As a practitioner of philosophical enquiry with children, I agree with Rex Kerr’s suggestion that offering and expecting good reasons can be highly instructive for children.
I’m glad to recommend some books which could help you lead your own philosophical discussions with your child. An excellent introduction is "The Philosophy Shop" (ed. Peter Worley, director of The Philosophy Foundation in the UK). It's a stimulating, accessible and well-structured repository of philosophical stories, questions and ideas. It includes many stories that are suitable for young children, though some others are more suitable for older children or adults. Peter Worley's book "The If Machine" (preview on Google books here) is also excellent.
Another helpful book is "The Philosophical Child", written for parents by Jana Mohr Lone (director of the Northwest Center for Philosophy for Children in the USA, and blogger at Wondering Aloud).
Whilst you can certainly stimulate your child's interest in philosophy and logic through one-to-one dialogue, there is nothing quite like the experience of collaborative enquiry among a group of children, where the kids are exposed to a diversity of views. Being part of a philosophical community of enquiring minds helps children to expand their thinking, and provokes them to question their own and each other's beliefs.
In certain parts of the world, there are extra-curricular opportunities for children to join a ‘philosophical community of inquiry’ or otherwise engage in collaborative philosophical thinking with their peers. I offer children such experiences in Australia through The Philosophy Club. I’m familiar with other Philosophy for Children organisations around the world, and would be happy to share my knowledge with anyone interested.