I'm going to lay out some rules of Greek grammar here that disprove what you are trying to say with your interpretation of the present participle of the Greek word for "believe," or πιστεύω. With each rule I spell out, I will give quotes from Greek grammars. (I have a bunch more I am not quoting. If these scholars don't convince you, nothing will.)
Rule 1: The present tense is often not continuative
Therefore, it is a great mistake to say that a present participle means we must keep believing or we will lose our salvation. The theologian must look at the context, in particular examining the main verb of the sentence to get the Aktionsart (a German word meaning “kind of action”).
“But it should be noted that the present indicative is also often aoristic and the future is usually so” (A. T. Robertson, A Short Grammar of the Greek New Testament,” p. 139).
“Some presents express punctiliar action and a few even express perfective action” (James Brooks and Carton Winbery, Syntax of New Testament Greek, p. 76).
Rule 2: When the article is before a participle, it is substantival.
This means it is working as a noun, not a verb. The Aktionsart is then diminished. In particular, ὁ πιστεύων (literally, “the believing one”) and other present participles of πιστεύω (“I believe”) should often be translated as just “the believer.” There is no other word in Koine Greek to mean believer.
“The participle, like an adjective, may be used in the place of a noun or other substantive. The participle itself then functions as a noun” (James Brooks and Carton Winbery, Syntax of New Testament Greek, p. 130).
“Like an adjective, the participle may also be used substantivally, that is, as a substantive, or noun” David Alan Black, Learn to Read New Testament Greek, 150).
Rule 3: The participle depends on the main verb for its Aktionsart, "kind of action'
Therefore, if the main verb is not a present tense, but a perfect tense or an aorist tense, the present participle points to that Aktionsart. In other words, participles cannot be the main verb. So, since there are numerous aorist or perfect main verbs in the examples given for “believing,” those verses prove just the opposite of the idea that one can lose his or her salvation. Those examples actually prove that salvation is one time only as per the aorist tense (looking at the action as a single whole) or perfect (past action with results remaining).
“It often turns out that a Present Participle alludes to an action with which the action of the main verb coincides” (C. F. D. Moule, An Idiom Book of New Testament Greek, p. 99).
“Generally speaking, the present participle denotes action taking place at the same time as the action of the main verb” (David Alan Black, It’s Still Greek to Me, p. 124).
“The ruling consideration in interpreting participles is that they express something that is dependent on the main verb” (David Alan Black, It’s Still Greek to Me, p. 125).
Here are some examples from your list along the lines of this third rule.
1.
John 3:16 Οὕτως γὰρ ἠγάπησεν ὁ θεὸς τὸν κόσμον, ὥστε τὸν υἱὸν αὐτοῦ τὸν μονογενῆ ἔδωκεν, ἵνα πᾶς ὁ πιστεύων εἰς αὐτὸν μὴ ἀπόληται, ἀλλ᾽ ἔχῃ ζωὴν αἰώνιον.
The verb “perish” is an aorist subjunctive, showing that a believer will not perish as an event. In other words, perishing is a one time event, not something that is possible over and over.
2.
will receive “remission” of sins (Acts 10:43),
Acts 10:43 Τούτῳ πάντες οἱ προφῆται μαρτυροῦσιν, ἄφεσιν ἁμαρτιῶν λαβεῖν διὰ τοῦ ὀνόματος αὐτοῦ πάντα τὸν πιστεύοντα εἰς αὐτόν.
The infinitive for “receive” is an aorist infinitive. Therefore, receiving remission (forgiveness) of sins is a one time event, not over and over as would be the case if a person lost his salvation.
3.
John 1:12 Ὅσοι δὲ ἔλαβον αὐτόν, ἔδωκεν αὐτοῖς ἐξουσίαν τέκνα θεοῦ γενέσθαι, τοῖς πιστεύουσιν εἰς τὸ ὄνομα αὐτοῦ·
“Receive” and “Gave” are both aorists. “Become” is an aorist infinitive. So it is a one time receiving to become a son of God.It is “given” one time only, not repeatedly, as would be the case if one kept losing his salvation.
4.
can “know they are saved” (1 Jn 5:13)
1 John 5:13 Ταῦτα ἔγραψα ὑμῖν τοῖς πιστεύουσιν εἰς τὸ ὄνομα τοῦ υἱοῦ τοῦ θεοῦ, ἵνα εἰδῆτε ὅτι ζωὴν αἰώνιον ἔχετε, καὶ ἵνα πιστεύητε εἰς τὸ ὄνομα τοῦ υἱοῦ τοῦ θεοῦ.
“Know” is a perfect subjunctive. The perfect tense indicates a single action in the past that produces results that continue. So we can know once and for all that we are saved. We don't have to repent over and over again, since we can know once for all.
5.
will be “born again” (1 Jn 5:1),
1 John 5:1 Πᾶς ὁ πιστεύων ὅτι Ἰησοῦς ἐστὶν ὁ χριστός, ἐκ τοῦ θεοῦ γεγέννηται· καὶ πᾶς ὁ ἀγαπῶν τὸν γεννήσαντα ἀγαπᾷ καὶ τὸν γεγεννημένον ἐξ αὐτοῦ.
“Born again” is a perfect passive participle. The perfect tense shows action repeated in the past with continuing results. Therefore we are only born again one time, and our salvation is the result that continues.
J of J: Rule 1: The present tense is
often not continuative
“… the present indicative is also
often aoristic.”
“Some presents express punctiliar action
and a few even express perfective action”
REK: “Often,” “some” and “a few” do not negate “normal.” The “normal” use of present participles is still progressively “continuous” and occasionally “habitual” or “regular.”
You make the exceptions the norm.
Rule 2: When the article is before a participle, it is substantival.
REK: It is still la present-tense progressive participle. In English it is called a verb-noun or an adjective-noun, or Gerund.
J of J: In particular, ὁ πιστεύων (literally, “the believing one”) and other present participles of πιστεύω (“I believe”) should
often be translated as just “the believer.”
REK: I disagree. “The believing one” still “normally” means “the one continuously believing.”
J of J: There is no other word in Koine Greek to mean believer.
REK: “Pis-ton” is male and “pis-tia” is female. If God had wanted those two words used, He would have inspired them.
J of J: “The participle, like an adjective, may be used in the place of a noun or other substantive. The participle itself then functions as a noun”.
REK: Yes, a “progressive” noun. The “one who is continuously swimming.”
J of J: Rule 3: The participle depends on the main verb for its "kind of action'
REK: In John 1:12 “even those who continuously believe on His name” is a clause which is
equal to the two aorist verbs, “received” and "gave.”
Context is the final arbiter.
J of J: … participles cannot be the main verb.
REK: Therefore, John 3:16 goes down the drain and is meaningless.
J of J: So, since there are numerous aorist or perfect main verbs in the examples given for “believing,” those verses prove just the opposite of the idea that one can lose his or her salvation. “It
often turns out that a Present Participle alludes to an action with which the action of the main verb coincides”.
REK: Again, “numerous” is NOT the majority of my examples.
Again, you make the exceptions the rule.
J of J: “Generally speaking, the present participle denotes action taking place at the same time as the action of the main verb.”
REK: Again, from John 1:12, God “received” and “gave” authority to those who were ”righteous by faith” because they “continuously believe” (Rom 3:22; 4:5).)
J of J: Quotes John 3:16 in Greek.
REK: “Perish” does not control “continuously believeth.” Plus, the “subjunctive” is correctly translated as “should not” instead of “would not.”
J of J: Acts 10:43
REK: “everyone who continuously believes in Him will receive the forgiveness of sins.” God did not inspire “believeth” as an aorist.
J of J: 3. John 1:12
REK: again, the last clause is EQUAL to the first two aorists.
J of J; 4. 1 Jn 5:13 The perfect tense indicates a single action in the past that produces results that continue.
REK: According to John 3:18, the perfect tense does not guarantee continuance. Our assurance comes from knowing that our justifying and sanctifying faith are the same.
J of J; 5. 1 Jn 5:1 The perfect tense shows action repeated in the past with continuing results.
REK: PERFECT means a thoroughly completed action in the past, but John 3:18 means it does not necessarily not change. PASSIVE means it is in flux. PARTICIPLE means progressive action. Because of the participle, this is a very strong assurance text teaching that those who are continuously believing can be assured they have been born again.