It is my understanding that for a Protestant baptism to be valid, there must be an intent to do what the Church does. I would assume this would mean that there is an intent to instill sanctifying grace and remove original and any later sins.
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No, not even. One needn't believe such things (grace, sin, etc.) even exist, ergo the validity of baptisms performed, say, by an atheist midwife or a Jєωιѕн doctor. Intention has to do with intending to do the actual act, not with what one thinks the act does (or doesn't) accomplish (this is the same reason that, say, a priest who doesn't believe in the Real Presence can still effect the Eucharist).
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The standard for intention of 'to do what the Church does' is admittedly vague, and that's because the bar for sacramental intention is itself incredibly low. The intention of an emergency worker (who isn't Catholic, and might not even be Christian) who issues an emergency baptism at, say, the request of an incapacitated mother whose newborn is dying, is probably just intending something like 'I intend to do this thing Christian people do to their babies' and that suffices.
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Now, when it comes to conditional baptisms after such baptisms (or after even baptisms performed under more 'ordinary' circuмstances, like the baptism a child might receive from Lutherans before later converting to Catholicism as an adult), such conditional baptisms have more to do with simply lacking any evidence that the baptism was performed, or performed correctly. In lieu of positive evidence, the Church conditionally baptizes.
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Technology makes it remarkably easier to track down the sufficient evidence. Baptismal records are digitized which makes their archival and retrieval far less susceptible to human error (this of course assumes the baptism was recorded in the first place). If the minister is still alive, they can be interviewed as well, just with a quick email or phone call. Photographic evidence is extremely common, and video evidence is becoming even more common. But, try to track down a Lutheran minister who performed a baptism thirty years ago in, say, 1899 and you'd have a much more difficult time. Hence, the preponderance of conditional baptisms.
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I hasten to add that even with all of this said, I
don't think that it would be correct to say that there is a presumption of the validity of non-Catholic baptisms (maybe there is, I've just never seen anything to indicate that's the case
per se). Such baptisms should still be investigated. If the investigation does not satisfy all reasonable and positive doubts, conditional baptism is performed.