I'm very sorry to hear that, Volk--dogs are some of my favorite people, and the ones I keep are part of my nuclear family, as far as I'm concerned.
I really do think there's an advantage to growing up around large animals. We had huge dogs when I was a baby (and all the rest of the way through my life) and I also spent a lot of time around horses, cows and pigs. When you learn, at five, to just push on the 1500 pound bull until he gets his big butt out of your way, there isn't really a dog in the world you ever fear, and the lack of fear makes for lack of incident, 99 times out of 100. Does that mean a dog couldn't hurt me? Of course not. The bull could have hurt me too (hell if it had stepped on me it could have killed me), but I seemed like I was in charge and for just about all animals this works. This same phenomena is why many people can walk right through a swarm of bees and not get stung, where others are in serious danger doing the same thing. You can't just ACT like you're in charge, you actually have to KNOW you're in charge.
I have only ever run from an animal encounter once, and that was cutting through somebody's private property when I was hiking in Colorado and had fallen and--I thought--broken my wrist (turned out to be a bad sprain). I was cutting across it instead of walking around the seemingly infinite fence to shave a few miles off of my return to my truck, and when I was about 100 yards from having crossed I heard the sonorous baying of a BUNCH of dogs in the distance. Knowing the difference between self confidence and stupidity, I turned around and bolted for that barbed wire, getting to it about fifteen seconds ahead of the pack of a dozen-or-so large mutts (I couldn't distinguish any particular breed among them, but a lot looked like each other so I'd guess a couple litters from the same parents). What really scared me was knowing for a fact that that little four foot fence was not a REAL barrier to a big dog---if I could jump it so could they, and probably without falling on their butts like I had. Fortunately, however, they seemed to genuinely recognize that this marked the end of their territory, and just consented to bark and growl at me until I'd walked far enough away.
I was armed, as always, but with a revolver--and still would have had a half dozen entirely unhurt pooches to deal with after making my absolute best argument, assuming I hadn't missed any which is a big assumption. Couple that with the fact that they were defending their property and I had no right to be on it, I would have had serious problems violently defending myself, both legally and within my own conscience. Plus, the dogs' owner might just grab his deer rifle in that situation...I know if somebody trespassed onto MY property and shot MY dog while "defending" himself, he'd have just opened himself up to a whole new threat.
That segway aside, the long and short of it is dealing with animals is an experiential thing, like pretty much everything else. My niece is almost four, and has two Catahoulas in her daily life that weigh about 90 pounds, and she loves them but has absolutely no patience for them not doing what she wants them to do. The old one will occasionally growl at her if she drops part of her snack and grabs it up again before he can get it, and she'll just point at him and angrily shout "No, Murphy!" which will invariably make him tuck his tail and move away. When I've visited them, it's funny to watch their mailman be almost too afraid to approach the door because of the barking dogs, while the three foot tall blonde munchkin inside rules them with a tiny iron fist. There isn't a doubt in her mind that she's in charge and, consequently, she IS in charge.
I think you would have stood a good chance, especially with a knife, but I can understand your concern. The good news is, they don't have hands. Had it attacked, it probably would have been the most terrifying 10 seconds of your life resulting in a dead dog, a bite mark, and possibly soiled undergarments 
