Synonyms for the word "under-pin" include support, brace, bolster, buttress, fortify, strengthen, reinforce, sustain, uphold, and prop up. All of these words suggest the idea of providing a foundation or structure for something else to rest on. Under-pinning can be used to hold up a physical structure like a building or bridge, or it can be used more figuratively to describe the support given to an argument or idea. In any case, the synonyms for under-pinning all have similar connotations of strength, stability, and reliability, and can be used interchangeably depending on the context.