What makes logic logical

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Antoinette Acosta September 20, Related Society Needs Science—and Science Needs Society Scientists must strengthen our connections with the public to better ensure that accurate and modern insights into human behavior are used to inform decisions.

In Their Own Words: Lives Lost in Excerpts from the research of a few of the remarkable psychological scientists we said goodbye to this year. Arena APS priorities are echoed in the U. We use technologies, such as cookies, to customize content and advertising, to provide social media features and to analyse traffic to the site. We also share information about your use of our site with our analytics partners. No one doubts that our eyesight has evolved, nor that it is species-specific.

We see the world very differently from the ways in which colour-blind animals do, or insects with their compound eyes. Animals with binocular vision see the world differently from those that have eyes set to look out the side for predators. But none of this is an argument that the world we see does not exist. The cat sees a bird on my lawn entirely differently to how I do. The blackbird, in turn, sees the cat in a way I would not recognise, while the worm it's eating doesn't see any of us at all.

But the worm, the cat, the bird and I all exist. Our visions of each other are imperfect but not delusional. I think the same goes for our intuitions about logic, rationality, and arithmetic. The really interesting question is whether the same is true of our moral intuitions or discoveries. Note that this reasoning is very general and independent of what our particular pretheoretic conception of logical truth is. An especially significant case in which this reasoning can be applied is the case of first-order quantificational languages, under a wide array of pretheoretic conceptions of logical truth.

It is typical to accept that all formulae derivable in a typical first-order calculus are universally valid, true in all counterfactual circumstances, a priori and analytic if any formula is. This means that one can convince oneself that both derivability and model-theoretic validity are extensionally correct characterizations of our favorite pretheoretic notion of logical truth for first-order languages, if our pretheoretic conception is not too eccentric.

The situation is not so clear in other languages of special importance for the Fregean tradition, the higher-order quantificational languages. We may call this result the incompleteness of second-order calculi with respect to model-theoretic validity. In this situation it's not possible to apply Kreisel's argument for 5. Different authors have extracted opposed lessons from incompleteness.

A common reaction is to think that model-theoretic validity must be unsound with respect to logical truth. This is especially frequent in philosophers on whose conception logical truths must be a priori or analytic.

One idea is that the results of a priori reasoning or of analytic thinking ought to be codifiable in a calculus. Wagner , p. But even if we grant this idea, it's doubtful that the desired conclusion follows.

Suppose that i every a priori or analytic reasoning must be reproducible in a calculus. From iii and i it follows of course that there are model-theoretically valid formulae that are not obtainable by a priori or analytic reasoning. But the step from ii to iii is a typical quantificational fallacy. From i and ii it doesn't follow that there is any model-theoretically valid formula which is not obtainable by a priori or analytic reasoning. The only thing that follows from ii alone under the assumptions that model-theoretic validity is sound with respect to logical truth and that logical truths are a priori and analytic is that no calculus sound with respect to model-theoretic validity can by itself model all the a priori or analytic reasonings that people are able to make.

But it's not sufficiently clear that this should be intrinsically problematic. After all, a priori and analytic reasonings must start from basic axioms and rules, and for all we know a reflective mind may have an inexhaustible ability to find new truths and truth-preserving rules by a priori or analytic consideration of even a meager stock of concepts. But this view is just one problematic idea about how apriority and analyticity should be explicated. See also Etchemendy , chs. Another type of unsoundness arguments attempt to show that there is some higher-order formula that is model-theoretically valid but is intuitively false in a structure whose domain is a proper class.

These arguments thus question the claim that each meaning assignment's validity-refuting power is modeled by some set-theoretic structure, a claim which is surely a corollary of the first implication in 5. The most widespread view among set theorists seems to be that there are no formulae with that property in Fregean languages, but it's certainly not an absolutely firm belief of theirs.

Note that these arguments offer a challenge only to the idea that universal validity as defined in section 2. The arguments we mentioned in the preceding paragraph and in 2. In fact, worries of this kind have prompted the proposal of a different kind of notions of validity for Fregean languages , in which set-theoretic structures are replaced with suitable values of higher-order variables in a higher-order language for set theory, e.

Both set-theoretic and proper class structures are modeled by such values, so these particular worries of unsoundness do not affect this kind of proposals. In general, there are no fully satisfactory philosophical arguments for the thesis that model-theoretic validity is unsound with respect to logical truth in higher-order languages.

Are there then any good reasons to think that derivability in any calculus sound for model-theoretic validity must be incomplete with respect to logical truth? There don't seem to be any absolutely convincing reasons for this view either. The main argument the first version of which was perhaps first made explicit in Tarski a, b seems to be this. It's certainly not a formula false in a proper class structure. The argument concludes that for any calculus there are logically true formulae that are not derivable in it.

From this it has been concluded that derivability in any calculus must be incomplete with respect to logical truth. On these assumptions it is certainly very reasonable to think that derivability, in any calculus satisfying 4 , must be incomplete with respect to logical truth. But in the absence of additional considerations, a critic may question the assumptions, and deny relevance to the argument.

The second assumption would probably be questioned e. The first assumption actually underlies any conviction one may have that 4 holds for any one particular higher-order calculus. Note that if we denied that the higher-order quantifiers are logical expressions we could equally deny that the arguments presented above against the soundness of model-theoretic validity with respect to logical truth are relevant at all.

It is often pointed out in this connection that higher-order quantifications can be used to define sophisticated set-theoretic properties that one cannot define just with the help of first-order quantifiers.

Defenders of the logical status of higher-order quantifications, on the other hand, point to the wide applicability of the higher-order quantifiers, to the fact that they are analogous to the first-order quantifiers, to the fact that they are typically needed to provide categorical axiomatizations of mathematical structures, etc.

See Quine , ch. Zalta for very helpful comments on an earlier version of this entry. On standard views, logic has as one of its goals to characterize and give us practical means to tell apart a peculiar set of truths, the logical truths, of which the following English sentences are paradigmatic examples: 1 If death is bad only if life is good, and death is bad, then life is good.

The Nature of Logical Truth 1. The Mathematical Characterization of Logical Truth 2. Wallies ed. Allison, H. Aristotle, Analytica Priora et Posteriora , W. Ross ed. Azzouni, J. Oxford: Oxford University Press. Beall, Jc and G. Belnap, N. Bernays, P. Mancosu, in Mancosu ed. Boghossian, P. Hale and C. Wright eds. Boghossian and C. Peacocke eds. Bolzano, B. George, Oxford: Blackwell, BonJour, L. Bonnay, D. Boolos, G. Capozzi, M. Haaparanta ed. Carnap, R. Schilpp ed. Carroll, L. Chihara, C. Schirn ed.

Coffa, J. Dogramaci, S. Dummett, M. Etchemendy, J. Patterson ed. Feferman, S. Field, H. Caret and O. Hjortland eds. Franks, C. Rush ed. Frege, G. Bauer-Mengelberg, in J.

McGuinness ed. Grice, P. Griffiths, O. Hacking, I. Hanna, R. Hanson, W. Hobbes, T. But even if the latter philosophical ambitions fail, a formal ontology can still be a most useful representational tool. One interesting view about the relationship between formal languages, ontology, and meta-ontology is the one developed by Carnap in the first half of the 20th century, and which is one of the starting points of the contemporary debate in ontology, leading to the well-known exchange between Carnap and Quine, to be discussed below.

According to Carnap one crucial project in philosophy is to develop frameworks that can be used by scientists to formulate theories of the world. Such frameworks are formal languages that have a clearly defined relationship to experience or empirical evidence as part of their semantics. For Carnap it was a matter of usefulness and practicality which one of these frameworks will be selected by the scientists to formulate their theories in, and there is no one correct framework that truly mirrors the world as it is in itself.

The adoption of one framework rather than another is thus a practical question. Carnap distinguished two kinds of questions that can be asked about what there is.

Such questions vary in degree of difficulty. What the philosophers aim to ask, according to Carnap, is not a question internal to the framework, but external to it. They aim to ask whether the framework correctly corresponds to reality, whether or not there really are numbers. The external questions that the metaphysician tries to ask are meaningless. Ontology, the philosophical discipline that tries to answer hard questions about what there really is is based on a mistake.

The question it tries to answer are meaningless questions, and this enterprise should be abandoned. Philosophers should thus not be concerned with O2 , which is a discipline that tries to answer meaningless questions, but with L1 , which is a discipline that, in part, develops frameworks for science to use to formulate and answer real questions.

One common criticism is that it relies on a too simplistic conception of natural language that ties it too closely to science or to evidence and verification. First they could change the theory, but stay in the same framework.

Secondly, they could move to a different framework, and formulate a new theory within that framework. These two moves for Carnap are substantially different. Quine would want to see them as fundamentally similar. Thus some such internal statements would be analytic truths, and Quine is well known for thinking that the distinction between analytic and synthetic truths is untenable. On the other hand, Quine and Carnap agree that ontology in the traditional philosophical sense is to be rejected.

Traditionally ontology has often, but not always, been an armchair, a priori, investigation into the fundamental building blocks of reality. As such it is completely separated from science. See his Quine See Yablo for more on the debate between Quine and Carnap, which contains many references to the relevant passages.

The view on ontological commitment discussed in section 4. In particular, we look at our best overall scientific theory of the world, which contains physics and the rest. For example, Stephen Yablo has argued that an internal-external distinction could be understood along the lines of the fictional-literal distinction.

And, he argues in Yablo , since there is no fact about this distinction, ontology, in the sense of O2 , rests on a mistake and is to be rejected, as Carnap did. On the other hand, Thomas Hofweber has argued that an internal-external distinction with many of the features that Carnap wanted can be defended on the basis of facts about natural language, but that such a distinction will not lead to a rejection of ontology, in the sense of O2.

See Hofweber and Hofweber Robert Kraut in Kraut has defended an expressivist reading of the internal-external distinction, and with it some Carnapian consequences for ontology.

See in particular Hirsch and Thomasson For various views about the effects of Carnap on the contemporary debate in ontology, see Blatti and Lapointe Although ontology is often understood as the discipline that tries to find out what there is, or what exists, this is rejected by many in the contemporary debate. These philosophers think that the job of ontology is something different, and there is disagreement among them what it is more precisely.

Among the proposed options are the projects of finding out what is real, or what is fundamental, or what the primary substances are, or what reality is like in itself, or something like this. Proponents of these approaches often find the questions about what there is too inconsequential and trivial to take them to be the questions for ontology.

Whether there are numbers, say, is trivially answered in the affirmative, but whether numbers are real, or whether they are fundamental, or primary substances, etc. See Fine and Schaffer for two approaches along these lines. But such approaches have their own problems. For example, it is not clear whether the question whether numbers are real is any different than the question whether numbers exist.

If one were to ask whether the Loch Ness monster is real, it would naturally be understood as just the same question as whether the Loch Ness monster exists. If it is supposed to be a different question, is this due to simple stipulation, or can we make the difference intelligible? Similarly, it is not clear whether the notion of what is fundamental can carry the intended metaphysical weight.

Thus to ask whether numbers are fundamental is not easily seen as a metaphysical alternative to the approach to ontology that asks whether numbers exist. See Hofweber and chapter 13 of Hofweber for a critical discussion of some approaches to ontology that rely on notions of reality or fundamentality.

Whether such approaches to ontology are correct is a controversial topic in the debate about ontology which we will not focus on here.

However, this approach gives rise to a special connection between logic and ontology which we will discuss in the following. The relation between the different approaches to ontology mentioned just above is unclear. Is something that is part of reality as it is in itself something which is fundamental, or which is real in the relevant sense?

Although it is unclear how these different approaches relate to each other, all of them have the potential for allowing for that our ordinary description of the world in terms of mid-size objects, mathematics, morality, and so on, is literally true, while at the same time these truths leave it open what the world, so to speak, deep down, really, and ultimately is like.

To use one way of articulating this, even though there are tables, numbers, and values, reality in itself might contain none of them. Reality in itself might contain no objects at all, and nothing normative.

Or it might. The ordinary description of the world, on this conception, leaves it largely open what reality in itself is like. To find that out is the job of metaphysics, in particular ontology. We might, given our cognitive setup, be forced to think of the world as one of objects, say. But that might merely reflect how reality is for us.

How it is in itself is left open. Whether the distinction between reality as it is for us and as it is in itself can be made sense of is an open question, in particular if it is not simply the distinction between reality as it appears to us, and as it really is.

This distinction would not allow for the option that our ordinary description of reality is true, while the question how reality is in itself is left open by this. If our ordinary description were true then this would mean that how reality appears to us is how it in fact is. But if this distinction can be made sense of as intended then it gives rise to a problem about how to characterize reality as it is in itself, and this gives rise to a role for logic in the sense of L1. If we are forced to think of the world in terms of objects because of our cognitive makeup then it would be no surprise that our natural language forces us to describe the world in terms of objects.

And arguably some of the central features of natural languages do exactly that. It represents information in terms of subject and predicate, where the subject paradigmatically picks out an object and the predicate paradigmatically attributes a property to it. If this is correct about natural language then it seems that natural language is utterly unsuitable to describe reality as it is in itself if the latter does not contain any objects at all.

But then, how are we to describe reality as it is in itself? Some philosophers have proposed that natural language might be unsuitable for the purposes of ontology. It might be unsuitable since it carries with it too much baggage from our particular conceptual scheme. See Burgess for a discussion. Or it might be unsuitable since various expressions in it are not precise enough, too context sensitive, or in other ways not ideally suited for the philosophical project. These philosophers propose instead to find a new, better suited language.

Such a language likely will be a major departure from natural language and instead will be a formal, artificial language. The task thus is to find the fundamental language, a language in the sense of L1 , to properly carry out ontology, in the new and revised sense of O2 : the project of finding out what reality fundamentally, or in itself, etc.

For a critical discussion of the proposal that we should be asking the questions of ontology in ontologese, see chapter 10 of Thomasson But this idea of a connection between L1 and O2 is not unproblematic.

First there is a problem about making this approach to O2 more precise. On this understanding it would simply be the world as it is except with no humans in it, which would in many of its grander features be just as it in fact is. But then what does it mean? Second, there is a serious worry about how the formal language which is supposed to be the fundamental language is to be understood. In particular, is it supposed to be merely an auxiliary tool, or an essential one? This question is tied to the motivation for a formal fundamental language in the first place.

If it is merely to overcome ambiguities, imperfections, and context sensitivities, then it most likely will merely be an auxiliary, but not essential tool.

After all, within natural language we have many means available to get rid of ambiguities, imperfections and context sensitivities. Scope ambiguities can be often quite easily be overcome with scope markers. Other imprecisions can often, and maybe always, be overcome in some form or other.

On other hand, the formal fundamental language might be taken to be essential for overcoming shortcomings or inherent features of our natural language as the one alluded to above. If the subject-predicate structure of our natural languages brings with it a object-property way of representing the world, and if that way of representing the world is unsuitable for representing how reality is in itself, then a completely different language might be required, and not simply be useful, to describe fundamental reality.

But then what do sentences in the fundamental language mean? Can we even make sense of the project of finding out which sentences in such a language are correct? A sample debate related to the issues discussed in this section is the debate about whether it might be that reality in itself does not contain any objects.

See, for example, Hawthorne and Cortens , Burgess , and Turner Here the use of a variable and quantifier free language like predicate functor logic as the fundamental language is a recurring theme. One way to understand logic is as the study of the most general forms of thought or judgment, what we called L4. One way to understand ontology is as the study of the most general features of what there is, our O3. Now, there is a striking similarity between the most general forms of thought and the most general features of what there is.

Take one example. Many thoughts have a subject of which they predicate something. What there is contains individuals that have properties. It seems that there is a kind of a correspondence between thought and reality: the form of the thought corresponds to the structure of a fact in the world. And similarly for other forms and structures. Does this matching between thought and the world ask for a substantial philosophical explanation?

Is it a deep philosophical puzzle? To take the simplest example, the form of our subject-predicate thoughts corresponds perfectly to the structure of object-property facts. If there is an explanation of this correspondence to be given it seems it could go in one of three ways: either the form of thought explains the structure of reality a form of idealism , or the other way round a form of realism , or maybe there is a common explanation of why there is a correspondence between them, for example on a form of theism where God guarantees a match.

At first it might seem clear that we should try to give an explanation of the second kind: the structure of the facts explains the forms of our thoughts that represent these facts.

And an idea for such an explanation suggests itself. Our minds developed in a world full of objects having properties. If we had a separate simple representation for these different facts then this would be highly inefficient. After all, it is often the same object that has different properties and figures in different facts, and it is often the same property that is had by different objects. So, it makes sense to split up our representations of the objects and of the properties into different parts, and to put them back together in different combinations in the representation of a fact.

And thus it makes sense that our minds developed to represent object-property facts with subject-predicate representations. Therefore we have a mind whose thoughts have a form which mirrors the structure of the facts that make up the world.

This kind of an explanation is a nice try, and plausible, but it is rather speculative. That our minds really developed this way in light of those pressures is a question that is not easy to answer from the armchair. Maybe the facts do have a different structure, but our forms are close enough for practical purposes, i.

And maybe the correspondence does obtain, but not for this largely evolutionary reason, but for a different, more direct and more philosophical or metaphysical reason. To explain the connection differently one could endorse the opposite order of explanatory priority, and argue that the form of thought explains the structure of the world.

This would most likely lead to an idealist position of sorts. It would hold that the general features of our minds explain some of the most general features of reality. This strategy for explaining the similarity has the problem of explaining how there can be a world that exists independently of us, and will continue to exist after we have died, but nonetheless the structure of this world is explained by the forms of our thoughts.

Maybe this route could only be taken if one denies that the world exists independently of us, or maybe one could make this tension go away. In addition one would have to say how the form of thought explains the structure of reality. For one attempt to do this, see Hofweber But maybe there is not much to explain here. Maybe reality does not have anything like a structure that mirrors the form of our thoughts, at least not understood a certain way. And all that is required for that is a world that contains John, but not also another thing, the property of smoking.

Thus a structural match would be less demanding, only requiring a match between objects and object directed thought, but no further match. Such a view would be broadly nominalistic about properties, and it is rather controversial. Another way in which there might be nothing to explain is connected to philosophical debates about truth.