Entailment
How do you understand entailment? Does it come down to necessity? Reasoning? Is a proposition entailed because of its status as a puzzle piece.. it just fits into a bigger picture?
Not the most intelligent question ever asked.. but maybe worth a ponder.
Not the most intelligent question ever asked.. but maybe worth a ponder.
Comments (55)
To jump then to what I think is the usual confusion is that most people want entailment to be a story of efficient causes. One particular thing causes that particular thing. This step dictates that step in mechanically necessary fashion.
But the larger view of deduction or causation is the holistic story where the process is one of constraints upon freedoms (or uncertainties). So the argument goes from the general to the particular, not the particular to the particular.
If Kermit is a frog, and all frogs are green, then Kermit is green by logical consequence. That is, the general constraint of "being a frog" is a limit on the colour some particular frog can be. But Kermit could be light green, forest green, aquamarine, and still meet the constraint. The actual shade of green becomes the residual freedom, the further fact about which the statements so far appear indifferent.
So reasoning deductively is about boxing in uncertainty. Information is added to limit the scope of the (Kantian) unknown. And even physical causality has this general nature - according to quantum physics now.
I have been thinking about contingency recently, how the only necessary notion is that everything is contingent. If p is given, then it must necessarily entails -p, as a contingent possibility. That's how I understand entailment.
Uncertainty on the part of whom?
What is given could have always been otherwise. What does bringing other worlds into this context add? Must the positing of an absolutely contingent world entail the possibility of an absolutely necessary world?
Depends. It could be a particular inquiring mind or it could simply be the world physically.
That would be the advantage of my semiotic approach. It applies the same way in either sphere.
The world is physically ignorant? That appears to be word salad.
I'll have to ponder this. Just as an aside.. you're sounding a little like Yoda.
Would it be possible to model the actual world?
I used to puzzle over this quite a lot - assuming that what you're actually asking is about the meaning of logic. In the end I dissolved the puzzle by concluding it's just a language game. We play the game because we have found it useful in the past and we are programmed by evolution to believe that things that have been useful in the past will be useful in the future.
Under that interpretation, the statement that A entails B just means that the two events, or propositions, satisfy a certain relationship that is specified in the language game we call logic.
You need to know what validity is to understand that. Traditionally, validity obtains when it's impossible that premises are true and a conclusion false, where traditionally, "and" in that definition is parsed as "either/or," so that validity obtains when either (a) it's impossible that premises are true, or (b) it's impossible that the conclusion is false, or (c) both (a) and (b).
This is relative to the particular species of logic being employed, so that validity, and thus entailment, are going to be different in traditional, bivalent logic with excluded middle versus relevance logics, paraconsistent logic, etc.
What this ought to entail is that 'entailment' is the relationship between ideas.
Well true propositions represent aspects of the actual world.. What do false propositions represent? Some of them could be said to represent other possible worlds. Just as the actual world stands out in thought against a background of other possible words, true propositions stand out against a background of false ones.
But rule following has to be anchored somewhere. Quine showed (in Truth by Convention) that in regard to application of logic, the anchor can't be anything external. It's apriori. Do you agree with that?
Someone might have that view, but it would be very unusual, including that it wouldn't be clear what they'd have in mind by entailment. On the surface of it, it would appear that maybe they see every fact as a necessary fact that is somehow causally connected to every other fact, and they're doing something that seems like a conflation of causality and implication (but they might have a different explanation that would make that not a conflation--again, it would partially depend on how they're thinking of entailment).
Say that we had a model (hopefully without needing to argue just what a model is) that encompassed mathematics. Well, that 2+2=4 doesn't imply that C=2?r, even though both are true in that model.
Do you think entailment is sense dependent or reference dependent. Sense dependent is epistemological and reference dependent is ontological. If you cannot understand a concept (A) without understanding another concept (B), then the concept (A) is sense dependent and a question of knowledge. If concept (A) cannot be without a concept (B) it is reference dependent, and a ontological issue. So, is entailment epistemological?
[as an aside, around midnight I feel like Yoda]
Clearly ;)
I think it's epistemological. For instance:
A: I said, "I have a dog."
If one knew everything about my dog, one would know all sorts of things about how she relates to aspects of the universe... that she likes tennis balls, that she weighs 15 lbs, how far she is from Neptune, and so on. These are truths entailed by A. Is that right?
That's sort of making use of Leibniz's complete individual concept.
I think we do have to have some model because just logical possibility will become so open-ended that it's meaningless.
A: I said, "I have a dog."
All sorts of things are logically possible here.. but I don't think all those things are entailed. I think entailment is more about how any particular thing is related to everything else. Logic is on the scene, but only because models necessarily employ some sort of logic.
It's not just whether something is logically possible. Entailment is implication, or that something follows from something else.
Things follow from "I have a dog," but it's easier to understand first if you understand it in terms of an argument, so that we have a set (>1) of premises.
So, for example, "I have a dog" and "all dogs are creatures with hearts." From that, "I have a creature with a heart" is entailed--that is, it's implied by the two premises, or it follows from them.
Or, "I have a dog" and "All dogs are animals named Fido." It's entailed by that that you have an animal named Fido. Of course, the second premise in this case isn't true--not all dogs are animals named Fido, but that doesn't matter for entailment. Entailment is about what is implied by or what follows from something. "I have a dog," "All dogs are animals named Fido," "Therefore I have an animal named Fido" is a valid argument. (It's just not sound, since soundness hinges on the premises being true.) But entailment obtains when we have a valid argument. In a valid argument, the conclusion is entailed by the premises.
So it's not just any logical possibility.
Re "I have a dog," by the way, as I mentioned before, there are things that are entailed by that, but they're limited to things like "There are dogs," "There is something called 'I'," "'I''s can have possession of things," and even "I have a dog" (that's simply "If P, then P").
I think the ability to see what is entailed by propositions and situations must be an intuitive capacity; entailment is simply knowable a priori and cannot be analyzed further.
Quoting Terrapin Station
This illustrates how entailment is "mechanical". It claims states of constraint that are absolute. And speaks to the way such states can be constructed.
So as I say, the physical reality is different. Constraints can never be absolute. Freedoms - either as ontic material entropy or epistemic informational uncertainty - can only be minimised, not eliminated. It is an important discovered fact of nature that it is indeterministic in the final analysis.
Also, constraints in nature tend to be contextual or holistic. A mountain or an earthquake are events produced by circumstances not of their own making. It is the accidents of plate tectonics and other geomorphic forces that entail the building of a mountain, the fissuring of a fault-line.
So there is a way that nature is.
Then life and mind come along and can play logical (or semiotic) tricks. They are modellers in a modelling relation with the world (a model being a formal system of entailment connected to the world by "acts of measurement").
So a modeller seeks to impose constraints on freedoms (ontic or epistemic, material or informational) in pursuit of some overarching purpose. There has to be a reason for being reasonable. And while it is still impossible for such constraints on nature to be absolute, it is not that hard for constraints to be "good enough" to achieve a purpose. A model can be indifferent to any difference which doesn't make a difference (to it).
And where a modeller really wins out over nature is the ability to construct states of constraint. A modeller can stick bits of an argument together to form some strait-jacket arrangement which forces nature into some tight corner.
That is the basis of the mechanistic view of reality. Petrol vapour explodes given a spark. But if you wrap that explosion around with pistons, cylinders, crankshafts and all the other bits of an engine - plus have control over the timing of the vapour puffed into a cavity, and the spark that ignites it - you are in business. You can drive right over nature in your SUV.
All life constructs these kinds of mechanisms. A bird makes a nest to protect its eggs. A spider spins a web to trap flies. At work is a mind that can build something that serves a purpose in mechanical step by step fashion.
So if we are looking for the origins of logic, for the reasons why it might be an abstraction that works, it is easy enough to see those origins in the rise of life and mind as a semiotic modelling relation.
First comes the ability to impose some state of constraint on nature (one that serves a purpose and is not merely an accident). And then comes the ability to assemble systems of constraint, step by step.
Thus entailment is indeed all about implication. It is about constructing states of constraint (material or informational) that restrict nature to such a degree it has no choice but to behave in a desired way. The rules of logic are all about encoding that biological imperative - the modelling relation - in the most abstract and universal set of rules we can imagine.
Again, the fact that nature is at base indeteministic - incapable of being completely constrained, is something that is left out of normal discussions of logic and thus results in great confusion when it comes to non-pragmatic "theories of truth".
But pragmatically, it's not a big deal as naturally all the attention of logic-users goes to what logical thinking can achieve. So it is the ability to construct arguments - formal systems of entailment - that gets celebrated. It is a remarkable fact that modellers can regulate the world to the degree that their desires can be reliably cashed out in systems of logical necessitation.
We can just get in our cars and ... drive.
Like usual, I haven't the faintest idea what you're talking about. Even things that seem to me should be simple I don't get. For example, "It claims states of constraint"--I'm not sure what "It" is given the way you've constructed your sentences. I just can't follow you most of the time.
That's a fascinating take. Why do you say that?
"It" is "entailment", of course. And the construction of the sentences indeed entails that interpretation on any reasonable view.
"Entailment" is the subject of the first sentence. So under normal grammatical conventions, the use of the pronoun "it" continues to refer to "entailment" unless some other information is introduced.
Your reading has already been epistemically constrained by the mention of "entailment" and so the meaning of "it" is logically entailed - even if, as you point out, what could stop and protest that your understanding of "it" is not absolutely constrained. There remains still a possibility of uncertainty.
Thus you illustrate my points nicely. Even if that is the last thing you want to do.
Quoting Terrapin Station
But that's not because I'm a poor writer. It's because you never seem to put much effort into understanding things before you tap out your replies.
I think insight is required to see entailment. It can't be just be a matter of following rules, because, for a rule to be usable, we must be able to see how to follow it, which would entail either insight or further rules about how to follow it. The latter would lead to an infinite regress, so I conclude there must be sheer insight (intuition) at work in all our thought.
I think it follows that insight is intrinsic to both analysis and synthesis. In the former we intuit how things may be broken down into parts and in the latter we can intuit how elements not obviously related to one another may possibly be related.
Insight, or rhetorical reason, which precedes the argument that entails. Like a skill that one acquires which enables identification of entailment.
Are you saying that insight is required in order to construct arguments (structures of entailment)? If so, I would agree; insight is certainly required at least as much to construct arguments as it is to follow them. The former (at least) requires both synthesis and analysis.
Your question about the dog lead me off in another direction (for a minute).
Do you think there are other types of entailment besides logical? I think entailment works within a Hegelian dialectic. A dialectical movement which preserves and negates both premises and in doing so generates a synthesis which is negatively determined. I guess what is entailed must be part of the synthesis.
Hegel dialectic has three moments:
1) understanding of the subject, its definition, what it means.
2) It cancels, negates and preserves 1) in a moment of self-sublation
3) the moment in which a new unity is grasped, the synthesis.
I also thought about entailment that might be involved in genealogical arguments, but these arguments are, it seems to me, to me more speculative reconstructions of history, which offer alternate explanations and suggest new possibilities. I not sure but don't think anything like logical or dialectical entailments are involved.
Well, I suppose one could identify entailment (say, as a recognizable pattern, possibility, or state of affairs) regardless of insight on what entails or why. An ability to identify the relation is sufficient.
Recognition of patterns equals intuition, no?
But entailment isn't something that can make claims about anything.
Anyway, I'm not suggesting for you to write any differently than you do. I'm just letting you know that your writing is often impenetrable for me.
Witty didn't use the word "anchoring" when he noted that rule following can't go on forever. Logical positivists sought to externalize it. Quine showed that we can't externalize it. It's apriori.
And you're committing the Evolution fallacy: "Evolution explains p" while p is used to explain evolution.
This is kind of trippy. The object of entailment isn't a proposition here. It's a pending comprehension of the dependence of the subject?
May be off track from your thoughts here, but I see a trail here leading to the object of entailment in all cases being Everything. That's also happening with my use of Leibniz's CIC to explain entailment.. and it's also the reason entailment is not a useful concept for describing truthmaking... every thing ends up being a truthmaker for every truth-bearer.
Quoting Cavacava
I happen to be reading Geneology of Morals right now. Ha!
Yeah... the very concept of entailment (that things are related) has to be apriori knowledge.
I don't know whether the ability to recognise something requires intuition or insight. To intuit, or see, are modes of perception, and what sets the intentional features of the entailment relation that you intuit might just be the present brute reality of the relation. For example, a sea urchin hardly intuits anything (it has no brain), yet acts as if it would intuit the entailment of present predators (e.g. scoops up gravel to hide).
No.
Evolution explains q, while p explains evolution.
q is the proposition 'Humans can't help but use logic'
p is 'Logic'
p=/= q
I would say that since the sea urchin is a living creature it perceives, which is the same as to say intuits, patterns, however minimally.
Also, there is a distinction between cognition and re-cognition. Although it might also be said that cognition must always already involve recognition. In any case recognition is not merely the registering of a pattern, but the knowing of that pattern as being the same as or alike to another. Such a thing obviously cannot be rationally deduced, so I conclude that it must be intuited.
Agreed. :)
Are you only claiming that evolution explains why people use logic, or are you claiming that evolution explains the form that logic itself takes?
To your question: it is the former I was thinking of. Now that you mention it, I think that evolution may possibly also have a role in the type of logic we mostly tend to use - eg a preference for including double-negative elimination in our rules rather than restricting ourselves to constructivist logic, but I am less sure of that.
Pattern recognition or pattern matching is more evolutionarily basic than cognition or deduction. And it follows from inductive learning. That is Hebbian association or Bayesian prediction.
You don't even have to think to recognise. It works at the level of habit.
So intuition would be what Peirceans would call abduction - the flash of insight which counts as inference to the best explanation. It is being able suddenly to see how a deductive account could supply the correct explanation. So rather than working it out step by step, the whole of the answer can be seen as if retrospectively.
And yes, that is an advanced form of recognition or pattern matching. Studies of creative thought show how we can juggle ideas about until they suddenly snap into place - finding a suitable fit with a schema or conceptual structure already in use for something else.
So we can recognise "this current problem" as a variant of "that old familiar problem". But it is the deductive structure we recognise as having a probable fit - so rather abstract features and relations, rather than concrete details, like the feathers, beak and tail that allow us to categorise a bird as a bird in a flash of pattern matching.
Strewth. If formal logic arose within the gene pool of ancient Greece, how on earth did all of us without Mediterranean bloodlines manage to master it? Incredibly speedy convergent evolution?
Yes, I agree with you that we can recognize abstract formal patterns as well as material formal patterns. And I do think you are right in the sense you put it that recognition is more basic than cognition. It does seem kind of strange to say that, though, since re-cognition seems to mean 'cognition again'.
Thanks Andrew, I don't have a ready response to your conjecture here; I'll need to think about it more. Or perhaps it would help if you fleshed it out a bit.