Jason Cole article: Brees - 3rd Franchise Tag? CBA Wording

Boiler Jim said

I think it likely that a truly independent source would conclude that the wording has to do with a player being designated as a franchise player for the third time. It could have been more clear as to intent, however. If the Saints don't sign Drew by July 15, it is probably going to court.

What else will probably happen is that, since this is CLEARLY the second tag that has been applied to Drew, it will be argued that it CAN NOT BE AN EXCLUSIVE tag... Which I believe would be fair to both the Saints and Drew...

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Perfect example of what I mean. If it were written like you suggest, then yeah it would mean a player designated for the third time regardless of the designating team. However, ITS NOT. It specifically states the any team that designates a player for the third time. The player is not the subject. With club being the subject of the sentence, the action expressed by the verb (designating a player for the third time) refers to just one club since the subject is "club".