In this critique of logical positivism that makes specific mention of the idea that what cannot be proven is not true Jacob Bronowski points out the fallacy of this.
Truth is not so simple and logic breaks down in complexity, so logic proves unreliable when we need it most, that is, for ultimate answers.
Turns out, the ultimate answers, if such things exist, may no be illogical but they cannot be proven true with formal logic.
Jacob Bronowski explain his objections much better than me.
