In many areas of mathematics there are some artifacts that can be used to represent a particular concept.
For example, in geometry, we can find plastic (or paper) models of the Mobius strip that represent the concept of non-orientable surface.
Are there examples of artifacts that can be used to represent concepts of mathematical logic?
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Martin Gardner's 1982 book Logic Machines and Diagrams discusses Carroll's “Game of Logic” (mentioned here by Michael Weiss) and other similar devices that perform logical calculations. For example, it discusses William Jevons' logical piano which I believe was the first use of electrical circuits to perform logical computations. The book is available for free on the Internet Archive.
Usually these machines can only handle propositional logic. For a deeper approach to mathematical logic you might look at Joachim Breitner's Incredible Proof Machine. I don't know if you would consider this an “artifact” though.
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There are several databases of formal proofs. See https://en.wikipedia.org/wiki/Proof_assistant for a list of tools that have been used to produce these formal proofs and follow the links there to find the databases. I'll offer the Isabelle Archive of Formal Proofs as one example.
These databases are more than just representations of concepts in mathematical logic. They are tangible and reusable artefacts allowing formalisation of mathematics to grow.
Lewis Carroll designed a board game called "The Game of Logic"; you will find a full description of the rules and an applet to play it at cut-the-knot's website.
Many games, like Mastermind, rely strongly on logical reasoning, but whether they count as an answer to your question is not so clear.