Romans 9 -- Scripture and Logic

Romans 9:7

6 But it is not as if the word of God had failed. For not all those who are descended from Israel are truly Israel,
7 nor are they all children because they are descendants of Abraham, but “In Isaac will your descendants be named.” (LEB)
“Decedents” – in the literal means “seed.”

Imagine there is a fig tree in my front yard. I take the seeds produced by this tree and then plant 20 of those seeds. Several years later these seeds produced 20 trees. However, only 2 of them are producing figs as their parent did, the other 18 trees are not fig trees. What happened? This is the context of the passage.
Paul, rather than making an argument, is heading off an argument he expects to get from his teaching. An argument is “UNSOUND,” if the structure is not valid or, if the premises themselves are false—that is, one or more premises describe an aspect about reality falsely. Here, Paul does not do a, reductio ad absurdum argument, or show an invalid structure; rather, Paul shows one of their premises has defined reality in a false way. This false statement can be exposed from MORE than just one angle. Often when one mistake is made in thinking, there are a few more with it. Most informal fallacies are not completely isolated.
At any rate, Paul rejects their argument by centering on falsifying their categorical proposition. Their premise quantifies with an “all,” (All S is P). Paul says, no; rather a true reflection of reality means, (some S is P), and (some S is not P). That is, to say “ALL [apples] are [low in sugar],” is a different-- (and in fact a contradiction to “some are not”)-- to “SOME [apples] are NOT [low in sugar], and, “SOME [apples] are [low in sugar].”
Paul’s opponent is saying,
All [ Israelites ] are [ children of Abraham ].

Paul’s corrects this false premise by saying,
Some [ Israelites] are [ children of Abraham ].
&
Some [ Israelites ] are not [ children of Abraham ].

Because Paul is saying, “Some Israelites are not children of Abraham,” then this becomes a contradiction to the opponent’s statement. Their statement is thus, a false statement of reality. Their argument is not sound.

Now, to prove this main point Paul says, yes, God promises to “bless” Abraham and his seed/decedents; however, there is more qualification of this promise that Yahweh reveals. God reveals the “blessing” is not on all of the physically descended children of Abraham, but rather, only those whom God elects/choose to be so. God elects some Israelites to inherit that promise. He also choses that some will not inherit that promise. The evidence of this is that God promises “in Isaac your decedents will be named/called.” It is not saying all of Issacs’ decedents will be called; rather, “some will,” and “some will not,” be called to be so.
Paul further proves this by illustrating Jacob and Esau. God chose Jacob to inherit Abraham’s blessing, and even “named” him Israel, but for Esau, God did not choose for him to inherit Abraham’s blessing. Just as one was chosen and the other was not, so also today, God is choosing who among the physically descended of Issacs’ children who will obtain Abraham’s blessing and who will not obtain it. We see this play out relative to man’s perspective via the person’s faith in Yahweh’s word, but that is for another time.
Verse 7
“Nor are they all children because they are [1] descendants of Abraham.”
“In Isaac will your [2] descendants be named.”

It will be easy to make an equivocation here, when Paul is using similar words in different ways depending on the context. Verse 7 uses descendants as the “physical descendants” first, then secondly, when quoted it is used as “Election descendant.” The context defines the term here. However, for simplicity when we make the syllogism we will use terms that define themselves for clarity.
The main dividing of point Paul is making here is a “Physical decedent,” verses a “Chosen decedent” —who receives the blessing of Abraham. The big idea is, who truly obtains Abraham’s blessing?
First, the term “children of Abraham,” in context is mainly about receiving the blessing God promised Abraham. Or that is, decedents of the blessing. Secondly, “Decedents/seed of Abraham,” is first refereeing to being physically descended from Abraham and then secondly as “elected descendant.”
Children of Abraham = Receivers of Abraham’s blessing.
Seed of Abraham (first time) = Physical descendants of Abraham.

We will construct the syllogism Paul’s typical hypothetical opponent wants to construct.

P.1. All [physical descendants of Abraham] are [receivers of Abram’s blessing].
P.2. All [Israelites] are [physical descendants of Abraham].
P.3. Thus, All [Israelites] are [receivers of Abraham’s blessing].

Paul refutes this argument by saying Premise 1 (P.1.), is a false premise. It is “only Some,” physical descendants of Abraham are receivers of his blessings. Remember categorically speaking, if one says, “only some are,” it is in realty TWO premises. “Some are not,” and “Some are.” Again, as a reminder, this is different when plainly saying, “Some A is B.” This could mean that, “All A is B,” however, as what is stated—at this point—it simply means that, “At least some A is B,” whether is it just some or all has not been revealed.
Now we will construct two syllogisms with premises that describe reality correctly and with a valid conclusion. These two syllogisms will reflect Paul’s teaching that some Israelites are, and some are not called in Isaac. This also concludes that God’s “promise” is indeed having unblemished effectiveness, because it is succeeding precisely how God promised it would. Some will be called, and some will be not. The hope and further elaboration of this promise Paul teaches, is that “this some,” will become an overwhelming percentage, at the end of the age, for the physical descendants of Jacob.

A.I.I.
Y.1. All [those called in Isaac] are [those who obtain Abraham’s blessing].
Y.2. Some [Israelites] are [called in Isaac].
Y.3. Therefore, Some [Israelites] are [those who obtain Abraham’s blessing].

The predicate of premise one (major term) in this syllogism could be exchanged with “[ those who obtain Abraham’s Promise].”

O.A.O.
U.1. Some [Israelites] are not [called in Isaac].
U.2. All [Israelites] are [physical descendants of Abraham].
U.3. Thus, Some [physical descendants of Abraham] are not [called in Isaac].