persist with -1/-1 and +1/+1 counters

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That persist mechanic video stated that a -1/-1 and a +1/+1 counter cancel each other out and go off the creature but could I not argue , by the rules that, they are both on the creature and get counted?
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420.5n If a permanent has both a +1/+1 counter and a -1/-1 counter on it, N +1/+1 and N -1/-1 counters are removed from it, where N is the smaller of the number of +1/+1 and -1/-1 counters on it.

They mutually annihilate. This rule was added somewhat recently so make sure you're looking at an up to date Comprehensive Rules.
That persist mechanic video stated that a -1/-1 and a +1/+1 counter cancel each other out and go off the creature but could I not argue , by the rules that, they are both on the creature and get counted?

This Q will be actual answered once this set coming out. For now, both counters cancelled each other.
There's no question that needs to wait for Shadowmoor to release to be answered. +1/+1 counters annihilate -1/-1 counters and vice versa. They're not planning to change that. The answer to the question is "No, you cannot use the rules to say that they are both there because the rules say they remove each other."

MTG Rules Advisor
 

The annihilation of a +1/+1 and a -1/-1 counter is actually a state-based effect:
State-based Effects

420.5. The state-based effects are as follows:

420.5a A player with 0 or less life loses the game.

420.5b A creature with toughness 0 or less is put into its owner’s graveyard. Regeneration can’t replace this event.

420.5c A creature with lethal damage, but greater than 0 toughness, is destroyed. Lethal damage is an amount of damage greater than or equal to a creature’s toughness. Regeneration can replace this event.

420.5d An Aura attached to an illegal object or player, or not attached to an object or player, is put into its owner’s graveyard.

420.5e If two or more legendary permanents with the same name are in play, all are put into their owners’ graveyards. This is called the “legend rule.” If only one of those permanents is legendary, this rule doesn’t apply.

420.5f A token in a zone other than the in-play zone ceases to exist.

420.5g A player who attempted to draw a card from an empty library since the last time state-based effects were checked loses the game.

420.5h A player with ten or more poison counters loses the game.

420.5i If two or more permanents have the supertype world, all except the one that has been a permanent with the world supertype in play for the shortest amount of time are put into their owners’ graveyards. In the event of a tie for the shortest amount of time, all are put into their owners’ graveyards. This is called the “world rule.”

420.5j If a copy of a spell is in a zone other than the stack, it ceases to exist. If a copy of a card is in any zone other than the stack or the in-play zone, it ceases to exist.

420.5k An Equipment or Fortification attached to an illegal permanent becomes unattached from that permanent. It remains in play.

420.5m A permanent that’s neither an Aura, an Equipment, nor a Fortification, but is attached to another permanent, becomes unattached from that permanent. It remains in play.

420.5n If a permanent has both a +1/+1 counter and a -1/-1 counter on it, N +1/+1 and N -1/-1 counters are removed from it, where N is the smaller of the number of +1/+1 and -1/-1 counters on it.

420.5p A planeswalker with loyalty 0 is put into its owner’s graveyard.

420.5q If two or more planeswalkers that share a planeswalker type are in play, all are put into their owners’ graveyards.
The rule is more than 18 months old.
That persist mechanic video stated that a -1/-1 and a +1/+1 counter cancel each other out and go off the creature but could I not argue , by the rules that, they are both on the creature and get counted?

Counted for what?

The creature's power and toughness after the event? No. After they annihilate each other they are gone.

Whether the creature has lethal damage? Yes. When the -1/-1 counter is placed, the game will check for all SBE conditions. If the -1/-1 counter(s) reduce the creature's toughness to the point where it has sustained lethal damage (or reduces the creature's toughness to zero) the SBE that sends the creature to the graveyard occurs simultaneously with the annihilation of the counters.

Whether the -1/-1 counter will prevent the creature returning? The creature will not return. Persist is (so the Shadowmoor rules primer tells us) a triggered ability that triggers specifically when the Persistent creature leaves play. Thus whether the trigger fires or not will depend on the gamestate immediately before the event. The "event" for this purpose is the SBE check. Therefore the game will see that the persistent creature did have one or more -1/-1 counters when it went to the graveyard, and the Persist ability will not even fire.
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There's no question that needs to wait for Shadowmoor to release to be answered. +1/+1 counters annihilate -1/-1 counters and vice versa. They're not planning to change that. The answer to the question is "No, you cannot use the rules to say that they are both there because the rules say they remove each other."

There is a bit of an intricacy here that should be clarified. Suppose a 2/2 with persist has a +1/+1 counter on it, some effect gives it -2/-2 until end of turn, and than some effect gives it a -1/-1 counter. Persist will not trigger even though the -1/-1 counter and the +1/+1 counter annihilate each other.

The reason is that both the counter annihilation and moving the creature to the graveyard for having 0 toughness are state-based effects that happen simultaneously. So, at the last moment the creature was in play, it had a +1/+1 counter and a -1/-1 counter on it. Since it had a -1/-1 counter on it, persist doesn't trigger.
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Counted for what?

The creature's power and toughness after the event? No. After they annihilate each other they are gone.

Whether the creature has lethal damage? Yes. When the -1/-1 counter is placed, the game will check for all SBE conditions. If the -1/-1 counter(s) reduce the creature's toughness to the point where it has sustained lethal damage (or reduces the creature's toughness to zero) the SBE that sends the creature to the graveyard occurs simultaneously with the annihilation of the counters.

Whether the -1/-1 counter will prevent the creature returning? The creature will not return. Persist is (so the Shadowmoor rules primer tells us) a triggered ability that triggers specifically when the Persistent creature leaves play. Thus whether the trigger fires or not will depend on the gamestate immediately before the event. The "event" for this purpose is the SBE check. Therefore the game will see that the persistent creature did have one or more -1/-1 counters when it went to the graveyard, and the Persist ability will not even fire.

There is a bit of an intricacy here that should be clarified. Suppose a 2/2 with persist has a +1/+1 counter on it, some effect gives it -2/-2 until end of turn, and than some effect gives it a -1/-1 counter. Persist will not trigger even though the -1/-1 counter and the +1/+1 counter annihilate each other.

The reason is that both the counter annihilation and moving the creature to the graveyard for having 0 toughness are state-based effects that happen simultaneously. So, at the last moment the creature was in play, it had a +1/+1 counter and a -1/-1 counter on it. Since it had a -1/-1 counter on it, persist doesn't trigger.

Merestil: Not true. If the counter is not there, it is not there; if the creature is killed well after a +1/+1 counter and -1/-1 counter remove each other, then it will return. There is not an invisible -1/-1 counter on the creature to stop Persist from triggering; there is no -1/-1 counter at all at this point.

CH: Do you mean if the creature has a +1/+1, then is given -2/-2, then receives a -1/-1 counter; or has a +1/+1 counter and is given a -1/-1 counter, and while that ability is on the stack, given -2/-2; or has a +1/+1 counter, is given a -1/-1 counter (the ability resolving), and then given -2/-2?

In the first two, the outcome is the same; the -1/-1 counter is put on it while it's a 1/1, so the SBE putting it into the graveyard will affect it before the two counters remove each other (I'm assuming this is because you don't check them in any order you want, you check them down the list); however, if it is to the third scenario you refer, then it will return, because the SBE that removes N +1/+1 and -1/-1 counters will be checked before anyone can play an ability giving it -2/-2, and so the counters will not be there when the resolution of the -2/-2 effect occurs; as such, the creature will return.

(I see the order you have it in, but since not everyone would understand the difference the order of the effects resolving would have, I wanted to clarify.)

MTG Rules Advisor
 

(I'm assuming this is because you don't check them in any order you want, you check them down the list)

That's not the reason. All applicaple SBEs resolve as a single event; the two counters are removed from the permanent at the same time that permanent is put into its owner's greveyard. The reason that persist triggers is because it's a put-into-graveyard-from-play trigger; it triggers based on the permanents existence just before it is put into a graveyard, i.e. just before SBEs are resolved. At that time it had a -1/-1 counter.
Ah, that makes sense as well. Alright. Well, still, the rest outside of that particular parenthetical statement is true.

MTG Rules Advisor
 

Merestil: Not true. If the counter is not there, it is not there; if the creature is killed well after a +1/+1 counter and -1/-1 counter remove each other, then it will return. There is not an invisible -1/-1 counter on the creature to stop Persist from triggering; there is no -1/-1 counter at all at this point.

I think Ecoris has cleared it up, but just to be sure you understand.

I was answering three questions, because I was not sure exactly what the OP meant.

The third and last was "If a creature with +1/+1 counters gets -1/-1 counters and is put into the graveyardas a result of those -1/-1 counters being placed, do they stop Persist from triggering?" The answer is yes they do.

This is because all SBEs are processed as a single event; therefore when the game checks the gamestate just before the event (the SBE check) for triggers, it sees the Persistent creature has -1/-1 counters, and Persist does not trigger.
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