Abstract
The theory of the logos that is typically considered the only existing logic and will be called here analytic logic is based on Aristotle’s work; in recent times it has gone through important developments, foremost among them a new and more satisfactory treatment of relations due to Frege. But what remains true of it is that language and meaning are organized by contraries: predicates that cannot be both true of the same thing (at the same time) and, by extension, sentences that cannot both be true (at the same time). Given contraries, words can be given definitive definitions, and arguments can reach conclusive conclusions—if they are judged to be correct, they are never going to be revised. Inferences are arguments of this sort: arguments from premises to conclusions. Because of their conclusive character, inferences can be formalized: one can abstract from them the minimum of content that makes it possible for the conclusions to be established and be assured that, whenever that same minimum of content is identified in an inference, a corresponding conclusion will be warranted.
Analytic logic has its most natural application to mathematical entities and to other entities (e.g., Platonic ideas) that philosophers have modeled on mathematical ones. Its most important anomalies (i.e., signs of imperfect adaptation between the theory and its subject matter) are (a) ambiguous words, which are associated with multiple (contrary) definitions; (b) vague predicates, which gradually merge with their opposites, thus contesting the very notion of a contrary; and (c) spacetime continuants, which constantly change all their identifying properties (into some of their contraries) while remaining self-identical. That these are anomalies, rather than problems to be eventually resolved within the theory by applying sufficient ingenuity, is suggested by their intractable nature; but accepting this view of them cannot follow from simple despair—it must be based on being presented with alternative theories of the same subject matter.
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Notes
- 1.
“That from the earliest times logic has traveled this secure course can be seen from the fact that since the time of Aristotle it has not had to go a single step backwards, unless we count the abolition of a few dispensable subtleties or the more distinct determination of its presentation, which improvements belong more to the elegance than to the security of that science. What is further remarkable about logic is that until now it has also been unable to take a single step forward, and therefore seems to all appearance to be finished and complete” (1998, Bviii). As I mentioned in the preface, there is more to be said about the relation of Kant to logic(s) than this passage suggests; I will get to it in due course.
- 2.
See, for example, Metaphysics 998b.
- 3.
A weaker version of the thesis alluded to here (our language determines how we think of the world) is relevant to what follows and will be addressed in Chap. 11.
- 4.
A whole diatribe has been running through the literature concerning what logical words are: a fake issue that has wasted people’s time. For the only sense that can be made of calling something a logical word is: a word whose logic is being studied (which makes the term dependent on the contingencies of what is being studied logically). Fortunately, as more and more words and phrases have become the subject of logical scrutiny, this issue has receded into the background, and it no longer has the currency it enjoyed at the time of Quine—that notorious issuer of draconian, unenforceable edicts. On this topic, see my (1999).
- 5.
The modality pertinent to a normative theory is deontic, as opposed to alethic: often what must happen does not happen (and “should” is then a more common term in it than “must”). So endorsing a logic is compatible with admitting that people often think, reason, and argue in violation of it, much as endorsing a theory of justice is compatible with admitting that people often do what the theory regards as unjust.
- 6.
If anything, the primacy of contraries has become even clearer in post-Fregean (analytic) logic. Consider, for example, the following: all (infinitely many) sentential connectives can be defined in terms of the single binary connective “either not A or not B” (the so-called Sheffer stroke), which states A and B to be contrary—or inconsistent with one another. Brandom (1979) identifies a variant of the Sheffer stroke that makes it possible to define both connectives and quantifiers. Strawson (1952), in addressing logical appraisal, selects the “notion of inconsistency or self-contradiction for detailed discussion” (p. 2).
- 7.
That logic only be answerable to normative intuitions is how I understand its traditional a priori status: its claims are independent of any empirical data and will not be impacted by them. It is an empirical fact that someone has those intuitions, but there is no empirical content to them: their normative character insulates them from the need to receive empirical confirmation, though there can still be an empirical debate among people empirically committed to the same logic—about what their intuitions in fact tell them. (So I reject a stronger notion of logic being a priori: that its correctness holds objectively, whatever people’s intuitions might be; and what I said on its setting up a program for the sciences must be understood in terms of what our intuitions warrant, not what the world objectively is.) On the other hand, if I had an intuition that now it is raining in Moscow, that descriptive intuition would have to be compared with the empirical data before it could be trusted. Whether there are any descriptive intuitions (say, the intuition that every event has a cause) that are as insulated from the empirical data as the normative ones and that can be trusted is, of course, the problem of whether synthetic a priori judgments are possible.
- 8.
Think of Russell’s theory of descriptions as a prime example of this kind of regimentation. And think of what the implications are of it having been called “a paradigm of philosophy” (by Ramsey 1990, p. 1).
- 9.
“Logic, as it is generally understood, is the organ with which we philosophize. But just as it may be possible for a craftsman to excel in making organs and yet not know how to play them, so one might be a great logician and still be inexpert in making use of logic. Thus we have many people who theoretically understand the whole art of poetry and yet are inept at composing mere quatrains; others enjoy all the precepts of da Vinci and yet do not know how to paint a stool. Playing the organ is taught not by those who make organs, but by those who know how to play them; poetry is learned by continual reading of the poets; painting is acquired by continual painting and designing; the art of proof, by the reading of books filled with demonstrations—and these are exclusively mathematical works, not logical ones” (2001, pp. 39–40).
- 10.
“When things have only a name in common and the definition of being which corresponds to the name is different, they are called homonymous. Thus, for example, both a man and a picture are animals. These have only a name in common and the definition of being which corresponds to the name is different; for if one is to say what being an animal is for each of them, one will give two distinct definitions” (1a).
- 11.
The introduction of new words is often justified by saying that what they refer to is also new and that the use of existing words might generate confusion with the existing meanings of those words. In the next chapter, we will see that the very notion of what is new here is controversial: depending on what logic you adopt, it might mean something radically distinct from whatever existed previously or, instead, a new phase of a continuing process (in which case the use of a new word would itself be confusing).
- 12.
“Logical illusion, which consists in the mere imitation of the form of reason (the illusion of fallacious inferences) arises solely from a failure of attentiveness to the logical rule. Hence as soon as that attentiveness is focused on the case before us, logical illusion entirely disappears” (1998, A296–97 B353).
- 13.
“The sea-anemones or sea-nettles, as they are variously called, are not Testacea at all, but lie outside the recognized groups. Their constitution approximates them on one side to plants, on the other to animals. For seeing that some of them can detach themselves and can fasten upon their food, and that they are sensible of objects which come in contact with them, they must be considered to have an animal nature. The like conclusion follows from their using the asperity of their bodies as a protection against their enemies. But, on the other hand, they are closely allied to plants, firstly by the imperfection of their structure, secondly by their being able to attach themselves to the rocks, which they do with great rapidity, and lastly by their having no visible residuum notwithstanding that they possess a mouth” (Parts of Animals 681a–b).
- 14.
In Chap. 9 we will see that this strategy, and the problem just mentioned about it, mirror the strategy of representing continuous structures like a geometrical line by the use of infinitely many numbers, and the problem that other strategy runs afoul of.
- 15.
“[T]wo bodies existing both at once would be one and the same numerical body. For if, for example, that ship of Theseus, concerning the difference whereof made by continued reparation in taking out the old planks and putting in new, the sophisters of Athens were wont to dispute, were, after all the planks were changed, the same numerical ship it was at the beginning; and if some man had kept the old planks as they were taken out, and by putting them afterwards together in the same order, had again made a ship of them, this, without doubt, had also been the same numerical ship with that which was at the beginning; and so there would have been two ships numerically the same, which is absurd” (1839, pp. 136–137).
- 16.
“An elastic ball that strikes another one in a straight line communicates to the latter its whole motion, hence its whole state (if one looks only at their positions in space). Now assuming substances, on the analogy with such bodies, in which representations, together with consciousness of them, flow from one to another, a whole series of these substances may be thought, of which the first would communicate its state, together with its consciousness, to the second, which would communicate its state, together with that of the previous substance, to a third substance, and this in turn would share the states of all previous ones, together with their consciousness and its own. The last substance would thus be conscious of all the states of all the previously altered substances as its own states, because these states would have been carried over to it, together with the consciousness of them; and in spite of this it would not have been the very same person in all these states” (1998, A363–364fn).
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Brandom, Robert Boyce. “A Binary Sheffer Operator Which Does the Work of Quantifiers and Sentential Connectives.” Notre Dame Journal of Formal Logic 20 (1979), pp. 262–264.
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Author information
Authors and Affiliations
Interlude: A Conversation Between Two Analytic Neighbors
Interlude: A Conversation Between Two Analytic Neighbors
- Jack::
-
Hi Don. How are you doing?
- Don::
-
Just the person I was looking for. I need to talk to you.
- Jack::
-
What about?
- Don::
-
It’s about our backyards. See, it’s been a while since we moved in…
- Jack::
-
A couple of months.
- Don::
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Indeed, and so far, we have been leaving everything open. Anyone can just wander back and forth between your property and mine.
- Jack::
-
Is there anything wrong with that?
- Don::
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Yes, I think a lot is wrong with that. We should build a fence and share the expense.
- Jack::
-
That’s some news. Why would I want to do that? I like the open space. I like the view. Why would I want to shut myself up in a cage?
- Don::
-
Well, to begin with, your dog always roams freely around here.
- Jack::
-
Immanuel is very friendly; he has never hurt anyone.
- Don::
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Hasn’t hurt me or my family. But suppose we want to have company, a barbecue for a few friends, a party.
- Jack::
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He wouldn’t bother them either.
- Don::
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Maybe not, but some people just don’t like dogs. They are afraid, and dogs can feel that and become aggressive.
- Jack::
-
Immanuel would never react that way.
- Don::
-
Still, I would not want my guests to be troubled.
- Jack::
-
Even if you knew that there would be nothing to worry about?
- Don::
-
Yes, because they wouldn’t know it, or they wouldn’t believe it, and that’s enough. If they don’t want your dog here, your dog shouldn’t be here, end of story. There should be a fence that keeps him inside your property.
- Jack::
-
Just tell me when you’re having company and I’ll keep Immanuel inside the house. I don’t see why we should pay for a major construction project because of something that will happen only once or twice a month.
- Don::
-
Well, it’s not just that. There’s also the issue of privacy.
- Jack::
-
What do you mean?
- Don::
-
In my own home, and in my backyard I should feel like I’m in my own home, I should be able to do whatever I please. Go around naked, if I want to, or make love in the grass. I shouldn’t constantly have the impression that I am being watched by my neighbor.
- Jack::
-
I don’t watch you, Don. I couldn’t care less what you do.
- Don::
-
It doesn’t matter if you watch me or not. What I said is that I should not have the impression that you are watching. And with all this open view, I have that impression. I don’t feel comfortable in my own home.
- Jack::
-
Don, what you are saying is ridiculous. See how our properties slope on this hill? See those other houses further up, no more than a hundred feet away? Even if you raise a fence to protect yourself from my eyes, what’s to prevent those other people from watching? Are you going to build a fortress all around you? For all you know, anyone of them could be directing a telescope into your backyard and observing every detail of what happens there, even recording it for posterity if they wish, and there is nothing you could do about it because they would be on their property, doing what makes them feel at home.
- Don::
-
If they did that, I’d sue their asses. But why don’t you let me worry about that and take care of it as I see fit, if and when I have a problem with it? I am trying to resolve an issue I have with you, and it’s irrelevant whether or not I have issues with anyone else.
- Jack::
-
Alright, alright. So what about this? When you want to go around naked or make love in the grass, you just send me a text that I should stay indoors, and I’ll oblige.
- Don::
-
You are unreasonable! You prefer to shut yourself up and your dog, just so that you can keep this space open. Why do you care so much about it?
- Jack::
-
People are different, we have to accept that. I enjoy the openness, the view, the sense of freedom it gives me. And I know that you will not always go around naked or make love in the grass; so most of the time I can be out here and even get to see you and have a nice conversation with you—not about a fence I hope.
- Don::
-
And because you want to enjoy the view and your sense of freedom, I should be forced to send you a text every time I want you out of the way? What about my freedom? The freedom to forget that you even exist?
- Jack::
-
There are pros and cons to every situation, Don. I am asking you to stand the little inconvenience of letting me know when you don’t want me here, but at the same time if you do want me, if you need any help, you can just call on me and I will come to assist you, without the trouble of ringing bells, getting keys, or opening doors.
- Don::
-
Actually all this easy coming and going you seem to value so much is another reason why we should have a fence.
- Jack::
-
Oh no… There are other reasons?
- Don::
-
Sure. Did you hear about the break-in last week?
- Jack::
-
No, I didn’t. What happened?
- Don::
-
A ne’er-do-well came into the neighborhood looking for something to steal.
- Jack::
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To buy drugs with, I would think.
- Don::
-
Most likely. A guy at the bottom of the hill left his house key under the mat for his cleaning lady, and the crook found it and entered the house to steal jewels and computers and other nice stuff. He even took a briefcase to put everything in.
- Jack::
-
Very thoughtful of him. But why are you telling me this story now? What does it have to do with us and with your fence?
- Don::
-
It has lots to do with it. The thug entered that guy’s house but only that house, because the house was fenced in on all sides. If the situation had been like ours he would have robbed us both.
- Jack::
-
Don, I don’t have a cleaning lady, you know it. And I never leave my key under the mat, or in a pot, or anywhere outside the house.
- Don::
-
That may very well be. But how do I know that you won’t forget to lock your door sometime or that someone else in your family won’t?
- Jack::
-
Do you think we are stupid or senile? Do you think we don’t care about being robbed, without you doing the worrying for us?
- Don::
-
You are probably the most responsible people in the world, but why should I base my own sense of safety on trusting you? Why should I make myself dependent on that trust? Every point of entrance into a house is also a weakness for it, and I have enough to worry about when it comes to the weaknesses of my own house without having to fret about the fact that every point of entrance into your house is also a point of entrance into mine.
- Jack::
-
Look, Don. You called me unreasonable, but it’s what you say that doesn’t make any sense. Because there may be times, rare times I take it, when you want to have a barbecue in the backyard, or even rarer times when you want to go around naked or make love in the grass, or because it is remotely possible, though highly unlikely, that sometimes one of us will leave our door open and that on just that occasion a crook will be prowling the neighborhood and be so lucky to find it open—just because of all these small contingencies that will never amount to more than (and I am being generous) 10 days out of the year, I should wake up all 365 days of the year to contemplate the wall of a cage! Don’t you see what an imposition you want to enforce on me?
- Don::
-
You don’t understand. It doesn’t matter how many days I actually do those things. I feel caged, constrained by your presence all the time, all 365 days; I feel that it is an imposition on me to have to inform you when I want some privacy. I told you: I don’t want to be forced to think about you all the time. If I care to see you and talk to you, I will come to your door and ring your bell; if I need your help, I will text you. But most of the time, I just want to be left alone, with you totally out of the picture.
- Jack::
-
As I said, Don, people are different. You clearly feel one way and I feel another, and our feelings are irreconcilable; we feel contrary things.
- Don::
-
But don’t you see that my feelings should count more?
- Jack::
-
Now, that’s great. Why would that be?
- Don::
-
Because the feelings that count the most are hurt feelings. Pain matters more than pleasure. Suppose someone takes pleasure in torturing another, and when the other guy tells him to stop he says: we are different; I would feel violated if you denied me this form of expression as much as you feel violated by the pain I inflict on you. Would that make any sense? Do you think there is any parity between these two contrary feelings? Should they be treated the same?
- Jack::
-
No, they should not. But who is the torturer here?
- Don::
-
You are. You are telling me that I should suffer my anxiety and my exasperation in silence so that you can have the pleasure of waking up to an open view.
- Jack::
-
But don’t you see that your description could be reversed? Then it would be my pain of waking up in a cage that counts less, for you, than your pleasure of finally being relieved of my presence!
- Don::
-
Just like the torturer. He could reverse the description too and say that his pain in not being able to express himself should count more than the pleasure his victim would feel in being relieved of his presence.
- Jack::
-
But that would be absurd!
- Don::
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No more absurd than your own redescription.
- Jack::
-
Wait a minute. The torturer is inflicting physical pain, is literally destroying the body of his victim. The pain he would feel if they made him stop, even admitting that we can allow such talk, would only be psychological; so clearly, there is no parity. But we are both talking about a psychological condition.
- Don::
-
Sure, Jack, but there are also psychological torturers. People who put others in double-bind situations, with no way out, no viable option for how to behave, and end up destroying them to an extent even more radical and irremediable than many physical torturers.
- Jack::
-
So I would be torturing you psychologically?
- Don::
-
That’s what I have been saying.
- Jack::
-
And your proposal for how to remedy that is to torture me psychologically.
- Don::
-
I don’t see it that way.
- Jack::
-
I know that you don’t see it that way, but it is your word against mine. So all you are saying is that your pain should count more because your word counts more.
- Don::
-
I see that there is no way to resolve this.
- Jack::
-
No, there isn’t. But I cannot prevent you from raising all the fences you want on your own property. I won’t like it, just as I wouldn’t if you planted a tree that blocked the view; but I can’t decide for you. So go ahead if that’s what you want; but be sure that you stay clear of everything that is mine. And be sure that I won’t pay a penny for it.
- Don::
-
If I had been happy to do what you just said, I would not have needed to talk to you, and I could have avoided all this aggravation.
- Jack::
-
Yes, you could have, and I could have too. I could have relaxed down there in the shade instead of listening to your complaints. We have been wasting our time. Rather, you have been wasting my time.
- Don::
-
I don’t think so. Now that we have agreed that there will be a fence, we must get to the sharing of the costs of it.
- Jack::
-
There will be no sharing, I told you. You do it on your own in your own backyard.
- Don::
-
I don’t think that’s fair.
- Jack::
-
Fair? You want to force me to do something I don’t want and I don’t like and want me to pay for it, and I am not being fair?
- Don::
-
Sure, because once the fence is up you will profit from it.
- Jack::
-
I will hate every inch of it!
- Don::
-
You say so, but it may be a tactic for having me take care of the whole project. Once it’s up, you will enjoy privacy and security as much as I will, at my expense.
- Jack::
-
I told you I don’t care about privacy and security—about what you call privacy and security.
- Don::
-
Yes, you told me. But the privacy and security the fence will provide you are objective goods, and for all I know, they might even increase the value of your house. That you say that they have no value does not make it so.
- Jack::
-
They have no value for me.
- Don::
-
Even if I give you the benefit of the doubt and believe that you not playing games with me, all I can grant you is that they have no value for you now. Tomorrow, or next month, you will see the light, or you will sell the house, at a higher price, and the new owner will see the light and appreciate the privacy and security the fence gives him, and I will have been paying for it.
- Jack::
-
OK, so suppose I agree that privacy and security are objective values. Do you agree that openness and freedom are also objective values? And that I have a right to set my own priorities among all these values? That I can decide to value freedom and openness more than your damned privacy and security?
- Don::
-
There must be an objective way of settling this dispute. There must be experts who can determine how to rank these values objectively.
- Jack::
-
Yes, and there must be laws against the kind of harassment you have been subjecting me to. There must be ways of protecting an honest citizen from this invasion of privacy—of my privacy, that you seem to have no respect for.
- Don::
-
I have a right to interrupt your lazy contemplation of openness to bring up an important matter.
- Jack::
-
I don’t find this matter at all important. I find it petty, not to mention intrusive.
- Don::
-
That’s another thing we see differently. But let me give it another shot. You have agreed that privacy and security are objective values, and I may agree that freedom and openness are too.
- Jack::
-
Now that’s making progress!
- Don::
-
You also said that you rank freedom and openness higher, but perhaps you can tell me how much higher. Do you think that freedom and openness are worth twice as much as privacy and security, or three times?
- Jack::
-
A hundred times!
- Don::
-
Don’t be foolish. Give me a reasonable figure.
- Jack::
-
Ten times.
- Don::
-
Then what if I asked you to pay for one tenth of the fence and to build one tenth of it on your property?
- Jack::
-
I would say, No.
- Don::
-
But why?
- Jack::
-
Because you should be compensating me for the loss of something that I value so much more!
- Don::
-
So no compromise is possible.
- Jack::
-
No. Build whatever cage you want, on your property, and leave me alone.
- Don::
-
You will be hearing from my lawyers.
- Jack::
-
My lawyers will.
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Bencivenga, E. (2017). Analytic Logic. In: Theories of the Logos. Historical-Analytical Studies on Nature, Mind and Action, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-63396-1_2
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