At what age should mathematical / logical concepts be introduced to a child?
I feel that parents should be held more responsible even than schools in the educational process.
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The best age is "when you need them". But let me elaborate:
Math starts way before school, because you need mathematical concepts to explain the everyday world. But we usually don't call it "Math", unless we are thinking about it.
Start with simple counting:
Count the apples when buying them at the store. Count the chairs at the table. Count fingers, toes, books and Legos. Count family members - is everyone home?
Ask for help when cooking: Give me two eggs. Measure four tablespoons of sugar.
Continue with easy calculations and comparisons:
How many glasses must be put on the table when every family member needs one? And when we have a guest? Or two?
How many cookies are left in the jar? Is that enough so that every family member gets one? Perhaps even two?
On to fractions:
Start with one apple, cut in half. Then quarters. Move on to a pie or a square/rectangular cake (use the correct terms!). Compare: Mommy's shoe is as long as two toddler shoes, so one toddler shoe is as long as half a mommy shoe. Introduce first times: half past, quarter to...
It's always a good idea to use proper names. Find hidden shapes. A square sandwich can be cut into two triangles. Or four. Or two rectangles. Or four squares again...
But always remember: These concepts should be introduced gradually. It doesn't make sense to think too hard and present these ideas as a lecture. Children are curious and learn "in passing". Use opportunities as they come up. And there will be plenty if you share your life with your child.
The current trend in test-driven school systems like in the United States is to introduce the formal academics too early. By "formal" I mean where you sit down with a page full of problems that you solve for its own sake. That means the role of parents today is largely helping your children keep up and not feel frustrated or stupid at not quite being ready for what they're expected to do.
The informal math usually comes fairly naturally to parents, and is introduced at the pace of the child's curiosity, as in Stephie's excellent answer. If you are homeschooling, you introduce the formal concepts alongside the informal when you see the child is ready to accept them. For example, your child asks good questions when you are doubling a recipe, so you schedule a formal lesson on multiplying fractions. For another example, we taught my son to subtract two-digit numbers when he started wondering how old he would be in different years. Consider the advanced math Homer Hickam learned in October Sky to further his rocket hobby.
Children learn best at their own individual pace, and they give off plenty of clues as to what that pace is. Try to learn to recognize those clues.
As early as possible (but not earlier) and at whatever pace the child can keep up with.
I find pocket money is an excellent tool to learn to count, add and subtract, provides incentives to get it right, but earlier on, there are many games you can use to tickle the math neurones.
When your child starts reasoning about his surroundings (which varies from one child to another but usually starts before the age of 5) you can start introducing basic logic into their reasoning (if A, then also B?).
Note, though, that my oldest finds me particularly annoying, especially when she asks me a question and I respond with a question, and often the same question.