Q&A: Questions about the model for non-deductive inferences
Questions about the model for non-deductive inferences
Question
Hello, a0
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Recently I came across two articles in Badad 23 and 24 that describe a mathematical model for non-deductive logic. a0
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I have a few questions about those articles: a0
1. First, I have a question about the topic itself why does it matter that it can be redeemed with money? Seemingly that is not relevant to the topic, which deals with marriage acquisition. The difficulty is strengthened by the fact that at the end of the passage they use this relevance claim in order to reject the refutation from the maidservant. a0
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2. Regarding the drawing of the model in almost all places the model is based on a column diagram. (Except in the passage about dayyo.) For a kal va-homer and a binyan av it is possible to present the model also by rows, but when looking at row diagrams of inferences of the common denominator, one can see that there is no difference between filling in 1 and filling in 0. What does that say about the model?
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3. Throughout the entire first part of the article there is no use of the microscopic parameters. In all places the decision is made only from the topological considerations of the graph. In fact, the only place where the dimension also has significance and perhaps not surprisingly is when they present the refutation that is based directly on a microscopic consideration.
Of course, the motivation for looking for microscopic elements that affect the result is clear, but is there an explanation for why considerations of connectedness or number of vertices matter?
If we imagine two diagrams with two vertices, one with an arrow between them and an explanation that one vertex has alpha, and the second has alpha and also beta, versus a diagram with no arrow, where one vertex has alpha and the other only beta why is the first diagram better? What is the intuition behind that?
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4. Regarding the rule that only one parameter can receive multiplicity seemingly this rule should be limited. After all, one can always look at two different kal va-homer inferences from different places as one diagram with two connected components. In such a case, in the first kal va-homer there will be alpha and then alpha-alpha, and in the second there will be beta, and beta-gamma.
A limitation that seems reasonable would be to say that multiplicity is limited to one microscopic parameter per connected component.
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5. Regarding a microscopic refutation of a kal va-homer seemingly what such a refutation does is cancel the arrow between the two vertices by presenting a microscopic parameter that exists on the lenient side but not on the stringent side. But one can always argue that in fact the opposite is true: on the stringent side there is an opposite microscopic parameter, and on the lenient side it disappears.
For example, one could refute the first kal va-homer in the passage by saying that with money there is benefit, which is not the case with canopy, and therefore money can effect betrothal while canopy cannot. But one could also say the opposite: canopy has lack of benefit (a formal legal act), and that is precisely what can effect marriage (and all the more so betrothal).
With a binyan av one cannot do this, because it does not matter who has the special parameter the point is that there is no similarity (and in the diagram there are two vertices instead of one).
Maybe this is the meaning of an any-old refutation, which can be made only against a binyan av and not against a kal va-homer?
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Thank you very much,
Answer
Hello Avner, and thanks for the comments.
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1. If money has the power to do something that a document or canopy cannot, then it is stronger. In my opinion that is a reasonable refutation. Especially since betrothal by money is learned from acquisition of a field.
I did not understand your comment from the refutation based on the maidservant.
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2. It seems to me that in the chapter dealing with dayyo we discuss the relation between column considerations and row considerations, and show that this should give the same result. There are subtleties there, and it is hard to get into that here. As I recall, when you go through the subtleties you will see that there is equivalence between the presentations.
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3. It seems to me that we explained why intuitively the topological parameters are significant (we defined them a priori and only afterward saw that this really works. That is, there is an internal logic to their relevance for deciding the inference). In principle they all point to simplicity (expressions of Occam's razor). Those parameters point to connections between parts of the graph, and the more connections there are, and the shorter and more directional they are, the simpler the picture.
As for the example of the diagrams with two vertices, I did not understand. When there is an arrow between the points there is one parameter (alpha and alpha-alpha) and not two (alpha and beta). I remind you that we solve for the minimal model the simplest one. The first is better both in terms of dimension and in terms of connectedness. And it is completely clear intuitively that the picture of the connected diagram is simpler. Here this is really just Occam's razor, since without an arrow there are two parameters and two independent columns, whereas with an arrow there is one parameter and the columns are dependent (that is, one is a function of the other). It is like a straight line being simpler than a parabola (a graph with two parameters).
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4. I did not completely understand the argument. But it is certainly possible that there are options for additional models. From your description I did not understand why the second kal va-homer has two parameters (beta and gamma). In the diagram of a kal va-homer there is only one parameter.
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5. Usually there are intuitive assumptions about the connection between the causes (or microscopic parameters) and the results (the halakhic results). Benefit adds an ability to effect betrothal; it does not detract. Contrary to what one might think, we do not come to the inference as a blank slate, if only because we decided which columns to put into the same table and which not.
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Questioner:
Hello, and thanks for the answers. a0
I will sharpen my questions a bit: a0
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1. At the end of the passage they try to refute the common denominator with the argument that in all of them there is coercion, and then Rav Huna rejects this by saying that money can work coercively only with a maidservant but not in marriage acquisition, and therefore it is not relevant. a0
It seems to me that the same argument could be made against the claim that money can redeem second tithe. It is not relevant, because at least in marriage acquisition we do not find it. a0
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2. Let us look, for example, at the following table (a refutation of the first kal va-homer): a0
You can see that the row diagram for filling in 0 and for filling in 1 is identical. In both there are two independent points, m and h. a0
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3. I understand why the topological consideration indicates simplicity in the graphic sense of the diagram, but if we try to think of the diagram as the metaphor and the set of halakhic events as the thing being represented, then I have trouble understanding the thing being represented.
For example, let us look at the following diagram (I did not write what the relevant data table is, but one can make one up): a0
What matters is that according to the model it is clear that we should choose option 1 on the left (which includes an arrow between C and A) because of connectedness. a0
But it is not clear why why should we prefer to say that C has a combination of alpha and beta? Why is that simpler than saying it has only beta? a0
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4. Let us look at the following table; it is made up of two different kal va-homer inferences that were joined together (this can be done with any two kal va-homer inferences in the world)
Filling in 1 in both places will give the diagram:
And we see that D has to receive a combination of two microscopic parameters instead of beta twice. That is a bit strange.
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Thank you very much, and sorry about the quality of my diagrams.
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Rabbi:
1. Money working coercively exists only with a maidservant and not in marriage acquisition. That is a positive datum (it is known that in marriage acquisition there is no money working coercively). What you proposed was a negative refutation, that perhaps redemption is not relevant to marriage acquisition.
2. If you look at the chapter on dayyo (I think that is where it is), you will see that the way the row diagram is constructed is different from the way the column diagram is constructed. It is a subtle calculation, and I do not have time right now to get into it.
3. The topological simplicity is not merely an additional tool whose purpose is to express the complexity of the parameters. The graph and the parameters (the dimension) are two different criteria. Therefore, the fact that in the two diagrams there are two parameters does not mean that they are equivalent. The complexity of the diagram itself is also a criterion, and it is the decisive one.
4. At first glance it seems to me that you are right, and one should allow higher multiplicity in one parameter in each subgraph. But this requires more thought.
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