I agree with Peter's response, and I'd like to pick up on the possibility that the machines in question are not computers. Although it is not clear what computation is, it seems plausible that not all machines are computers. A claim that such non-computational machines can have knowledge would escape Turing's or Searle's arguments. One might argue that human beings are such machines: we work in mechanical ways, we have knowledge, but we are more than mere computers. John Searle has a mechanistic, non-computational, view along these lines. A potential challenge that such a view faces is to explain what this broader sense of 'mechanical' means. It must mean something different from 'performs a computation', but one might be reluctant to broaden the notion so far that it applies it to all possible systems: that would render it trivially true that machines can understand. Finding an intermediate ground is not obvious.
I think most would agree that there are multiple forms of intelligence. However, is there one particular form - for example, logic - which is foundational to all others?
I think that the question that you ask is still an open one: it is not known to what extent our mental life is underwritten by logical reasoning. The question may eventually be resolved by cognitive science. However, one worry that might face someone in answering your question is how broadly 'intelligence' should be understood. If you mean our general ability to cope with difficult situations that confront us in day-to-day life, then it is highly unlikely that logic is ultimately responsible for our degree of success. If you mean our ability to solve problems in IQ tests, then logic is highly likely to be responsible. I'm personally rather sceptical of any approach that divides up human cognitive life into different general 'intelligences'. To my mind, a more interesting approach would be to pick a particular cognitive process---say, language learning---and try to determine, for that particular cognitive process, the extent to which logical inference plays a role.