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Logic, Non-Classical

For example,. We often omit the superscript, when no confusion will result. They correspond to free-standing sentences whose internal structure does not matter.

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And so on. The non-logical terminology of the language consists of its individual constants and predicate letters. In taking identity to be logical, we provide explicit treatment for it in the deductive system and in the model-theoretic semantics.

Non-Classical Logic

Most authors do the same, but there is some controversy over the issue Quine [, Chapter 5]. Examples of atomic formulas include:. If an atomic formula has no variables, then it is called an atomic sentence. If it does have variables, it is called open. In the above list of examples, the first and second are open; the rest are sentences. Clause 8 allows us to do inductions on the complexity of formulas. If a certain property holds of the atomic formulas and is closed under the operations presented in clauses 2 — 7 , then the property holds of all formulas.

Here is a simple example:. Theorem 1. Moreover, each left parenthesis corresponds to a unique right parenthesis, which occurs to the right of the left parenthesis. Similarly, each right parenthesis corresponds to a unique left parenthesis, which occurs to the left of the given right parenthesis. If a parenthesis occurs between a matched pair of parentheses, then its mate also occurs within that matched pair. In other words, parentheses that occur within a matched pair are themselves matched.

Proof : By clause 8 , every formula is built up from the atomic formulas using clauses 2 — 7. The atomic formulas have no parentheses. Parentheses are introduced only in clauses 3 — 5 , and each time they are introduced as a matched set. So at any stage in the construction of a formula, the parentheses are paired off.

We next define the notion of an occurrence of a variable being free or bound in a formula. We do not even think of those as occurrences of the variable. All variables that occur in an atomic formula are free. That is, the unary and binary connectives do not change the status of variables that occur in them. Although it does not create any ambiguities see below , we will avoid such formulas, as a matter of taste and clarity.

An Essay in Classical Modal Logic

These, too, will be avoided in what follows. Some treatments of logic rule out vacuous binding and double binding as a matter of syntax. That simplifies some of the treatments below, and complicates others. Free variables correspond to place-holders, while bound variables are used to express generality. If a formula has no free variables, then it is called a sentence. If a formula has free variables, it is called open. Before turning to the deductive system and semantics, we mention a few features of the language, as developed so far. This helps draw the contrast between formal languages and natural languages like English.

We assume at the outset that all of the categories are disjoint.

Logic, Non-Classical

For example, no connective is also a quantifier or a variable, and the non-logical terms are not also parentheses or connectives. Also, the items within each category are distinct. For example, the sign for disjunction does not do double-duty as the negation symbol, and perhaps more significantly, no two-place predicate is also a one-place predicate. Consider the English sentence:. It can mean that John is married and either Mary is single or Joe is crazy, or else it can mean that either both John is married and Mary is single, or else Joe is crazy.

If our formal language did not have the parentheses in it, it would have amphibolies. The parentheses resolve what would be an amphiboly. Can we be sure that there are no other amphibolies in our language?

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Our next task is to answer this question. Lemma 2. Each formula consists of a string of zero or more unary markers followed by either an atomic formula or a formula produced using a binary connective, via one of clauses 3 — 5. Proof : We proceed by induction on the complexity of the formula or, in other words, on the number of formation rules that are applied. The Lemma clearly holds for atomic formulas. Lemma 3. Proof : Here we also proceed by induction on the number of instances of 2 — 7 used to construct the formula.

Clearly, the Lemma holds for atomic formulas, since they have no parentheses. The proof proceeds by induction on the number of instances of 2 — 7 used to construct the formula, and we leave it as an exercise. Theorem 5. If the latter formula was produced via one of clauses 3 — 5 , then it begins with a left parenthesis. Theorem 6. Moreover, no formula produced by clauses 2 — 7 is atomic. In this case, it must have been produced by one of 3 — 5 , and not by any other clause. Similar reasoning takes care of the other combinations. It shows that each formula is produced from the atomic formulas via the various clauses in exactly one way.

We apologize for the tedious details. We included them to indicate the level of precision and rigor for the syntax. As above, we define an argument to be a non-empty collection of sentences in the formal language, one of which is designated to be the conclusion. If there are any other sentences in the argument, they are its premises. Again, we define the deducibility relation by recursion. We start with a rule of assumptions:. We next present two clauses for each connective and quantifier.

The elimination rule is a bit more complicated. They cannot both be true. However, we do not have the converse yet. There is some controversy over this inference. It is rejected by philosophers and mathematicians who do not hold that each meaningful sentence is either true or not true. Intuitionistic logic does not sanction the inference in question see, for example Dummett [], or the entry on intuitionistic logic , or history of intuitionistic logic , but, again, classical logic does.

It is not valid in intuitionistic logic. That is, anything at all follows from a pair of contradictory opposites. Some logicians introduce a rule to codify a similar inference:. The inference is sometimes called ex falso quodlibet or, more colorfully, explosion. A small minority of logicians, called dialetheists , hold that some contradictions are actually true. For them, ex falso quodlibet is not truth-preserving. Deductive systems that demur from ex falso quodlibet are called paraconsistent. Most relevant logics are paraconsistent. See the entries on relevance logic , paraconsistent logic , and dialetheism.

Or see Anderson and Belnap [], Anderson, Belnap, and Dunn [], and Tennant [] for fuller overviews of relevant logic; and Priest [],[a] for dialetheism. Deep philosophical issues concerning the nature of logical consequence are involved.

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  5. Far be it for an article in a philosophy encyclopedia to avoid philosophical issues, but space considerations preclude a fuller treatment of this issue here. Suffice it to note that the inference ex falso quodlibet is sanctioned in systems of classical logic , the subject of this article.

    Recall that the only closed terms in our system are constants. The introduction clause for the universal quantifier is a bit more complicated. It could be any object, and so anything we conclude about it holds for all objects. This elimination rule also corresponds to a common inference. The introduction rule is about a simple as can be:. Suppose, for example, that Harry is identical to Donald since his mischievous parents gave him two names. Again, this clause allows proofs by induction on the rules used to establish an argument. To illustrate the level of rigor, we begin with a lemma that if a sentence does not contain a particular closed term, we can make small changes to the set of sentences we prove it from without problems.

    Lemma 7. The cases for the other rules are similar. Theorem 8. The rule of Weakening. Most of the other cases are exactly like this. Theorem 8 allows us to add on premises at will.

    Andrew Bacon

    Some systems of relevant logic do not have weakening, nor does substructural logic See the entries on relevance logic , substructural logics , and linear logic. Theorem 9. Theorem The rule of Cut. The remaining cases are similar. Theorem 11 allows us to chain together inferences. This fits the practice of establishing theorems and lemmas and then using those theorems and lemmas later, at will. The cut principle is, some think, essential to reasoning. In some logical systems, the cut principle is a deep theorem; in others it is invalid. The system here was designed, in part, to make the proof of Theorem 11 straightforward.

    That is, a set is consistent if it does not entail a pair of contradictory opposite sentencess. The Lindenbaum Lemma. Notice that this proof uses a principle corresponding to the law of excluded middle. Intuitionists, who demur from excluded middle, do not accept the Lindenbaum lemma.

    It is what the variables range over. The interpretation function assigns appropriate extensions to the non-logical terms. In particular,. Thus we assume that every constant denotes something. Systems where this is not assumed are called free logics see the entry on free logic. The role of variable-assignments is to assign denotations to the free variables of open formulas. If the language contained function symbols, the denotation function would be defined by recursion. This is about as straightforward as it gets.

    This takes care of the atomic formulas. We now proceed to the compound formulas of the language, more or less following the meanings of the English counterparts of the logical terminology. The final clause is similar. Theorem 6, unique readability, assures us that this definition is coherent. At each stage in breaking down a formula, there is exactly one clause to be applied, and so we never get contradictory verdicts concerning satisfaction.

    As indicated, the role of variable-assignments is to give denotations to the free variables. We now show that variable-assignments play no other role. So we define. A straightforward induction establishes the following:. The definition corresponds to the informal idea that an argument is valid if it is not possible for its premises to all be true and its conclusion false. Our definition of logical consequence also sanctions the common thesis that a valid argument is truth-preserving--to the extent that satisfaction represents truth.

    Validity is the model-theoretic counterpart to deducibility. A sentence is logically true if and only if it is a consequence of the empty set. Logical truth is the model-theoretic counterpart of theoremhood. So a set of sentences is satisfiable if it has a model. Satisfiability is the model-theoretic counterpart to consistency. This is a model-theoretic counterpart to ex falso quodlibet see Theorem We have the following, as an analogue to Theorem We now present some results that relate the deductive notions to their model-theoretic counterparts. The first one is probably the most straightforward.

    So one would expect that an argument is deducible, or deductively valid, only if it is semantically valid. The other cases are about as straightforward. Corollary But this is impossible, given the clause for negation in the definition of satisfaction. For all we know so far, we may not have included enough rules of inference to deduce every valid argument. The converses to soundness and Corollary 19 are among the most important and influential results in mathematical logic.

    We begin with the latter. Proof: The proof of completeness is rather complex. We only sketch it here. One interesting feature of this construction, due to Leon Henkin, is that we build an interpretation of the language from the language itself, using some of the constants as members of the domain of discourse. Three-place predicates, etc. The variable assignments are similar. The other cases follow from the various clauses in the definition of satisfaction.

    Soundness and completeness together entail that an argument is deducible if and only if it is valid, and a set of sentences is consistent if and only if it is satisfiable. So we can go back and forth between model-theoretic and proof-theoretic notions, transferring properties of one to the other. Compactness holds in the model theory because all derivations use only a finite number of premises. The interpretation we produced was itself either finite or denumerably infinite. Thus, we have the following:. There is a stronger version of Corollary Notice that if two interpretations are equivalent, then they satisfy the same sentences.

    Proof: Like completeness, this proof is complex, and we rest content with a sketch. But we cannot rest content with the Skolem-hull, however. This also relies on the axiom of choice. Unfortunately, space constraints require that we leave this step as an exercise. In this paper I attempt to make good on this metaphor. In order to do this I introduce a modality that, put informally, stands to propositions as logical truth stands to sentences.

    The resulting theory, formulated in higher-order logic, also vindicates the Humean idea that fundamental properties and relations are freely recombinable and a variant of the structural idea that propositions can be decomposed into their fundamental constituents via logical operations.

    Indeed, it is seen that, although these ideas are seemingly distinct, they are not independent, and fall out of a natural and general theory about the granularity of reality. PDF In Prior proved a theorem that places surprising constraints on the logic of intentional attitudes, like "thinks that", "hopes that", "says that" and "fears that". Paraphrasing it in English, and applying it to "thinks", it states: If, at t, I thought that I didn't think a truth at t, then there is both a truth and a falsehood I thought at t.

    In this paper I explore a response to this paradox that exploits the opacity of attitude verbs, exemplified in this case by the operator "I thought at t that", to block Prior's derivation. According to this picture, both Leibniz's law and existential generalization fail in opaque contexts. In particular, one cannot infer from the fact that I'm thinking at t that I'm not thinking a truth at t, that there is a particular proposition such that I am thinking it at t.

    Moreover, unlike some approaches to this paradox the failure of existential generalization is not motivated by the idea that certain paradoxical propositions do not exist, for this view maintains that there is a proposition that I'm not thinking a truth at t. Several advantages of this approach over the non-existence approach are discussed, and models demonstrating the consistency of this theory are provided.

    Finally, the resulting considerations are applied to the liar paradox, and are used to provide a non-standard justification of a classical gap theory of truth. One of the main challenges for this sort of theory to explain the point of assertion, if not to assert truths can be met within this framework. PDF In this paper two paradoxes of infinity are considered through the lense of counterfactual logic, drawing heavily on a result of Kit Fine. I will argue that a satisfactory resolution of these paradoxes will have wide ranging implications for the logic of counterfactuals.

    I then situate these puzzles in the context of the wider role of counterfactuals, connecting them to indicative conditionals, probabilities, rationality and the direction of causation, and compare my own resolution of the paradoxes to alternatives inspired by the theories of Lewis and Fine. PDF An increasing amount of twenty-first century metaphysics is couched in explicitly hyperintensional terms.

    A prerequisite of hyperintensional metaphysics is that reality itself be hyperintensional: at the metaphysical level, propositions, properties, operators, and other elements of the type hierarchy, must be more fine-grained than functions from possible worlds to extensions. In this paper I develop, in the setting of type theory, a general framework for reasoning about the granularity of propositions and properties.

    The theory takes as primitive the notion of a substitution on a proposition property, etc. A class of structures are identified which can be used to model a wide range of positions about the granularity of reality; certain of these structures are seen to receive a natural treatment in the category of M-sets. PDF This paper formulates some paradoxes of inductive knowledge.

    Two responses in particular are explored: According to the first sort of theory, one is able to know in advance that certain observations will not be made unless a law exists. According to the other, this sort of knowledge is not available until after the observations have been made. Certain natural assumptions, such as the idea that the observations are just as informative as each other, the idea that they are independent, and that they increase your knowledge monotonically among others are given precise formulations.

    Some surprising consequences of these assumptions are drawn, and their ramifications for the two theories examined. Finally, a simple model of inductive knowledge is offered, and independently derived from other principles concerning the interaction of knowledge and counterfactuals.

    PDF Generalising on some arguments due to Arthur Prior and Dmitry Mirimanoff, we provide some further limitative results on what can be thought. PDF Individuals play a prominent role in many metaphysical theories. According to an individualistic metaphysics , reality is determined at least in part by the pattern of properties and relations that hold between individuals. A number of philosophers have recently brought to attention alternative views in which individuals do not play such a prominent role; in this paper I will investigate one of these alternatives.

    PDF In this paper I explore the logic of broad necessity. Definitions of what it means for one modality to be broader than another are formulated, and I prove, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. I show, moreover, that it is possible to give a reductive analysis of this necessity in extensional language using truth functional connectives and quantifiers.

    This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. I formulate this conjecture precisely in higher-order logic, and examine concrete cases in which it fails. I end by investigating the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. I give some philosophical reasons to think that the logic of broad necessity does not include the S5 principle.

    PDF We explore the view that Frege's puzzle is a source of straightforward counterexamples to Leibniz's law. Taking this seriously requires us to revise the classical logic of quantifiers and identity; we work out the options, in the context of higher-order logic. The logics we arrive at provide the resources for a straightforward semantics of attitude reports that is consistent with the Millian thesis that the meaning of a name is just the thing it stands for. We provide models to show that some of these logics are non-degenerate. I use this thesis about language and the negative result about disquotation to motivate the view that people do say things with utterances of paradoxical sentences, although they do not say the proposition you'd always expect, as articulated with a disquotational principle.

    Please note that the following is not the final draft: PDF The fact that physical laws often admit certain kinds of space-time symmetries is often thought to be problematic for substantivalism -- the view that space-time is as real as the objects it contains. The most prominent alternative, relationism, avoids these problems but at the cost of giving abstract objects rather than space-time points a pivotal role in the fundamental metaphysics.

    This incurs related problems, I shall argue, concerning the relation of the physical to the mathematical. In this paper I will present a version of substantivalism that respects Leibnizian theses about space-time symmetries, and argue that it is superior to both relationism and the more orthodox form of substantivalism. 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. In this critical discussion we examine one the books central claims: to have provided a theory of truth that avoids the revenge paradoxes. In the first part we assess this claim, and in the second part we investigate some features of Scharp's preferred theory of truth, ADT, and compare it with existing theories such as the Kripke-Feferman theory.

    In the appendix a simple model of Scharp's theory is presented, and some potential consistent ways to strengthen the theory are suggested. Penultimate draft: PDF Those inclined to positions in the philosophy of time that take tense seriously have typically assumed that not all regions of space-time are equal: one special region of space-time corresponds to what is presently happening. When combined with assumptions from modern physics this has the unsettling consequence that the shape of this favored region distinguishes people in certain places or people traveling at certain velocities.

    The foundations for an alternative classical non-linguistic approach is outlined which is not subject to the same kinds of problems. Although revenge paradoxes of different strengths can be formulated, they are found to be indeterminate at higher orders and not inconsistent. Reprinted in the Philosopher's Annual In this paper I present a precise version of Stalnaker's thesis and show that it is both consistent and predicts our intuitive judgments about the probabilities of conditionals.

    The thesis states that someone whose total evidence is E should have the same credence in the proposition expressed by 'if A then B' in a context where E is salient as they have conditional credence in the proposition B expresses given the proposition A expresses in that context. The thesis is formalised rigorously and two models are provided that demonstrate that the new thesis is indeed tenable within a standard possible world semantics based on selection functions.

    Unlike the Stalnaker-Lewis semantics the selection functions cannot be understood in terms of similarity. A probabilistic account of selection is defended in its place. I end the paper by suggesting that this approach overcomes some of the objections often leveled at accounts of indicatives based on the notion of similarity.

    It is suggested that this is problematic for theorists who endorse the principle that 'P' and ''P' is true' are always intersubstitutable. Penultimate draft: PDF fair causally isolated coins will be independently flipped tomorrow morning and you know this fact. I argue that the probability, conditional on your knowledge, that any coin will land tails is almost 1 if that coin in fact lands tails, and almost 0 if it in fact lands heads.

    I also show that the coin flips are not probabilistically independent given your knowledge. These results are uncomfortable for those, like Timothy Williamson, who take these probabilities to play a central role in their theorizing. PDF This paper presents a counterpart theoretic semantics for quantified modal logic based on a fleshed out account of Lewis's notion of a 'possibility'. According to the account a possibility consists of a world and some haecceitistic information about how each possible individual gets represented de re.

    A semantics for quantified modal logic based on evaluating formulae at possibilities is developed. It is shown that this framework naturally accommodates an actuality operator, addressing recent objections to counterpart theory, and is equivalent to the more familiar Kripke semantics for quantified modal logic with an actuality operator. The result of combining classical quantificational logic with modal logic proves necessitism -- the claim that necessarily everything is necessarily identical to something.

    The standard way to avoid these consequences is to weaken the theory of quantification to a certain kind of free logic. However, it has often been noted that in order to specify the truth conditions of certain sentences involving constants or variables that don't denote, one has to apparently quantify over things that are not identical to anything.

    In this paper I defend a contingentist, non-Meinongian metaphysics within a positive free logic. I argue that although certain names and free variables do not actually refer to anything, in each case there might have been something they actually refer to, allowing one to interpret the contingentist claims without quantifying over mere possibilia.

    This work improves on Hartry Field's recent results establishing consistency and omega-consistency of truth-theories with strong conditional logics.


    A novel method utilising the Banach fixed point theorem for contracting functions on complete metric spaces is invoked, and the resulting logic is shown to validate a number of principles which existing revision theoretic methods have so far failed to provide. Penultimate draft: PDF In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox.

    This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naive truth theory.

    The rise and fall and rise of logic | Aeon Essays

    On this basis I identify two natural subsystems of Pukasiewicz logic which individually, but not jointly, lack the problematic feature. Penultimate draft PDF A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical meta-theory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere.

    The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the meta-theory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated.

    It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete.

    Penultimate draft: PDF I consider two puzzles in which an agent undergoes a sequence of decision problems. In both cases it is possible to respond rationally to any given problem yet it is impossible to respond rationally to every problem in the sequence, even though the choices are independent. In particular, although it might be a requirement of rationality that one must respond in a certain way at each point in the sequence, it seems it cannot be a requirement to respond as such at every point for that would be to require the impossible. In Progress Vagueness and Thought. Book project with Oxford University Press.

    In progress. This paper has mostly be subsumed by the paper "Stalnaker's Thesis in Context" see above and other work that is currently in progress. However this paper does contain a discussion of the relation between Lewis's triviality results and the failures of conditional proof with side premisses that occur in the possible world semantics for indicatives, which is not in this later work. It also discusses a putative counterexample to the principle CSO. Since this paper has already been cited a few times I shall leave it up here.