For a sunny walk in the park, you might not need one
Yeah, but we all carry one, just in case!
I'd like to address Beef's OP (but paraphrase the hell out of it):
Can you go on a day hike and not need a knife? - YES!
When I go for a day hike should I leave my knife behind? - NO!
The thing is that for certain activities like a day hike you probably don't need a knife, though most of us will take one because we want to. But just because you 'probably' don't need a knife doesn't mean that you 'definitely' don't - look up the 2 words in the dictionary if you are confused.
I could leave the house without a knife or pants, but either way I'd feel a bit naked. A knife might just be a SAK or even less (I have a 0.2oz folding knife with VG-10 steel blade & titanium scales). On the 'General Knife Discussion' section of this forum there was a thread about cases of knives saving lives and someone posted an example of a woman that died because no one had a knife - this was in an urban environment with many people about including some that tried to help her (clothing caught in an escalator). I EDC a SAK & a Multitool even though most of the time I don't need them and don't use them - but just because 9 out of 10 days I don't need them doesn't mean that not having them on hand EVERY day doesn't matter. What if I don't have them on me on the day I need to use one of them? My EDC provides me with many useful tools including 3 plain blades, 1 serrated blade, a file, scissors, saw, pliers, screwdrivers, bottle opener, can opener, etc. Why would I not want a variety of useful tools available when I leave the house - at home I have a house full of useful stuff, when I go out I need to have at least a few basic tools available.
I do think that for a day hike a SAK is probably going to be enough - there are plenty of different models available and you could get one with the most useful tools for an emergency situation (blade, saw, file).
Of course for camping there are many more possible tasks for a knife and my BK-7 has capabilities that a SAK can't match. I would really like to have my BK-7 or BK-9 on hand if I was in a survival situation too, a strong full tang fixed blade that can chop or baton can also do many other tasks as needed with little worry about failure. There is a lot of psychological benefit to be had from being equipped with a knife that you know you can rely on!
At the end of the day it is up to the individual to decide on how much knife they want to carry, I would always consider a SAK to be the minimum in wilderness equipment and feel much happier with a solid fixed blade of some sort as well. Let's face it - unless I really needed to travel light I'll have a full tang fixed blade + Mora + folder + SAK + Multitool for my minimum knife equipment. Maybe I don't need all that - but can I be sure that I won't?