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In one form or another, attempts to construct a logic of everyday reasoning go back to antiquity. The classical, Aristotelean, bivalent logic may be viewed as a product of such attempts. However, Aristotelean logic does not qualify as a logic of everyday reasoning because it does not come to grips with the core issue — the intrinsic imprecision of everyday reasoning. In modern times, logical systems — driven by a quest for the ultimate in precision, rigor and depth — have become more and more estranged from everyday reasoning. But as we move further into the age of machine intelligence and automated decision-making, the need for a logic of everyday reasoning and natural language understanding becomes increasingly apparent. Today, a logic of everyday reasoning is not in existence. In my lecture, I sketch two attempts to construct one. The first attempt relates to what may be called precisiable everyday reasoning. A very simple example is: Icy roads are slippery; slippery roads are dangerous; therefore icy roads are dangerous. In this case, it can be shown that the seemingly reasonable conclusion is incorrect, and that a correct conclusion can be inferred. A simple example of unprecisiable reasoning is the following. I hail a cab and ask the driver to take me to address A the fastest way. The driver takes me to A but it is not possible to prove or disprove that the route which he chose is the fastest way. Construction of a logic of unprecisiable everyday reasoning requires a radical departure from traditional approaches to logical reasoning.