A Logic Puzzle about Sophists

I love knights-and-knaves puzzles: where knights always tell the truth, and knaves always lie. The following puzzle has a new type of person: a sophist. A sophist only makes statements that, standing in their place, neither a truth-teller nor a liar could make. For example, standing next to a liar, a sophist might say, “We are both liars.” Think about it. If the sophist was a truth-teller, then the statement would have been a lie, thus creating a contradiction. If the sophist was a liar, the statement would be true, again creating a contradiction.

Here is the puzzle with sophists. And by the way, this one is intended for sixth graders.

Puzzle. You are on an island inhabited by knights, knaves, and sophists. Once upon a time, a sophist made the following statements about the island’s inhabitants:
1. There are exactly 25 liars on this island.
2. There are exactly 26 truth-tellers on this island.
3. The number of sophists on this island is no less than the number of truth-tellers.
How many inhabitants were on the island once upon that time?

I love this new sophist character in logic puzzles, but I have no clue why they are called sophists. Can anyone explain this to me?



  1. Sasha Volokh:

    Probably because most people know about sophists through their unflattering portrayal in Socrates’ dialogues. They were debaters and rhetoricians in ancient Athens, but Socrates criticized them for being relativists and not seeking truth as such. (Though today, “sophistry” might have migrated even further and have a connotation of misleadingness that would make sophists out to be liars! But I think Socrates’s view was that sophists’ positions were orthogonal to truth/lying, much like Harry Frankfurt’s definition of “bullshit”.)

  2. tanyakh:

    Thanks, Sasha.

  3. Derek:

    Great puzzle! I’ve never seen “Sophists” as a part of these.

  4. George R:

    Sasha’ s approach on sophists is nice and correct. I would just add – being myself a Greek- that in modern Greek language sophism (Σόφισμα) is basically an argumentation which tries to support false values. It looks “impressive” but in reality is empty of content and thus value. A “non-argument” in essence.

  5. George R:

    I don’t really understand the puzzle’s definition of a sophist. Is their statement always neither true nor false? They have to always state something that is ill-defined? Non falsifiable?

  6. tanyakh:

    George. The sophist’s statement has to be not true if we replace the sophist with a trutheller and has to be not a lie if we replace the sophist with a liar. For example, a sophist might say: “I am a liar”.

  7. Ivan:

    If one were a sixth-grader, one might have answered that the number of inhabitants was 78. If one were a philosopher, one would have asked who was the interviewer who reported the Sophist 1’s statement. Was he local or not? If NOT, was he included in the headcount (how)? If LOCAL, was he a liar, a truth teller, or a sophist (all reporting Sophist 1’s statement differently)? Now we are entering the garden of the forking paths …

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