Kripke’s thesis is that rigid designators are true, we have an intuition of them, and that they are the same in every possible world (Kripke 48). A designator is a common term that covers names and definitions (24). Specifically, names are rigid designators (48).
Kripke also has a lucid discussion on what a “possible world” is (and isn’t). We imagine a situation that could have been otherwise. What properties of x would remain in that world and which would be different?
Example: “The man who invented bifocals is Benjamin Franklin.”
“Benjamin Franklin” is a rigid designator. Benjamin Franklin is Benjamin Franklin in every possible world. But the phrase “the man who invented bifocals” is a nonrigid designator. One can imagine a world where someone other than Franklin invented bifocals.
His most notorious and ground-breaking argument is that there can be both contingent a priori truths and necessary a posteriori truths. How? Take Goldbach’s conjecture: every even number greater than two is the sum of its primes. This appears to be necessary, per mathematics, but is only known a posteriori.
Conclusions
*Kripke agrees with Mill that singular names are non-connotative (127).
*General terms, those of natural kinds, have a greater kinship with proper names that normally realized (134).
*a priori truths can be contingent, meaning the fixed reference for a term isn’t always synonymous with a term (135).
*the relationship between a brain state and a mental state is a contingent one, and relations of identity cannot be contingent (154).
Criticisms
Kripke sometimes spends several pages analyzing a minor point with little payout.
Evaluation
One can see why this book broke new ground. I read it after I read Plantinga’s The Nature of Necessity, so I didn’t see what was objectionable about possible worlds semantics. Much of the book, however, was beyond my pay grade.
Logical Investigations 
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