Here’s a good reason to study logic 🙂

Very clear, thank you very much

Is the following argument valid:
1. Yosef ben Shimon got married
2. Yosef ben Shimon did not get divorced
Conclusion: YBS is married.

On the one hand, it seems to me that it is valid because the conclusion follows from the premises, and if the premises are true then the conclusion is true. But on the other hand, one could say that YBS was widowed, so he is a widower and not married, and therefore the argument is not valid.

You need to add the premise, false though it is, that anyone who got married and did not get divorced is married. Then the argument would be valid, even though its conclusion would still be false.

That is the meaning of the concepts. This is what is called content-based logic, as opposed to formal logic. But as you wrote, the argument is of course not valid, because widowhood is possible. If you understand that yourself, then what is the question?

In other words, the question is:
Do we say that the argument is valid in formal logic but not in content-based logic, or that the argument is not valid at all? Or, when there are several logics in play, do we treat each one separately for discussion, and then say: according to this logic the argument is such-and-such, according to that logic it is such-and-such, etc., or do we consider them all together?

These are just words. A valid argument, whether formal or content-based, is an argument such that if you accept its premises you must accept its conclusion. That is not the case here.
By the way, there is no formal validity here either; your formalization is simply incorrect. There can be content-based validity that is not formal validity, for example: no bachelor is married, but there cannot be formal validity that is not also content-validity. Here the formalization is defective in any case.

If there isn’t even formal validity here either, as I wrote above, because the step is missing that whoever got married and did not get divorced is married, then why did the Rabbi bring in the issue of content-based validity at all?

What’s the complication?
Imagine that P is a gate such that if it opens, what is outside fills the inside.
When P is closed (“false”), what is inside (Q) does not depend on P, because P is closed. And inside can be full or empty.
So there is no problem at all when P is “false” (closed) even when the inside is full.

Because you can formalize the argument in a way that will be formally valid. But when you check it, it won’t be content-valid, because you did not formalize it correctly.

A. Simba is a cub
B. Every lion is a mammal
-> therefore Simba is a mammal

Can one say that such an argument is valid? Is it necessary to add the premise that “every cub is a lion,” or is that self-evident?

Another question: following from what you always define as a proposition, namely a sentence that says something about the world that is either true or false, is a sentence of the form “X or not X,” such as “right now it is either light outside or dark outside,” a proposition? Because apparently it doesn’t say anything; it doesn’t add any information for me. Of course it can be either X or not X, no?

What do you mean “necessary”? According to the law? Nothing is necessary. Do what you think and what will be clear.
If the meaning of the word “cub” is lion, then that’s just a dictionary matter. There is no need to attach a dictionary to the argument. If there is an additional premise here, then an additional premise is needed. Should you state it or not? Decide for yourself.
Usually, in everyday wording people are not careful to state all the premises explicitly. When moving to precise logical formulation, one does what is called “completing enthymemes,” that is, adding all the implicit premises.

As for your question about propositions, that is semantics. Usually it is defined as a proposition that is always true. It does say something about the world. The fact that it adds nothing for you is no different from the proposition “it is dark outside now,” which also adds nothing for me because I already know it.

More about material implication: there is one small thing I don’t understand.
Suppose Aharon has horses, black ones and white ones and so on. And we checked and found that every black horse is also gentle. According to this one can formalize it as p—>q, that is, if the horse is black then it is gentle.
But I don’t understand: the horse’s being black is not the reason it is gentle! It just happens that every one of his horses, if it is black, is also gentle, but one is not the reason for the other. In short, in material implication is the antecedent of the conditional sentence the cause of the consequent? Apparently not. So what exactly is the meaning of “if… then…”?

I explained this above. Logical implication does not express causation but only a logical relation between the truth values of propositions. If we were looking for a logical relation that expresses causal production, we would not be able to define it formally, because that depends on the content of the propositions. Therefore one defines material implication, which is the minimal relation that defines a relation of implication between propositions.

1. I am hungry
2. I am fat
Conclusion: either it is raining today or it is not raining

In logic we learn that this argument is valid because its conclusion is a tautology. Fine, but that is really strange. What connection is there between the premises and the conclusion? The conclusion does not follow from the premise. Why should it matter to me whether it is a tautology or not?!

(I feel that this is connected to what we discussed above about material implication, so I’m asking something similar.)

It’s not just connected. It’s exactly that. Material implication is about ignoring the connection between the premises and the conclusion. Otherwise it cannot be formalized. The relation of implication is built on the basis of the truth values of the propositions alone, without relating to the content-based connection between them. That is exactly what I explained above.

Apparently it’s a bit different. Even in material implication, for any conclusion you can take only the minimal set of premises required for it, and that doesn’t harm the formalization. One can debate how to define a minimal set, but the minimum is to filter out individual premises such that even with their negation the conclusion is still implied. No?

I didn’t understand. Try suggesting a truth table that defines the relation of implication without regard to content — after all, we are speaking about formal logic and not content-based logic — in such a way that the causal relation between the premises and the conclusion is preserved. That is a content relation, and it cannot be preserved in formal logic. Do you have such a truth table? Material implication is the only logical way out on the formal plane.

I’m the one who didn’t understand. In formalization one can filter out unnecessary premises — for example, a premise will be filtered out if the argument remains valid without it. Therefore, in formalizing the argument presented by EA, both premises would be filtered out, and the tautology would remain on its own. Even in non-content implication one can, and should, make do with a minimal argument. Either this is trivial or I didn’t understand.

You are jumping too far. I will repeat my question/request to you: when you want to formalize the claim “if there is rain then there are clouds” as an implication, P->Q, you need to define the relation of implication. I ask how you would define it in a non-material way. Give me its truth table. In other words, show me how you formally express the difference between “if there is rain, there are clouds” and “if the prime minister is Bennett, then there are more than a million residents in Israel.” In both cases the antecedent and the consequent are true. In the first there is an essential connection, and in the second there is not. But from the standpoint of a truth table and truth values, the two claims are identical. Do you have a formalization that can distinguish between them?

Obviously there is no difference in a truth table. I wasn’t talking about that. I hold only by material implication. And within that I am talking about filtering out superfluous premises. If I am given only one premise, “if A then C,” then there is nothing to be done and the table cannot distinguish the nature of the connection. But if I am given a set of three premises: A, B, if A then C, then although proposition C follows from the premises, one can omit premise B and build a smaller argument with a set of premises of size 2: A, if A then C, and derive C from that. In the case where proposition C is a tautology, the case EA presented, the minimization process will filter out all the premises.

Obviously one can drop unnecessary premises. That was not the question. The question was why the argument as it stands is valid, and to that I answered. Even the argument that puts the tautological conclusion without any premises at all is valid. This is the rule of weakening in logic: you can always introduce a premise as the antecedent of an implication, and then from a valid argument you will obtain a tautology.

[Fair enough. I thought, and still think, that the question was about the emotional terminology: it is clear that the argument is valid according to the truth-table definition, but why do logicians deal with arguments in which there are unnecessary premises and declare “if the premises are true then the conclusion is true,” when the conclusion is true even without the premises? And the answer is that in fact there is no need to deal with arguments with unnecessary premises, only with minimal arguments.]

1. Most students wear pants
2. Sarah is a student
Therefore it is reasonable to assume that Sarah is wearing pants

This is a valid argument. But is it deduction or induction?

That is of course deduction.
If the conclusion were: Sarah is wearing pants, that would be a kind of induction.

If the conclusion were as you said, that would be an argument that is not valid deductively but is strong inductively, right?

I don’t know what “strong” means. It depends what majority wears pants: 51%? 99%?

Right, right, I just wanted to make sure that the argument would be deductively invalid, because it could be that she isn’t wearing pants, but in any case we still “have something to say” from an inductive standpoint, right?

That is the meaning of induction.

I always learned, especially from you, that a factual proposition can be either true or false.
And here is a proposition that does not fit that: my son is fat.
It is not true, since I have no son, and it is also not false, since the proposition “my son is not fat” is also not true, since I have no son.
So here are two contradictory propositions that have the same truth value, namely false.

Ah no, the contradiction of the proposition is not “my son is not fat” but “it is not true that my son is fat,” as you explained in the column on positive commandments and passive omission. Right?

That is also true, but here you are entering the issue of counterfactuals, statements contrary to fact. Claims about my son when I have no son are not propositions in the simple definition, and therefore they are not subject to the law of the excluded middle, either true or false. It seems to me that I once gave you the example of the current king of France being bald. Check among the bald people and you will not find him, but neither will you find him among the hairy ones, because France currently has no king.

Okay, so the law of the excluded middle applies only if we assume existence, right? And if we do not assume it, in any case one should say that the contradiction of the proposition “X is Y” is not “X is not Y” but “it is not true that X is Y”?

Can one say that factual propositions such as “my son is fat” when you have no son, or “the current king of France is bald,” are propositions only about potential facts? If so, does it follow that there is a special way of dealing logically, and perhaps metaphysically, with such facts?

EA, absolutely. That is what I wrote.
Definitely. This is a whole field in philosophy that I mentioned: counterfactuals.

To sharpen the point: a counterfactual sentence is a claim that deals with what would have been if France had a king, or if there had not been a president there. Claims about the current king of France are not claims at all, because this is a sentence without a subject.

1. I think
therefore I exist.
Is this a logically valid argument? Or does one need to add the premise: 2. Everything that thinks exists.

And regarding that premise, can one deny it philosophically? That is, does it make sense to claim that there is something that thinks and does not exist? I think not… what do you think?

מבט על הקוגיטו: Cogito Ergo Sum (טור 363)

A=B
B=C
Therefore A=C.

Where is the begging of the question in this valid inference?

When you assume the two premises, you are also implicitly assuming the conclusion. You just split it into two parts.

All human beings are mortal
Socrates is mortal
Therefore it is probable that Socrates is human

What is the name of this inference? Analogy, right? Because I read somewhere that it is induction.

We would translate the first premise as: some mortals are human beings.
Socrates is mortal.
Conclusion: Socrates is human.
There is no special name for this. It is a very dubious inference. Its quality depends on how many other mortals there are besides human beings. That will determine the probability of the conclusion.

Are the laws of logic regulative or constitutive? What about the laws of nature?
It seems to me that the former are constitutive, because they are necessary; without them there would be no world. And the latter are regulative, because there is a possible world in which they would have been different; it just happens that here anything with mass falls. Is that correct?

It’s a matter of definition. From the Holy One’s point of view, one could say that the laws of nature are regulative, though even here one can argue about it. But this discussion is merely lexical and not important.

Why not important? You yourself insisted that there are regulative laws and constitutive laws, and that the distinction helps us understand the laws of the Torah. Sometimes they are regulative, sometimes constitutive.

I didn’t say the distinction is not important. I said that the discussion regarding the laws of logic and nature is not important. With respect to them, the logical status is clear, and the whole question is just what to call it.
With regard to commandments, when one says that some law is constitutive or regulative, that is a substantive point that clarifies the character of the law.

To EA

Regarding the claim “my son is fat” when you have no son: in mathematics they say that this sentence is false, but vacuously. That is, the sentence “it is not true that my son is fat” is true, but vacuously. In other words, your son is not found within the set of fat people. And that is indeed true regardless of the fact that he is also not in the complementary set. So indeed there is no contradiction here. The claim “my son is not fat” — that is, that he belongs to the set of non-fat human beings — is also true in the case where you have no son. It simply means that your son belongs to the empty set, that is, that he does not exist.

It seems to me that this clarifies the matter. It is indeed connected to the discussion of logical implication, where there is no content-based connection between the propositions, and its truth depends only on the truth values of the propositions that compose it. There can be such an implication; one simply says that it is vacuously true.

Regarding a sufficient and necessary condition: in logic they say that if A is a sufficient and necessary condition for B, then the reverse is also true. There is symmetry.
But I don’t understand: how can B be a cause of A, if A is the cause of B — that is, if A is what brings about B and not the other way around?
I didn’t understand in what sense there is symmetry. What does it mean that B is also a sufficient and necessary condition for A?

First, in my opinion a cause is a sufficient condition and not a necessary and sufficient condition, though there are philosophers who think otherwise. But regarding your question itself, you are mixing planes of discussion. A cause is a logical condition, sufficient, but it is not true that a logical condition is a cause. Rain is a condition for clouds — if there is rain, there are clouds — but it is not their cause; rain does not produce the clouds. I have pointed this out many times before, including here on the site.

I forgot to ask at the end of the class: why is “being alive” not a property? After all, it is a property of a certain object that it is alive, just like its property of having four legs, for example. Why not? In other words, what is a property?

Who said it isn’t a property?

As I understood content-based logic, it deals with a sentence such as “red is a color,” but not with a sentence such as “the dwarf before us is rich,” because regarding the first sentence one can say whether it is true or false solely by its concepts, without recourse to experience — that is, it is an analytic sentence — whereas the second is true or false if it corresponds to the state of the world we are discussing. Did I understand correctly?

If neither formal logic nor content-based logic deals with the sentence about the dwarf, then what does deal with it?

I do not think there is a field called “content-based logic.” This is a term whose purpose is to sharpen the essence of formal logic. The argument “Moshe is a bachelor, therefore Moshe is not married” is valid, but not because of its form; rather because of its content. In contrast, the argument “all human beings are mortal and Socrates is a human being, therefore he is mortal” is valid because of the form. Formal logic deals with the second type. The first type is called “content-based logic” because it is not formal, but I do not know of significant discussions about it. I assume there are some, but I doubt whether there is really such a field.
Arguments or claims about the world, like your dwarf, are claims and not arguments. And they are not connected to logic but to science; observation determines their truth. This is synthetic versus analytic, but not content versus form.

I didn’t quite understand the difference between a proposition and an argument, when every argument can be turned into a proposition by means of implication. For example, I assert the proposition that “if all human beings are mortal and Socrates is a human being, then he is mortal.”
Moreover, for example the sentence, or proposition, “the color red is a color,” one can say of it that it is true from the standpoint of formal logic. So we see that logic also deals with propositions and not only arguments.

You do not turn an argument into a proposition. A proposition of implication is equivalent to an argument in the sense that if the argument is valid then the implication proposition is a tautology, and vice versa. Still, an argument is not a proposition. The proposition establishes only the relation of implication, whereas the argument establishes the truth of the conclusion on the basis of the truth of the premises. The distinction between analytic and synthetic applies to propositions, not to arguments.
The proposition “the color red is a color” is similar to “no bachelor is married.” Its truth is a result of the content, and therefore it is not formal. In predicate calculus you may perhaps formalize it, though even that would not be so simple.