walter Pedriali

Project Students: Laura Celani, Spencer Johnston, and Bruno Jacinto.

Affiliated Students: Noah Friedman-Biglin.

Network Members: JC Beall (Connecticut), Roy Cook (Minnesota), Philip Ebert (Stirling), Hannes Leitgeb (Center for Mathematical Philosophy at Munich), Øystein Linnebo (Birkbeck), Jeff Ketland (Center for Mathematical Philosophy at Munich/Oxford), Vann McGee (MIT), Peter Milne (Stirling), Catarina Dutilh Novaes (Groningen), Dag Prawitz (Stockholm), Agustín Rayo (MIT), Greg Restall (Melbourne), Marcus Rossberg (UConn), William Stirton (Edinburgh), Ian Rumfitt (Birkbeck), Peter Schroeder-Heister (Tübingen), Neil Tennant (OSU), Heinrich Wansing (Dresden), Timothy Williamson (Oxford) and Elia Zardini (Aberdeen).

The Research Problem

Logical consequence is the relation between premises and conclusion of a valid piece of reasoning (an argument). The Foundations of Logical Consequence project concentrates on two principal positive approaches to explicating this notion, model-theoretic and inferentialist. According to the former, a conclusion is a logical consequence of some premises if it is true in every model in which they are true. (A model is an interpretation which assigns meanings to non-logical expressions uniformly.) According to inferentialism, the conclusion is a logical consequence of the premises if it may be derived from them by step-wise application of primitive inference-rules, conceived (according to some inferentialists) as implicitly defining the logical expressions they contain, whose acceptance (some hold) is constitutive of understanding those expressions.

Certain very general problems attend both approaches. Model-theoretic accounts have characteristically been proposed in a reductive spirit: logical consequence, an apparently modal notion, is explained in terms of an apparently non-modal, purely extensional apparatus of models. How can any illuminating account ‘cross levels’ in this fashion? Another important concern reflects on the quantification over interpretations standard in this approach: how might anyone know that an argument is truth-preserving under all interpretations of a certain sort, save by knowing by some independent means that it is valid? And by what principle do we select the vocabulary (the ‘logical constants’) to be held invariant under re-interpretation?

The classic objection to inferentialism is posed by Arthur Prior’s demonstration that not every characterisation of inferential role determines an admissible logical operation. Responses to Prior’s challenge have tended to provoke further puzzles: a recurrent proposal has been to demarcate acceptable cases by additional proof-theoretic constraints—e.g., harmony, conservativeness, uniqueness, consistency, or normalisation. But the proper characterisation of these notions remains stubbornly controversial, as does their claim to be proof-theoretic notions at all.

Our over-arching focus in the first half of the project will be to explore the resources that both positive views have to address the above difficulties; and to establish whether they are really in competition. The second half of the project will turn to two large groups of issues on which a satisfactory characterisation of consequence might be expected to shed light, and which in turn might constrain its form. One concerns the absoluteness of logic. Frege believed that the ‘laws of thought’ were universal, topic-invariant, and certain. This conception is challenged by a number of tendencies in more recent philosophy of logic. Some philosophers have argued that the logic introduced by Frege fails to accommodate a requirement of relevant connection between premises and conclusion, others that it is unsuitable as a vehicle for reasoning about infinite totalities, vague concepts, quantum phenomena, or the logical paradoxes. A major concern of our research will be to explore the interaction between these broadly revisionary views and the model-theoretic and inferentialist approaches. Is it a constraint on a satisfactory account of logical consequence that it leave space for a revisionary debate? Or might the correct account teach us that these debates are fundamentally misguided?

The final part of the project will turn to the epistemology of logical consequence. In particular, it will compare the strengths of the two principal approaches regarding such issues as: (i) our apparent knowledge of the validity of simple principles of inference, e.g., Modus Ponens; (ii) the phenomenon of ‘blind inference’ – inference uninformed by explicit beliefs about validity, with which we regularly credit children, and perhaps intelligent animals; (iii) the question of the nature of inference itself – what it is for a thinker to have inferred a particular conclusion from other beliefs, whether rightly or wrongly.

Essentially, then, there are two overarching problems involved in our research: the metaphysics problem of explaining what constitutes logical consequence, and the epistemological problem of how we recognise instances of it. At its most general level, our project is thus concerned with an ‘integration challenge’ (Peacocke): that of providing a unified response to both problems.

Research Context

The fundamental idea of the model-theoretic approach is rooted in Frege’s truth-conditional semantics of the late 19th century, receiving its full development in the 1930s and 1940s by Tarski and Vaught. Inferentialism’s antecedents reach back to the work of Gentzen in the 1930s, but it has only been investigated closely in the past twenty years or so. Etchemendy claimed in 1990 that the model-theoretic approach was fundamentally misguided (see Phase 1); responses have been either to defend the account by further clarification showing that his criticism was itself mistaken; or to abandon the approach and look for an inferentialist account immune to his objections. But the inferentialist approach is still in the process of being properly worked out. Challenging objections to the inferentialist programme have been presented, notably by Timothy Williamson (see Phase 4). The debate is well-developed but the notion of logical consequence remains deeply controversial and poorly understood.

Research Methods

The project runs through four phases, each preparatory for the next:

Conceptions of Logical Consequence

The Structure of Logical Consequence

Revisionism in Logic

The Epistemology of Logic

PHASE 1: CONCEPTIONS OF LOGICAL CONSEQUENCE (January-August 2009)

The traditional model-theoretic notion is reductionist, attempting to replace the intuitive notion of necessary following-from by truth-preservation over all models. How, if at all, does this reduction capture the modal nature of logical consequence? (Etchemendy)

Is the model-theoretic approach to logical consequence ‘epistemically impotent’? In addition to the well-known unsurveyability of the class of models, the model-theoretic approach leaves unclear what role reinterpretations of an argument are supposed to play in our reasoning practice.

A further issue is whether it is extensionally adequate. Models are constrained in size on pain of paradox. Moreover, on standard semantic treatments either the Continuum Hypothesis or its negation, for example, rank as logical truths of second-order logic, thereby stretching instances of logical consequence beyond recognisability by reasoning.

The model-theoretic account of logical consequence requires a demarcation of logical terms. The main proposal has been that logical constants are invariant under all 1-1 permutations (Tarski). Is the Tarskian account satisfactory? If not, can inferentialism give a better answer?

For inferentialism, the first issue concerns the autonomy of the rules governing a logical expression (Read)–the crucial idea that they are, in effect, freely stipulated and thereby determine meaning. This idea needs close examination: what is it for a basic logical rule to be stipulated, and how can such stipulation provide for a rule’s being self-justifying (Dummett)?

Prior showed that the inferentialist needs to constrain the set of rules which can determine a logical constant. One such constraint, harmony, controls the relationship between the grounds for asserting a proposition (containing the constant in question) and the consequences of doing so (Dummett). But the exact nature of harmony remains stubbornly unsettled: is it conservativeness (Belnap), normalization (Prawitz), deductive equilibrium (Tennant) or general-elimination harmony (Read)? Can the informal notion of harmony be captured inferentially at all?

The relations between harmony and autonomy need to be explored. What do harmonious inferential rules do for an expression? Does harmony ensure autonomy? Or that the expression is properly logical (and what does that mean?)–or that it is coherent in a way that Prior’s ‘tonk’ is not? Does it ensure consistency, and is this necessary for harmony?

Inferentialism confronts issues about incompleteness. Is the inferentialist committed to completeness, and, if so, how does this affect the inferentialist picture of higher-order logic? Must the notion of proof be somehow augmented to include a non-recursive aspect? (Dummett, Read)

Inferentialism has its own version of the demarcation problem above. The notion of a logical constant, and with it the notion of logical form, are in jeopardy if all or a very wide class of expressions can be characterised inferentially (Brandom).

PHASE 2: STRUCTURE OF THE RELATION OF LOGICAL CONSEQUENCE (September 2009-August 2010)

Logical consequence is standardly thought of as a relation whereby one thing follows from others. But what are the relata? Theoretical pressure leads to the suggestion that the premises might be any of sets, multi-sets, sequences, pluralities, bunches and other possibilities. Which account is to be preferred?

What are the components of the relata: sentences, propositions, utterances, or what? The question interacts closely with issues concerning the occurrence of semantically context-sensitive devices, like demonstratives and tenses, in inferences. How is this phenomenon to be treated systematically, and how does it impact on the nature of logical consequence (Kaplan)? Our researches here will be informed by work in Arché’s concurrent Contextualism and Relativism project.

Tarski claimed that any viable consequence relation must be reflexive, monotonic (that is, closed under augmentation of premises or conclusion), transitive, compact (that is, consequence is consequence of a finite subset) and closed under uniform substitution. Is each of these indeed necessary? Is any other condition necessary?

A connected but more specific issue is the theoretical pressure to treat the second member of the relation (the conclusion, or succedent) as also involving some combination of elements. This is the philosophical problem of interpreting multiple-conclusion logics, and the notions of assertion and denial in general (Rumfitt, Restall).

Are there different–extensional and intensional–ways of combining premises and assumptions? Certain versions of linear logic and of relevant logic make such a distinction. But what does it amount to, and how does it relate to pre-theoretic intuition?

PHASE 3: REVISIONISM IN LOGIC (September 2010-August 2011)

It is quite consistent with absoluteness (the idea that there is a single absolutely correct set of patterns of logical inference) to allow that a theory of correct inference can be defective in various ways, and so open to revision. And revision of a defective theory may feed back into revision of the practice that it tries to codify. Relevantism, intuitionism and dialetheism can each be interpreted as proposing revisions in this absolutist spirit.

How strong are these various revisionary views when so interpreted?

There are further distinctions. Revisionism per se need not challenge a preferred conception of what logical consequence consists in; the dispute may rather concern which logical principles correctly track it. But revision may also be driven by differing conceptions of logical consequence, consistently with accepting that there is but one correct conception. So

Which revisionary logics are candidates to incorporate a revised conception of logical consequence–and how does this relate to the two broad approaches? (Case studies will embrace intuitionist, relevance, linear, and other paraconsistent logics.)

Much consideration of logical consequence in recent years has focussed on substructural logics, i.e., logics with restricted structural rules. Structural rules, e.g., weakening, contraction, exchange, and cut, proof-theoretically reflect some of the Tarskian constraints listed above. Are these logics genuine rivals, or merely usefully instrumental in providing a framework for identifying presuppositions and assumptions?

A third form of revision, still consistent with the absolutist spirit, holds that the absolutely correct logic may vary with subject matter (consistently with an invariant overarching conception of logical consequence). Thus, do e.g., vagueness, infinity, the semantic paradoxes or quantum phenomena demand special logical treatment? Do they do so under the aegis of a single conception of logical consequence?

Finally, logical pluralism, properly so termed, is the view that there is more than one legitimate conception of logical consequence. This is essentially opposed to Fregean absolutism, raising three major research questions:

Is there a defensible irenic position which so regards the broad opposition between model-theoretic and inferentialist approaches?

Beall and Restall have argued that the best uniform account of logical consequence contains a parameter (the notion of a ‘case’) whose different admissible values generate different ranges of valid inference. We will assess this form of logical pluralism in depth and detail.

Is pluralism a form of relativism? We will draw on work in Arché’s concurrent Contextualism and Relativism project to address this question and explore the theoretical possibilities for assessment-relativist views of logical consequence more generally.

PHASE 4: THE EPISTEMOLOGY OF LOGIC (September 2011-June 2012)

This phase will draw on the investigations of the previous phases to explore certain fundamental questions concerning the epistemology of inference. Ordinary thinkers extend their knowledge by reasoning, and so can be presumed to be sensitive to at least basic instances of the consequence relation. It cannot be required that this sensitivity be informed by any explicit or conscious theory of what the relation consists in.

How do an inferentialist conception of logical consequence, and of the meanings of the logical constants, respectively, connect with the justification of deductive practice?

In what light do the competing accounts of logical consequence place the phenomenon of blind inference—basic inferential competences seemingly exhibited by intelligent animals and young children in advance of any explicit beliefs about logical validity or the conceptual resources needed to articulate them?

What light do the competing accounts shed on our basic explicit logical intuition—the shared experience that simple principles like modus ponens, conjunction introduction, and reductio are immediately convincing? Does either better explain how such impressions can be knowledgeable?

If inferentialism is right, and some basic inference patterns are concept-constituting or analytic, how can anyone rejecting such a pattern (e.g., McGee on Modus Ponens) have the very concept? (Williamson)

What light do the accounts respectively shed on the possibilities more generally for the justification of logical principles? Does inferentialism distinctively make space for rule-circular justifications?