Waseem Andrabi/Hindustan Times via Getty Images ISIS is part of the story of domestic terror, but it's not the end of it.

Law enforcement officials reported last week that the man who allegedly ambushed a Philadelphia police officer had declared his allegiance to the militant group that calls itself the Islamic State, saying he'd committed the crime "in the name of Islam."

Philadelphia Mayor Jim Kenney later downplayed the question of whether Islam genuinely played a role in the Jan. 7 attack, which left the officer with three gunshot wounds to the arm. The suspect, 30-year-old Edward Archer, had been suffering from mental illness. Archer's mother said her son had been "hearing voices" and was in need of medical help. Archer is a Muslim and attended a local mosque, where he was known as "Abdul Shaheed." But thus far, no evidence has emerged that Archer was actually in contact with anyone from ISIS.

Still, in the current climate of fear surrounding Islamic terrorism, the mere mention of the militant group triggered widespread alarm. The idea that Archer might in some way be linked to the terrorist organization quickly made its way into national headlines, reinforcing a frightening narrative about the power and influence of the Islamic State within the U.S. In the rush to seize on this angle, hysteria overshadowed any levelheaded attempts to examine Archer's claims and figure out whether they have any substance.

Police say Archer told them "I pledge my allegiance to the Islamic State, and that's why I did what I did." But it's not clear what actual ties Archer has to the terrorist group, if any. The FBI is reportedly investigating trips Archer took to Saudi Arabia in 2011 and to Egypt in 2012, and Philadelphia police said they'd received an anonymous tip over the weekend that Archer was part of a small group of local radicals that had previously threatened violence. Philadelphia police did not respond to a request for comment.

ASSOCIATED PRESS This undated photo provided by the Philadelphia Police Department shows Edward Archer, who police say ambushed a Philadelphia police officer at point-blank range with a stolen gun on Jan. 7, 2016.

It's possible that Archer had formally pledged his allegiance to the jihadist group and its leader, Abu Bakr al‑Baghdadi. That pledge, known in Arabic as bay’ah, serves as an affirmation of loyalty to the caliphate -- the political entity the Islamic State claims to have formed in the Middle East.

While these offerings of bay'ah often come from other jihadist groups looking to declare their allegiance to the Islamic State, individuals can also make such pledges.

And while Islamic tradition holds that bay'ah must be validated by the caliph -- in this case, al-Baghdadi -- the Islamic State appears to be taking a more expansive approach in its effort to promote radicalism worldwide.

"ISIS has made it pretty clear that they are accepting anyone who pledges allegiance to them," said William McCants, a senior fellow at the Brookings Institution's Center for Middle East Policy. "You don't have to wait for some kind of reciprocity, and of course they have to do that in order to continue to encourage these lone-wolf attacks, because lone wolf attackers usually aren't going to be known ahead of time. ISIS isn't going to know about them or have the time to confirm each pledge. In the ISIS world, having a pledge is enough, and in theory all pledges are accepted."

In the eyes of some law enforcement officials, not all statements of support for ISIS are equal.

"Somebody who says that they support the Islamic State is a lesser degree than someone who says, 'I pledge allegiance, or give bay'ah, to the Islamic State," said Michael P. Downing, deputy chief of the Counter-Terrorism and Special Operations Bureau of the Los Angeles Police Department. "In that case of 'I pledge allegiance,' basically what they are saying is this: 'I am a citizen of the State and I pledge to God, who told me to be a deputy or soldier of the Islamic State, and my allegiance is to God.’ And that is a much deeper commitment and degree."

In the U.S., even a full-throated pledge to the Islamic State is not a crime in and of itself. But it will raise red flags, for obvious reasons, and that can prompt law enforcement to look deeper into a person's relationships and activities to see if if they are attempting to “support, join, recruit, fund or provide material support” to the Islamic State, Downing said.

If someone commits a crime and claims to have done it in the name of a terror group, clearly that will affect how law enforcement officials proceed. But it's vital not to overstate the significance of a suspect's professed support for, or allegiance to, the Islamic State. Such declarations do not magically open up a line of communication between the person and the organization itself. In other words, just because someone says they acted on behalf of the Islamic State doesn't mean the group ever provided them with any sort of direction before the attack -- or even that the group necessarily knows this person exists.

ASSOCIATED PRESS In this frame from a video provided by the Philadelphia Police Department, Edward Archer appears to run with a gun toward a police car driven by Officer Jesse Hartnett in Philadelphia, Jan. 7, 2016.

In Archer's case especially, McCants said, it's important not to ignore the role that mental illness may have played.

"If this was really an ISIS supporter, you would want to see some kind of history of actual support before they were caught, either verbal or otherwise," he said. "Someone talking to the police, especially if they have a history of mental illness, and they're just reeling off reasons they did [something] and they just happen to mention ISIS -- that is different."

It's crucial for law enforcement and the media to be as forthright as possible when reporting on the extent of a terror suspect's relationship with the Islamic State -- especially considering how often people misconstrue and overreact to such threats, using them to reinforce Islamophobia and other dangerous stereotypes.

In a blog post at War on the Rocks last year, Clint Watts broke down the different ways to describe a suspect's ties to the Islamic State, or any other terror organization, in the context of a specific plot.

The most devastating plots, Watts wrote, have been fully "directed" by the Islamic State, like the November attacks in Paris. Others, like the Charlie Hebdo shooting last January, were "networked" by the group, meaning they likely arose within a community of terrorist affiliates, veterans and sympathizers, but were not formally organized by the leaders of any one group.

At the low end of the spectrum, according to Watts, are the various types of violence "inspired" by the Islamic State. These attacks often involve lone actors who have no legitimate connection to the terror group, but who may have become radicalized in part due to its influence. The disjointedness of these schemes often limits their ability to inflict mass casualties, but it also makes them harder for law enforcement to identify and thwart. As Watts writes:

'Inspired' plots are essentially a function of fame seeking in a highly individualistic social media age. These 'inspired' plots will continue and occur in spikes as one lone jihadi wannabe, often witnessing the attention gained by another, seeks to join in the media splash of whatever group currently holds international attention. Globally, 'inspired' plots will be more of a detriment than a benefit to al Qaeda and the Islamic State over the long run. While 'inspired' plots spark fear and in some cases a heavy-handed backlash against Muslims (bringing more recruits into the jihadi movement), these uncoordinated attacks usually fail, are aimed at targets of limited strategic significance, or kill innocent Muslims. These failures and fizzles can reduce popular support for the group.

Jae C. Hong/ASSOCIATED PRESS Investigators in San Bernardino, California, examine the scene where two ISIS-inspired attackers engaged in a shootout with police, Dec. 3, 2015.

McCants said the Islamic State's ability to inspire these kinds of attacks is troubling -- even if they are ultimately less of a threat than other, more sophisticated operations.

"We have to take at least some measure of their ability to inspire people to do these kind of things," he said. "They have been a lot more successful at doing it than, say, al Qaeda has been. Al Qaeda had tried an awful lot to get people to carry out attacks in its name in this country, and did not have many high-profile successes."

There are hundreds of Americans actively expressing support for the Islamic State on social media, according to a recent report by George Washington University. That accounts for just a tiny fraction of the estimated 3.3 million Muslims now living in the United States (and indeed, it's not at all clear whether every single American voicing support for ISIS actually identifies as a Muslim). But law enforcement must work to identify the dangers these people might pose.

At the same time, we've seen what can happen when authorities and the media don't exercise due caution in explaining the apparent motives behind certain terrorist attacks.

Outlets ran a number of inaccurate stories in the aftermath of the Islamic State-inspired shooting in San Bernardino, California, last month, suggesting that the female shooter had publicly expressed support for the Islamic State on social media. After inviting accusations that law enforcement had failed in the impossibly vast duty of catching and responding to every single instance of Islamic State support online, authorities clarified that the woman had only made her statements privately.

Many of these initial reports failed to contextualize the nature of the attackers' relationship with the Islamic State. In other cases, readers simply may not have bothered with such details. As a result, many people -- including presidential candidates -- used stories about the Islamic State connection to accuse all Muslims, and especially immigrants, of potentially being beholden to the terror group.

Sensationalism triumphed again last week in the scramble to describe the shooting in Philadelphia simply as an act carried out by an "ISIS-linked" shooter. A more nuanced examination of this case would emphasize that violence and extremism are products of a complex set of social and political conditions, often inflamed by personal factors including, but not limited to, a suspect's mental health. Instead, we got an anecdote that invited bigoted responses from people unwilling to look beyond religion and race in search of motivations for the shooting.

It's convenient to explain the complicated phenomenon of radicalism using only two letters -- ISIS, ISIS, ISIS -- but this sort of reductionism fails to educate or make anyone safer. In fact, it actually helps the Islamic State in its mission to terrorize the U.S., stoke Islamophobia and further radicalize young people. And with hateful acts against Muslims on the rise around the country, we've already seen one peril of such oversimplification.

This article has been updated to clarify that Kenney himself did not reference reports of Archer's mental illness.

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