Maverick Philosopher

Footnotes to Plato from the foothills of the Superstition Mountains

Broadly Logical Modality

David Brightly has difficulty with the notion of broadly logical modality.  Let me see if I can clarify this notion sufficiently to satisfy him.  It might be best to begin with the notion of narrowly logical impossibility.  I'll number my paragraphs so that David can tell me exactly where he disagrees or finds obscurity.

1.  There are objects and states of affairs and propositions that can be known a priori to be impossible because they violate the Law of Non-Contradiction (LNC).  Thus a plane figure that is both round and not round at the same time, in the same respect, and in the same sense of 'round,' is impossible, absolutely impossible, simply in virtue of its violation of LNC.  I will say that such an object is narrowly logically (NL) impossible.  Hereafter, to save keystrokes, I will not mention the 'same time, same respect, same sense' qualification which  will be understood to be in force.

2.  But what about a plane figure that is both round and square?  Is it NL-impossible?  No.  For by logic alone one cannot know it to be impossible.  One needs a supplementary premise, the necessary truth grounded in the meanings of 'round' and 'square' that nothing that is round is square.  We say, therefore,  that the round square is broadly logically (BL) impossible.  It is not excluded from the realm of the possible by logic alone, which is purely formal, but by logic plus a 'material' truth, namely the necessary truth just mentioned.

3.  If there are BL-impossible states of affairs such as There being a round square, then there are BL-necessary states of affairs such as There being no round square.  Impossibility and necessity are interdefinable: a state of affairs is necessary iff its negation is impossible.  It doesn't matter whether the modality is NL, BL, or nomological (physical).  It is clear, then, that there are BL-impossible and BL-necessary states of affairs.

4.  We can now introduce the term 'BL-noncontingent' to cover the BL-impossible and the BL-necessary.

5.  What is not noncontingent is contingent.  (Surprise!) The contingent is that which is possible but not necessary.  Thus a contingent proposition is one that is possibly true but not necessarily true, and a contingent state of affairs is one that possibly obtains but does not necessarily obtain.  We can also say that a contingent proposition is one that is possibly true and such that its negation is possibly true.  The BL-contingent is therefore that which is BL-possible and such that its negation is BL-possible.

6. Whatever is NL or BL or nomologically impossible, is impossible period.  If an object, state of affairs, or proposition is exckluded from the realm of possible being, possible obtaining, or possible truth by logic alone, logic plus necessary semantic truths, or the (BL-contingent) laws of nature, then that object, state of affairs or proposition is impossible, period or impossible simpliciter.

7.   Now comes something interesting and important.  The NL or BL or nomologically possible may or may not be possible, period.  For example, it is NL-possible that there be a round square, but not possible, period.  It is BL-possible that some man run a 2-minute mile but not possible, period.  And it is nomologically possible that I run a 4-minute mile, but not possible period.  (I.e., the (BL-contingent) laws of anatomy and physiology do not bar me from running a 4-minute mile; it is peculiarities not referred to by these laws that bar me.  Alas, alack, there is no law of nature that names BV.) 

8.  What #7  implies is that NL, BL, and nomological possibility are not species or kinds of possibility. If they were kinds of possibility then every item that came under one of these heads would be possible simpliciter, which we have just seen is not the case.  A linguistic way of putting the point is by saying that 'NL,' 'BL,' and 'nomological' are alienans  as opposed to specifying adjectives:  they shift or 'alienate' ('other') the sense of the noun they modify.  From the fact that x is NL or BL or nomologically possible, it does not follow that x is possible.  This contrasts with impossibility.  From the fact that x is NL or BL or nomologically impossible, it does follow that x is impossible.  Accordingly, 'NL,' 'BL,' and 'nomological' do not shift or alienate the sense of 'impossible.' 

9.  To appreciate the foregoing, you must not confuse senses and kinds.  'Sense' is a semantic term; 'kind' is ontological.  From the fact that 'possible' has several senses, it does not follow that there are several species or kinds of possibility.  For x to be possible it must satisfy NL, BL, and nomological constraints; but this is not to say that these terms refer to species or kinds of possibility.

 

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Bill Vallicella

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13 responses to “Broadly Logical Modality”

  1. william of woking Avatar
    william of woking

    >>>2. But what about a plane figure that is both round and square? Is it NL-impossible? No. For by logic alone one cannot know it to be impossible. One needs a supplementary premise, the necessary truth grounded in the meanings of ’round’ and ‘square’ that nothing that is round is square.
    I don’t follow this.
    (*) Some bachelor is married
    Are you saying that by logic alone we cannot know this to be impossible? Do we need a supplementary premiss grounded in the meaning of ‘bachelor’ in order to know that it is?

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  2. peterlupu Avatar
    peterlupu

    William,
    “I don’t follow this.
    (*) Some bachelor is married
    Are you saying that by logic alone we cannot know this to be impossible?”
    Of course not! (*) looks like this:
    1) (Ex) (Bx & Mx)
    (1) is not formally a contradiction. To get one you need a clause which defines ‘bachelor’ as ‘unmarried male’; now you substitute that clause into (1) for ‘Bx’ so as to get:
    2) (Ex) (MLx & ~Mx & Mx)
    (2) is a formal contradiction.

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  3. David Brightly Avatar
    David Brightly

    Thanks, Bill. All is well until para 7. Let’s keep to ‘impossibility’ as it seems to make sense to categorise this rather than ‘possibility’ (para 8). You want to say that it is not nomologically impossible that BV runs a 4-minute mile but that it is nevertheless impossible that BV runs a 4-minute mile. Let’s suppose that there is a physiologico-anatomical (PA) law of the form ‘x runs a 4-minute mile –> x’s body mass index is less than K’. Now I agree that there is no contradiction deducible from our putative law (together with others of similar form) and ‘BV runs a 4-minute mile’. But as soon as we add the PA relevant fact ‘BV’s body mass index exceeds K’, we obtain the contradiction that renders it impossible that BV runs a 4-minute mile. Perhaps we can characterise the difference between our positions by saying that you would withhold this PA fact about the object BV while we are discussing PA-impossibility whereas I would introduce it. You are a ‘late-binder’ whereas I am an ‘early-binder’.
    Again, returning to the volcanos example from the earlier post, we have a ‘supplementary’ object, namely the earth. If we bring in facts about the Earth, such as its surface area being less than some given constant K, then that would appear to render ‘there are 1,000,0006 active volcanos on earth now’ impossible. If we don’t introduce such a fact, or perhaps better, we introduce the area constraint but leave K unspecified, then we don’t obtain the contradiction.
    So modal talk is about solving equations under constraints. That’s wonderful! I can understand that!

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  4. Bill Vallicella Avatar
    Bill Vallicella

    Peter and William,
    Peter’s response to William is exactly the one I would make, and it strikes me as decisive.
    Logic is purely formal. It deals with the forms of propositions and implication relations between propositions that rest on these propositional forms. If so, then one cannot know by logic alone that *Some bachelor is married* is impossible. The impossibility is not grounded in logic alone, but in logic + semantics. That is why we ought to distinguish between the narrowly and the broadly logical.

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  5. Bill Vallicella Avatar
    Bill Vallicella

    You make a good point, David. There are more specific laws that make it nomo-impossible for me to run a 4-min mile. So the example I gave was no good. I cannot say that it is nomo-possible for me to run a 4-min mile but impossible, full stop.
    But it seems there ought to be an example of an event or state of affairs that is nomologically possible according to every law of nature, no matter how specific, but is nevertheless impossible.
    What I am trying to show is that ‘x is nomologically possible’ does not entail ‘x is possible.’
    I am not clear what we are arguing about. You originally objected to ‘The merely possible is not actual.’ Is your point that there is nothing possible that is not actual? That everything is actual? That the actual and the possible coincide?

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  6. David Brightly Avatar
    David Brightly

    Hello Bill,
    Yes, my original objection was to the effect that the assumption underlying your aporia, that states of affairs could be partioned into the actual, the possible, and the impossible, didn’t fit with my conception of possibility as relative to a set of constraints. As the set of constraints increases the range of possibilities reduces. Broadly logical constraints are relatively weak and hence broadly logical possibility is relatively large. As we include constraints of more and more ‘types’ possibility shrinks. Eventually we must reach some maximal set of constraints. What is possible under this set of constraints is in a sense ‘absolutely’ possible. I don’t understand your intuition expressed in your last comment that the possible is yet smaller..
    Perhaps it’s the sense that we have reached the limit of this process and can’t make the possible any smaller that gives the feeling of reality to the absolutely possible. There really is nothing more that would prevent your desk being two inches from the wall. Yet it isn’t.

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  7. peterlupu Avatar
    peterlupu

    David,
    What is the source of the constraints you mention in your recent reply to Bill? Do you simply mean that Logical Possibility, for instance, is only constrained by the laws of logic, Physical Possibility by the laws of physics, Biological Possibility by the laws of biology, etc.,?

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  8. Kyle Avatar
    Kyle

    Where can I send an email to Maverick Philosopher? This was the only place that I found that I could comment. I would like to make a couple of points on a different posting.

    Reply
  9. David Brightly Avatar
    David Brightly

    Hello Peter,
    I’m not quite sure what you are asking. In an important sense the constraints come from us. They are our ideas expressed in language and a proof of (im)possibility is a purely formal, symbol-shuffling process. We might be investigating a fictional crime, say, or the viability of an engineering or business project. On the other hand, surely we hope that the propositions we use reflect the law-governed nature of the world, and in so far as the predictions we make tally with what we find to be possible and impossible we tend to believe that we have so captured natural regularity. So I’d say that what we take to be the laws of physics, etc, define what we mean by physical (im)possibility, etc.

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  10. William Avatar
    William

    >>Logic is purely formal. It deals with the forms of propositions and implication relations between propositions that rest on these propositional forms.
    It’s probably too late to contribute to this debate – but I will anyway. As you know, I have a difficulty with the notion of ‘formal’. If ‘form’ relates purely to the physical shape of the tokens that make up a sentence, we can find all sorts of counterexamples to your point. For example the proposition
    (*) you [pointing to a man] are a man, you [pointing to a woman] are not a man
    is of the form p & ~p, and so by your criterion above is ‘narrowly logically’ impossible. For it to be impossible in the sense you require, it has to be specifed that both tokens of ‘p’ have the same meaning. And so we cannot do without the semantics, as you suggest above.
    A further difficulty. Ordinary language allows ‘composite’ predicates, i.e. predicates with an internal structure, as in
    (**) Dave is an unmarried man
    Is the sense
    (***) Dave is an unmarried man and Dave is married
    NL impossible or not? If it is, how do you justify this, given that it has the ‘form’
    (****) Uman(d) & M(d)
    Note that ‘UM’ is simply a symbol with no internal semantics. If you reply that we can unpack UM(x) into ‘~M(x) & man(x)’, thus the sentence above, unpacked, involves a logical contradiction, I reply: this is a semantic manoeuvre, and in that case why can’t we unpack ‘bachelor(x)’ in a similar way.
    I do agree with you that logic is purely formal. I do not agree with you on your interpretation of the concept ‘form’. ‘Form’ has to include an element of semantics, in order to be coherent.

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  11. David Brightly Avatar
    David Brightly

    Hi WW,
    I have some sympathy with what you are saying. We have been discussing the BL possibility of propositions like (1) ‘DB has blue hair’. This has similar form to (2) ‘DB has blue moods’ and (3) ‘DB has blue trousers’. (1) asserts that a property of part of an object has a certain value. (2) makes a category error. (3) asserts that a property of an object bearing a certain relation to another object takes a certain value. What do you make of the Quinean idea that these surface differences can be regimented away to reveal the ‘logical structure’ of these propositions. This seems to cope with your composite predicate example. However, even if we can do this it seems that we still get different answers to ‘is it BL-possible that DB has blue hair?’ according to different classifications of the object DB. As a physical object, yes; as a mammal, yes (some cats and monkeys are blue); as a human, maybe not. Or what about ‘DB has red hair?’ As a human yes; as a child of JB and AB, yes, if one of my parents has the right gene. As the person with the genetic composition I actually have, well, no.
    Having said that, I think your indexical example is rather weak. Either the gestures are taken as tokens in the language, in which case the sentence has the form p & ~q and there is no NL-impossibility; or the gestures are excluded and, assuming that ‘you’ has a fixed referent, we have NL impossibility as Bill and Peter suggest. Surely we start by assuming fixity of reference and only relax this constraint if we can’t make sense of the sentence, either in itself, or in its context. Is it your view perhaps that ‘form’ can only be imposed once the sentence is understood? Does this matter for the problem of understanding modality?

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  12. William Avatar
    William

    Hi David,
    >>Either the gestures are taken as tokens in the language, in which case the sentence has the form p & ~q and there is no NL-impossibility; or the gestures are excluded and, assuming that ‘you’ has a fixed referent, we have NL impossibility as Bill and Peter suggest.
    Point taken, but the question is (a) in what sense the inclusion of gestures involves samenesss of ‘form’, as opposed to signifying intention or meaning. And (b) even if we allow that gestures can have a ‘form’, we can still suppose that God miraculously changes the meanings of terms with the same ‘form’, so that the proposition expressed is different. E.g.
    (*) p & p
    Suppose that God changes the meaning of the second token so that it means ‘not p’, i.e. the contradictory of what is expressed by the first ‘p’. Then what is expressed is necessarily false (rather than contingent upon the truth of p).
    I suppose you could reply that what is expressed by the physical tokens is irrelevant, and that it is the formal rules governing their use that is relevant only.

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  13. David Brightly Avatar
    David Brightly

    W,
    I don’t understand your issue with fixity of reference. This seems to me to be a presupposition, else we have chaos. I’d say that what is expressed by tokens is contained in the rules that govern their use, including rules of convention. I don’t know what it would be for God to ‘change the meaning of a token’.
    I think we are on firmer ground with your original example. How do we know that there is a connection between ‘married’ and ‘batchelor’ so that the latter can be rewritten in terms of the former. In other words, how much of the meaning of the terms are we to take into account? Hence my ‘blue hair’ example. I think it’s generally accepted that it’s BL-possible that DB has blue hair. My feeling is that it’s hard to draw a firm boundary around BL-possibility that stops it from sliding into other ‘types’ of possibility, just as you (if I’ve got you right) think that NL-possibility slides into BL-possibility.

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