Environmental contamination of property is a common subject of negotiation when parties are contemplating a real estate transaction, not least because of the expansive scope of liability and the costs for clean-up *142that may be imposed under the Comprehensive Environmental Response, Compensation and Liability Act of 1980 (“CERCLA”), 42 U.S.C. § 9601 et seq. In this case, LaSalle National Trust, trustee of an Illinois Land Trust for Narco Tower Road Associates (“Narco”), and ECM Motor Company (“ECM”) executed not one but two agreements in conjunction with ECM’s sale of property to Narco. Those agreements attempted to address the question of who would bear the responsibility for site cleanup, when clean-up was to occur, and who would bear the cost. After Narco unsuccessfully tried to collect certain clean-up expenses from ECM, it brought suit in federal court raising claims under both CERCLA and state law. The question before us is whether the CERCLA claim was so utterly without merit that the district court lacked federal subject matter jurisdiction. Because we conclude that the allegations exceeded that low threshold, we reverse the district court’s jurisdictional dismissal of the case and remand for further proceedings.
I.
On September 9, 1991, Narco and LaSalle National Trust executed a real estate sales contract with ECM, under which Narco was to buy certain property from ECM. Paragraph 9.10 of the contract addressed the problem of possible environmental contamination due to hazardous waste:
The entire Property, all Improvements thereon, and the Personal Property will at closing be free from contamination from any substance or materials presently identified to be toxic or hazardous, including asbestos, (“Hazardous Material”) according to any applicable federal, state or local statutes, rule or regulation.... Seller agrees to remove any Hazardous Material, including asbestos, which exists on the Property prior to closing in an expeditious manner.... In the event that prior to the Closing, any such Hazardous Material is found on the Property or in the improvements, the remedy and termination provisions of Paragraph 9.12 shall apply.
The referenced provision of Paragraph 9.12 stated that “[i]n the event any of the foregoing representations and warranties are not true, correct and accurate as of the closing Date, the Purchaser, at its exclusive discretion, may terminate the transaction....” Section 18.6 of the contract specified that it was to be governed by Illinois law. Finally, section 14 of the contract detailed the consequences of default by either party. Nothing in section 14 or in any other section of the September 9 agreement indicated that the contractual remedies were to be exclusive.
Before the closing date, ECM retained Braun Intertee (“Braun”) to evaluate the extent of contamination on the property. Braun removed some 75 cubic yards of contaminated soil and reported that no further action was necessary. Skeptical, Narco retained its own expert, Hygienetics, to review Braun’s findings. Hygienetics eventually concluded that some remaining contamination existed, that it was not a health threat, but that further clean-up work was advisable. It estimated that the recommended measures would cost about $50,000.
Because ECM and Narco did not expect that the additional work would be completed by the planned closing date, they concluded a supplemental agreement on December 31, 1991. In that agreement, ECM acknowledged the existence of further contamination on the property and agreed to complete the clean-up after closing. The parties placed $50,000 in an escrow account to fund the work, and they agreed to seek the approval of the Illinois Environmental Protection Agency (“IEPA”) on the question of appropriate remediation levels for the property. Finally, they addressed the subject of remedies in Paragraph 12, which is the primary focus of the case before us:
Purchaser’s remedies under this Agreement shall be exclusive and in lieu of, and not in addition to, any remedies or rights which it may have, pertaining to the presence of Hazardous Material on the property, in the Real Estate Sales Contract dated September 1,1991.
On the same day that they signed the supplemental agreement, the parties closed the deal.
*143Eight months later, in August 1992, the IEPA established remediation standards for the property. Although Narco and ECM discussed the work ECM would need to do to satisfy the IEPA standards (apparently over a long period of time), they failed to come to terms with one another. ECM continued to investigate, and Narco grew weary of waiting for action. On July 19, 1994, LaSalle and Narco filed this suit against ECM and other parties, seeking damages for ECM’s failure to complete the promised work. Count I of the complaint raised the claim against ECM based on CERCLA § 107, 42 U.S.C. § 9607, Count II charged a breach of contract, Count III claimed that Hygienetics had negligently evaluated the contamination on the property, and Count IV asserted that Hygienetics and two other firms had negligently misrepresented the contamination and necessary corrective measures.
The district court concluded that CERC-LA liability, and thus federal subject matter jurisdiction, depended on whether Paragraph 12 of the December 31 supplemental agreement operated as an exclusive remedy for all environmental claims. Construing the contract under Illinois law, the court found that Paragraph 12 indeed barred all statutory claims, including those like Narco’s that might be raised under CERCLA. Because the CERCLA claim failed, the court concluded that federal jurisdiction also failed, and it accordingly dismissed Count I. It also dismissed Counts II, III, and IV, finding no justification for keeping the state claims when the federal claim had dropped out of the case so early.
II.
When a complaint sets forth a claim that allegedly “arises under” federal law, and thus invokes jurisdiction under 28 U.S.C. § 1331, a federal court should not accept the plaintiff’s word for the proposition that federal jurisdiction exists. As the Supreme Court noted in Hagans v. Lavine, 415 U.S. 528, 94 S.Ct. 1372, 39 L.Ed.2d 577 (1974), it “has repeatedly held that the federal courts are without power to entertain claims otherwise within their jurisdiction if they are ‘so attenuated and unsubstantial as to be absolutely devoid of merit.’” Id. at 536, 94 S.Ct. at 1378-79 (citations omitted) (quoting Newburyport Water Co. v. Newburyport, 193 U.S. 561, 579, 24 S.Ct. 553, 557, 48 L.Ed. 795 (1904)). See also Crowley Cutlery v. United States, 849 F.2d 273, 276 (7th Cir.1988). Through its choice of language, however, the Court has also made clear that only the most extreme cases will fail the jurisdictional test of substantiality. Thus, in Oneida Indian Nation v. County of Oneida, 414 U.S. 661, 666-67, 94 S.Ct. 772, 776-77, 39 L.Ed.2d 73 (1974), the Court overturned a jurisdictional dismissal by the lower courts, with the following comment:
Moreover, we think that the basis for petitioners’ assertion that they had a federal right to possession governed wholly by federal law cannot be said to be so insubstantial, implausible, foreclosed by prior decisions of this Court, or otherwise completely devoid of merit as not to involve a federal controversy within the jurisdiction of the District Court, whatever may be the ultimate resolution of the federal issues on the merits.
Just in case there was any doubt on the matter, the Court noted again in Duke Power Co. v. Carolina Environmental Study Group, 438 U.S. 59, 70, 98 S.Ct. 2620, 2628-29, 57 L.Ed.2d 595 (1978), that “the test is whether the cause of action alleged is so patently without merit as to justify ... the court’s dismissal for want of jurisdiction.” (Emphasis and ellipsis in original) (quotations and citations omitted.) See also Mid-American Waste Systems, Inc. v. City of Gary, 49 F.3d 286, 292 (7th Cir.1995); Gammon v. GC Services Ltd., 27 F.3d 1254, 1256 (7th Gir.1994); Argento v. Village of Melrose Park, 838 F.2d 1483, 1491 (7th Cir.1988).
Given the generosity of this standard, it is obviously possible that a plaintiff may successfully invoke federal jurisdiction, for purposes of 28 U.S.C. § 1331, and yet may lose on some other ground, such as a failure to state a claim on which relief can be granted, see Bell v. Hood, 327 U.S. 678, 66 S.Ct. 773, 90 L.Ed. 939 (1946), or failure to prove the merits after a full trial. A plaintiffs claim may appear to be sufficient on its face, but a defendant may have a compelling affirmative *144defense. In some cases, the distinction between a claim that is wholly frivolous for jurisdictional purposes and a claim that is doomed on the merits may seem medieval in its fineness, and may be of little practical importance. Nonetheless, the distinction is one that the Supreme Court has always recognized, and it rests on both sound policy grounds as well as practical considerations. From a policy standpoint, it underscores the importance of giving a full hearing to those who are attempting to raise claims in federal court, even if those claims are eventually unsuccessful. From the more pragmatic side, the way in which the facts are handled under Fed.R.Civ.P. 12(b)(1) differs significantly from the correct approach for purposes of Rule 12(b)(6), and the Rule 12 procedures differ again from those used for Rule 56.
It is plain that the factual posture of these kinds of pretrial dispositive motions can be critical. In the 12(b)(1) case, the district court is entitled to find jurisdictional facts itself; in the 12(b)(6) case, the court is required to keep the case if there is any set of facts under which the plaintiff might prevail, see Conley v. Gibson, 355 U.S. 41, 45-46, 78 S.Ct. 99, 101-02, 2 L.Ed.2d 80 (1957); and for Rule 56, all disputed material facts must be viewed in the light most favorable to the party opposing the motion, although that party has the burden under Celotex Corp. v. Catrett, 477 U.S. 317, 106 S.Ct. 2548, 91 L.Ed.2d 265 (1986), to come forward with evidence showing the genuineness of any fac tual disputes that it asserts must be tried. Although, as we noted in Pratt Central Park Ltd. Partnership v. Dames & Moore, Inc., 60 F.3d 350 (7th Cir.1995), the district court must conduct a preliminary inquiry into the complaint’s allegations to decide whether it is nonfrivolous, we also warned that this inquiry cannot be a “doppelganger of the inquiry on the merits.” Id. at 351. The inquiry must be confined to the jurisdictional issue, as it was in Pratt Central Park, where the court had to resolve the question whether the contract at issue absolutely precluded the plaintiff from satisfying the $50,000 amount-in-controversy requirement of 28 U.S.C. § 1332. If a complaint asserts that a contract gives rise to rights under federal law, and a fair reading of the contract indicates that the claim meets the test of Hagans, Oneida Indian Nation, and Duke Power, jurisdiction exists, and the case may proceed.
With these principles in mind, we turn to the CERCLA complaint that Narco and La-Salle attempted to bring. This Court has made it clear that section 107(e) of CERCLA precludes efforts to divest a responsible party of its liability, but that it also permits indemnification or other kinds of expense sharing agreements. See Harley-Davidson, Inc. v. Minstar, Inc., 41 F.3d 341, 342 (7th Cir.1994), cert. denied, — U.S. -, 115 S.Ct. 1401, 131 L.Ed.2d 289 (1995). ECM alleges that the supplemental agreement of December 31 is just such an indemnification agreement, and it argues furthermore that the agreement so plainly precludes remedies under CERCLA that it defeats the very jurisdiction of the federal court. In order to resolve this question, we turn to Illinois law, which governs the interpretation of this contract. See Harley-Davidson, supra (question whether indemnification provision of contract covers CERCLA claims is to be determined by applicable state law).
Illinois, like most jurisdictions, requires that contracts be interpreted as a whole, in a way that gives effect to all terms, in the light of their ordinary and natural meanings. In re Halas, 104 Ill.2d 83, 83 Ill.Dec. 540, 545, 470 N.E.2d 960, 965 (1984); Illinois State Toll Highway Authority v. Gust K. Newberg, Inc., 177 Ill.App.3d 6, 126 Ill.Dec. 355, 360, 531 N.E.2d 982, 987 (1988), appeal denied, 125 Ill.2d 565, 130 Ill.Dec. 480, 537 N.E.2d 809 (1989); Medcom Holding Co. v. Baxter Travenol Laboratories, Inc., 984 F.2d 223, 227 (7th Cir.1993). Furthermore, the court must give effect to the intention of the parties at the time of the agreement. In re Doyle, 144 Ill.2d 451, 163 Ill.Dec. 515, 522, 581 N.E.2d 669, 676 (1991); Lenzi v. Morkin, 103 Ill.2d 290, 82 Ill.Dec. 644, 645, 469 N.E.2d 178, 179 (1984); Schek v. Chicago Transit Authority, 42 Ill.2d 362, 247 N.E.2d 886, 888 (1969). Where the terms of the contract are clear, the court must ascertain that intent solely from the language, of the agreement. La Throp v. *145Bell Federal Savings & Loan Ass'n, 68 Ill.2d 375, 12 Ill.Dec. 565, 370 N.E.2d 188 (1977), cert. denied, 436 U.S. 925, 98 S.Ct. 2818, 56 L.Ed.2d 768 (1978); Kennedy, Ryan, Monigal & Assoc. v. Watkins, 242 Ill.App.3d 289, 182 Ill.Dec. 391, 394, 609 N.E.2d 925, 928 (Ill.App.1993). If, on the other hand, the court concludes that the contract is ambiguous, the resolution of the ambiguity becomes a question of fact, to be decided by the trier of fact. Farm Credit Bank of St. Louis v. Whitlock, 144 Ill.2d 440, 163 Ill.Dec. 510, 513, 581 N.E.2d 664, 667 (1991) (threshold determination of existence of ambiguity is question of law for the court); AM International, Inc. v. Graphic Management Associates, Inc., 44 F.3d 572, 575 (7th Cir.1995).
In this case, the initial problem is what exactly constitutes "the agreement," or, to put it more precisely, it is whether we have two distinct agreements that are independent of one another, or two interrelated agreements that should be construed as a whole. The district court appears to have looked at the December 31, 1991, supplemental agreement as a stand-alone document. The language of that agreement does not, however, support such an interpretation. The very first "whereas" clause of the December 31 agreement referred to the real estate sales contract of September 1, 1991. The second clause noted that the real estate sales contract had promised that the property would be delivered free of contamination, and the third clause noted that more remediation work needed to be done. The balance of the December 31 agreement then supplements and alters the commitments of the parties under the September 1 contract, replacing the rigid obligation to complete remediation prior to closing with a more flexible arrangement that included post-closing work by the seller, ECM. Because the timing of the clean-up was altered, the remedies clause of Paragraph 9.12 of the September 1 agreement-which, as noted above, provided that the deal was off if any hazardous material was found prior to the closing-also had to be altered. In short, the December 31 agreement makes no sense unless it is read in context with the September 1 agreement.
Looking at the language of Paragraph 12 of the December 31 agreement in that light, we conclude that Narco and La-Salle have not alleged CERCLA claims that are so "implausible," "patently without mer it," or "insubstantial" that they fail for jurisdictional purposes. One could read Paragraph 12 to say that "[p]urchaser's remedies under [the December 31 agreement] shall be exclusive and in lieu of ... any remedies or rights which it may have ... in the Real Estate Sales Contract dated September 1, 1991," deleting only non-essential language from the text. That in turn could mean that Paragraph 12's remediation measures substitute for the contract termination rule of Paragraph 9.12 of the September 1 agreement. Since there is nothing in the September 1 agreement itself that purports to exclude CERCLA liability, or to exclude any and all other remedies for the parties, this indicates that a facially plausible argument exists that the two agreements taken as a whole did not have that effect.
The language of these agreements does not even approach the conclusiveness of the agreements that this Court construed in Harley-Davidson, supra, 41 F.3d at 344. In that case, the agreement said that "[Harley-Davidson] shall ... indemnify AMF INCORPORATED against all debts, liabifities, and obligations, without any limitation, relating to AMF INCORPORATED's AMF York Division, its operations and products, whether known or unknown, •.. and whether existing on the date of this agreement or coming into existence hereafter." Id. In addition, we note that neither the district court nor this Court in Harley-Davidson believed that there was a failure of subject matter jurisdiction (although that would not be dispositive if no one had noticed an actual flaw). Instead, this Court found that the indemnification agreement was enforceable and applicable, and thus that it was a valid defense to Harley-Davidson's claim for contribution.
III.
On remand, the district court is free to consider the question whether, in light of the key contractual language, ECM has the same kind of valid defense to the Narco/LaSalle *146action that AMF enjoyed. In light of the fact that the two agreements must be interpreted together, we also leave open the question whether the agreements are sufficiently plain that they can be interpreted without resort to additional evidence, or if ambiguities exist that require further exploration. We hold today only that it was error to dismiss on the ground that there was no federal subject matter jurisdiction to support the CERCLA claim. Since the district court’s reason for dismissing the supplemental state claims was the lack of federal jurisdiction over the CERCLA claim, the state claims must also be reinstated on remand, so that the case as a whole will again be before the district court.
REVERSED AND RemaNDED for further proceedings consistent with this opinion.