In the hallway of their house in Toronto's East End, Matthew Carver and his partner Joy Tan slip a tiny purple harness onto their cat, Eleven. Eleven submits with dignity. She does not display the crazed enthusiasm of an outdoor-bound dog, but crazed enthusiasm is not the cat's way. Reserve, stealth, calculation, occasional use of lethal force – these are the ways of the cat.

Once outside, Eleven's curiosity is piqued by butterflies, insects and, especially, lavender. The adorable, striped 14-month-old – possibly part Bengal, but who knows with a rescue cat – walks calmly beside Mr. Carver, stopping occasionally to rest in a bed of succulents. Mr. Carver waits patiently at the end of a purple leash. Across the street, a woman walking a rambunctious border collie calls, "Is she just learning? I love to see that!"

Mr. Carver, a painter, is used to the curiosity that a cat on a leash brings. He and Ms. Tan began training Eleven for the harness soon after they picked her up from the cat rescue (they would also like to buy her a tiny life jacket so they can go kayaking). Mr. Carver had never owned a cat and was smitten by the kitten; Eleven now features in some of his paintings.

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He and Ms. Tan, worried that their small cat would be threatened by predators both four-legged and four-wheeled, decided Eleven would go outside only with supervision. It's an idyllic picture – man, cat, sunny street – that does not hint at the larger conflict raging in neighbourhoods and among animal-lovers: Should all cats be kept indoors and only allowed outside, such as Eleven, in the company of a human?

One compelling piece of evidence is all around on Mr. Carver's street: the sound of birdsong. It's estimated that cats kill between 70 million and 300 million birds each year in Canada and between 1.4 billion and 3.7 billion birds in the United States (feral cats, a problem in many Canadian cities, are thought to kill three times more animals than house cats). That's on top of voles, chipmunks, rabbits, mice and rats, who are also prey, but whose interests are less fiercely protected.

Mr. Carver scoops up his cat – in typical fashion, she has decided enough is enough and demands that her human bend to her will and carry her home. "We mainly didn't want her going outside because we were worried about her," he says. "But when you think about all the birds that might be saved, it just makes sense."

Rationality, unfortunately, is not the cornerstone of our relationship with our pets. We project onto them our notions of freedom and happiness and independence – especially cats, animals that we love not just for their cute faces, but for their aloofness and spectacular hunting skills. Cats are pretty much the living embodiment of the Seinfeld dictum, "no hugging, no learning."

For many cat owners, restricting their pets' movements means destroying some essential element of catness. Conflict, in the past few years, has escalated between outdoor-cat owners and people who own indoor cats and love birds: Researching this story, I heard from one man whose friendship with a bird lover ended over his free-roaming cats, and a woman whose East End neighbourhood was engaged in a hissing, spitting fight over stray cat turds and backyards ominously strewn with feathers.

The forces of captivity are in the ascendant. About one-third of households in Canada owns at least one cat and nearly 20 municipalities across the country have "no-roam" bylaws restricting cats to their owner's properties (Toronto is not one of them, though the city's website notes that "an indoor cat lives a longer, healthier life than that of an outdoor cat"). High-profile authors such as Jonathan Franzen and Margaret Atwood have written about the threat posed by cats to wild birds. Ms. Atwood teamed with the advocacy group Nature Canada to produce her first graphic novel, Angel Catbird, a humorous look at the perils of life on the street for birds.

Younger pet owners may be more sensitive to the issue: In a survey done by Nanos Research in 2016, slightly more than 40 per cent of cat owners said they let their cats run free, but the group most likely to supervise their cats was young people under the age of 29. The tide may be turning and it's sweeping cats inside.

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Not everyone wants to go along with the tide, of course. In a hilly pocket in Toronto's West End, on a short, secluded street perfect for ambling, sunning and mooching food from strangers, is a neighbourhood that one of its residents describes as "a cat's paradise." Here, York University professor Julia Creet lives with her outdoor cats Indy (black, almost human) and Winter (white, not really sure about humans). Ms. Creet rescued both cats – Winter was living on the street and Indy was homeless after his owner's death.

When Ms. Creet rescued Indy, he'd been living in a condo for weeks. "He was so ecstatic about being outside again," she says. That ecstasy, unfortunately, came at the expense of some local birds. Ms. Creet tried putting a bell on Indy and then found something even more effective: A Birdsbesafe collar, which is essentially a bright ruff that fits around the cat's neck and warns away prey. (Although it cost $35 and had to be ordered from the United States, it worked, and Indy stopped bringing his kills home.)

For Ms. Creet, whose house is fitted with a door flap activated by the cats' microchips, the benefits of the pets' freedom outweigh potential risks to their health: "They're animals. They have intense senses. When you restrict them to small spaces and sterile environments, there's some part of them that's not allowed to live."

As humans, we anthropormorphize the animals we love (less so the ones we loathe; there aren't a lot of stuffed scorpions out there). You only have to look at the worldwide success of the hit documentary Kedi, about the feral cats of Istanbul, to understand how strong that pull is. As one man says in the film, "The cat embodies the indescribable chaos, the culture, the uniqueness that is Istanbul." What isn't said, but is obvious to the viewer, is the way the residents of the city love and envy the cats for their freedom.

A small, shame-faced personal confession here: I have always been in the outdoor camp. I've been the personal doorman for a dozen cats over the years (As T.S. Eliot wrote of Rum Tum Tugger, "When you let him in, then he wants to be out;/ He's always on the wrong side of every door"). My current two were born in London, England, where house cats roam more often than not, and some cat rescues will even make sure the adoptive owner has a garden for the cat to use.

It never occurred to me to keep them inside the house, because they seem so happy to leave. It gives me huge pleasure not only to see them outside, but to watch the neighbours interacting with them – the elderly lady who leans down from her walker to talk to them, the toddler who stops crying when his mother says, "Look at the kitties!" They are part of the streetscape of my neighbourhood.

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Of course, as with all cats, they are languid until they're not. The older one, Perdu, is like Tony Soprano – a big, deceptively genial killer. The younger, Athena, is Christopher Moltisanti – jittery, territorial, largely luckless. I praise them when they kill mice and rats, and shriek at them on the extremely rare occasions they bring in birds. You don't need to tell me that it makes no sense.

This lack of logical consistency does not surprise Ted Cheskey, a conservationist and ornithologist with Nature Canada. As part of Nature Canada's Keep Cats Safe and Save Bird Lives program, Mr. Cheskey conducted focus groups with cat owners and discovered that there's a lot of projection going on between owners and their cats. People who valued their freedom, in particular, "didn't want to be constrained and felt their cats had the same needs," Mr. Cheskey says. Telling them that cats could lead perfectly happy lives indoors, with lots of play, didn't work. Neither did chastising them or taking an antagonistic approach. The thing that did change cat owners' attitudes was educating them about the risks to wildlife, particularly the threat to songbirds.

So how many birds do house cats actually kill? It's hard to know for sure. When the University of Georgia tracked the movements of 55 suburban kitties for a year in 2010-11, using the wonderfully named Kittycam, they discovered that our big-eyed friends really are like the Sopranos, blithely strewing carcasses all over the countryside. About a quarter of the prey was brought home; half was left at the site of the kill. A little more than a quarter was eaten.

An equally fascinating BBC documentary called The Secret Life of the Cat revealed how little we know about the extent of cats' roaming, especially nocturnally: It tracked some 50 cats in a village called Shamley Green and discovered that the most footloose feline (a hermaphrodite named Hermie) had a territory of eight acres. Most of them brought home frogs, birds and small mammals a couple of times a week. Who knows what they left in the great rubbish bin of the countryside?

The problem, obviously, is not limited to Toronto's borders. The International Union for Conservation of Nature "ranks house cats as one of the world's worst invasive species," writes Abigail Tucker in her wonderful book The Lion in the Living Room: How House Cats Tamed Us and Took Over the World. Cats adapt quickly to new environments, can eat a wide variety of prey and are quite stellar breeders. The problem is worse on islands, and in countries where cats drive extinctions. A particularly fervent anti-cat mood has gripped Australia, where the environment minister once blamed the world's favourite pet for "a tsunami of violence and death."

There are 600 million house cats in the world, Ms. Tucker writes, making them the most popular house pet: "The lion's little jester of a cousin, once an evolutionary footnote, has become a force of nature." The problem is that we still think of this pet as a lion, free to roam the veldt, and not as a dog – a companion creature whose movements should be constrained, for their own good.

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The cat-safety argument is one that often gets lost, according to Jen Edwards, who has been fostering kittens for Ontario's Uxbridge Cat Rescue for three years. Indoor cats are not only safe from predators, but also a variety of illnesses. Ms. Edwards lives in the countryside northeast of Toronto and does not like to think about the number of times she's seen cat corpses on the road, or scrolled past "Lost Cat" postings on Facebook. As with many cat rescuers, she insists that adopters promise to keep their cats inside.

"Whether my cats would choose to be inside or outside honestly doesn't matter to me," Ms. Edwards said in an e-mail interview. "I chose to make them part of my family. And every day I choose to keep them safe and loved. It is my responsibility as a pet owner to ensure they have lots to do to keep them busy. And truly, it's not hard to keep a cat busy and stimulated."

Indeed, the range of cat distractions available for indoor cats rivals the array of toys for bored children. The helicopter feline-parent can buy "catios" – fenced outdoor runs – as well as climbing towers that rival the Burj Dubai, chic loungers that double as scratching posts, and every manner of simulated bug on a stick. There are "cat dancers" for those cats not too proud to dance. There are harnesses for cats, such as Eleven, who can be coaxed to walk in them. It may not be a walk on the wild side, but it's still a walk and, in the words of the great Canadian pop song, nobody hurts and nobody dies. We may be looking at a very tame future.