In 1973, the U.S. Supreme Court said in McDonnell Douglas Corp. v. Green that once the plaintiff (court uses the word petitioner if you want to read the case; respondent means defendant) meets the burden of showing a prima facie case, then the burden shifts and the defendant must offer evidence (yes, it can be testimony) to rebut the prima facie case of discrimination. Testimony or not, the evidence or reason offered by the defendant for the conduct should be based on an objective standard, not a subjective standard, opinioin, or personal judgment.
However, although they can "claim anything" as you put it, the burden then shifts back to you and at that point all the plaintiff will be given an opportunity to show that the defendant's reason is 1) pretext (pretext in this case means testimony or evidence put forward to conceal a true purpose) or 2) that the the reason or system that you are claiming caused the discrimination against you is in fact discriminatory in its effect. This means that you have an opportunity to show that it has a discriminatory effect on other protected classes of employees such as yourself. If you prove either of those things, 1) employer's reason was pretext; or 2) that employers reason or method is discriminatory in application, you should prevail.
So the short answer to your question is yes and no. The defendant may say what he wants and if the defendant's reason appears to be non-discriminatory, you will recieve an opportunity to prove that they are lying to cover up the discrimination or that their reasoning although not specifically aimed at discriminating against you, has a discriminaotory affect.
If you want further details, google the case I mentioned previously.
The defendant employer has to cite or identify (articulate) a non-discriminatory reason for the action against the plaintiff only after the plaintiff has introduced evidence that the employer's action constituted unlawful discrimination *if it is not otherwise explained.* Once the employer's stated non-discriminatory reason for the action against the employee has been identified, the plaintiff must prove that reason (or the claim that it is not discriminatory) is bogus. The plaintiff always has the ultimate burden of proving the unlawful discriminatory character of the disputed action.
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The other attorneys are correct. It means in this sense "to say." Proof is not required and you then have to submit evidence that their legitimate, non-disctiminatory reason is wrong. While it is true the Employer can try to say anything, in practice, the Employer's articulation of a reason "pins them down" and if you have a case you can marshall the existing evidence to prove the reason wrong and simply a pretext to let you go.