This sentence is false. The previous sentence is, if true, false and, if false, true. Either way asserting it appears to commit us to a contradiction, a sentence that is both true and false. What ought we to infer from a contradiction? Current orthodoxy answers “anything whatsoever”. So from the first sentence of this paragraph it follows that the moon is made of cheese. Unsurprisingly, many people find this view counter-intuitive on first encounter. Drawing on discussions in ancient and especially medieval philosophy of logic, this project proposes a new challenge to the prevailing position, defending the claim that

*nothing*follows from a contradiction. It draws out the significance of this issue within philosophy and cognate disciplines, such as mathematics and computer science.

An existing heterodoxy insists that some things, but not everything, from a contradiction, and a debate has developed between this position and the orthodoxy. Yet historically, neither position commanded dominance in the ancient or early to mid-medieval periods. A host of significant thinkers – including Aristotle, Boethius, and Abelard – held that nothing whatsoever follows from a contradiction. My project is a long-overdue examination of this position, known as

*Ex Falso Nihil (EFN)*, including a thorough examination of its history and philosophical foundations, a formalisation using the tools of modern mathematical logic, and an assessment of the mathematical strength of the resulting formalisation. The principal output of the research will be a monograph, aimed at a wide interdisciplinary audience with the aim of furthering interest in and debate about EFN.

The project will unpick the history of EFN before suggesting that modern developments in logical understanding allow us to see clearly how it can be motivated independently of less plausible ideas (concerning 'if.. then' statements) with which it was associated historically. I will develop a novel case for EFN on the basis that it is the only account of contradictions that satisfies constraints on an adequate theory of meaning. I will present a system of logic which validates EFN and reflects my preferred philosophical motivations, and will discuss philosophical issues arising out of this system. Finally, a technical strand to the project will assess how much classical mathematics can be done with the new system. I conjecture that the result here will be considerably more impressive than with existing logics adopting unorthodox approaches to contradiction, making it a useful tool for investigating inconsistent historical mathematical theories and pointing to potential computer science applications (for instance, in knowledge representation systems).

There are three key tasks to be completed in this project, each of which will be undertaken as a distinct phase. The methodologies of each combine individual thought and reading, structured discussion with colleagues, and formal work in logic:

**(1) Motivating EFN**

I will defend the claim that EFN is an historically significant position, present in the writings of important ancient and medieval thinkers. From the 14th century onwards arguments for anything whatsoever following from a contradiction consequence signal the prehistory of modern logic. Critically examining the motivations offered for EFN in the past, I will develop a new one based in a certain account of language use. If P is a sentence then,

*for bilateral proof-theoretic semantics*, I am entitled to assert ‘it is not the case that P’ just in case some

*denial conditions*associated with P are met. From this perspective, were a contradiction assertable some proposition is such that it ought to be both asserted and denied. This places a speaker in a position where she is committed to incompatible speech-acts, so that no subsequent inferential move is licit. If we can motivate the background proof-theoretic semantics, there is a strong case for EFN; and such motivation is forthcoming. My work in this phase will develop this line of argument and defend it against anticipated criticisms.

**(2) Formalising EFN**

Drawing on existing work of mine, I will show that an elegant system of mathematical logic can be developed which treats contradictions according to EFN. The basic idea here is that the system ‘blocks’ any further reasoning once a contradiction has been asserted (by means of a so-called ‘structural rule’). I want to develop the suggestion that principled motivation can be had for this system by recourse to Rumfitt’s idea that the recognition of a contradiction is a ‘logical punctuation mark’, representing the impossibility of taking reasoning any further (although Rumfitt doesn’t follow the idea in the direction of EFN and defends orthodoxy: my approach is novel). I will fill out proofs of technical results (soundness and completeness relative to a Kripke semantics), which will be important in applications of the system.

One major challenge during this phase is that we want to be able to reason from assumptions with contradictory implication. So, for example, I want to be able to assume that 0=1 in order to conclude that it is not the case that 0=1 (so-called proof by

*reductio*). How can we motivate this in a principled fashion given that we want to block reasoning from straightforward assertions of contradictions? The response I plan to develop here is once again linguistic in focus, drawing out the differences between assertion and supposition as distinct speech-acts.

**(3) Testing EFN**

The logic from phase two is a

*paraconsistent logic*(that is, one in which not every proposition follows from a contradiction). It is well-known that such logics have potential for application in both mathematics – the sympathetic investigation of historical theories now known to be inconsistent, for instance – and in computer science, particularly in knowledge representation systems. As plausibly modelling everyday reasoning more closely than classical logic and providing new options for formal theories of truth, they are also of philosophical interest. The problem with existing paraconsistent logics is that they are often extremely weak, i.e. their capacity to prove things is severely limited, one example here being that it can take a great deal of work to recover

*modus ponens*(the inference from A and ‘if A then B’ to B). Our logic is considerably stronger than currently extant paraconsistent logic. The third phase will gauge the extent of this strength. The investigation will have two parts: (a) whether characteristic classical patterns of inference are valid, with particular attention being paid to those that commonly fail in paraconsistent logics, (b) whether the standard formalisation of the arithmetic of the natural numbers 0, 1, 2… (Peano Arithmetic), if so this is deeply significant given the extent to which working classical mathematics can be implemented in Peano Arithmetic.

The aim of the research overall is to present EFN as a genuine contender as a response to the problems presented by inferences from contradictions and to demonstrate its formal tractability. After the three-year project I intend to investigate potential applications, particularly in the theory of truth.