# No honor amongst liars

I am asking this question because I am being increasingly perplexed by what constitutes a 'lie' and the truth, and also indeterminate, in the tag.

Is there a formal definition of a 'lie' - I am finding some of the logical arguments difficult to follow.

Is it:

assume P is true, then a liar always declares 'P is false', and vice versa?

What if the premise P is not so clearly defined?, for example 'what is your age?'

### UPDATE (3/feb/2017)

Looking at several of the liar puzzles, the main cause of concern appears to be what the negation of a statement is.

There appear to be, for example, several negations of:

both Harry and Sally got wed

such as:

• neither Harry nor Sally got wed
• Harry got wed, Sally didn't
• Harry didn't get wed, Sally did
• both Harry and Sally did something else instead
• the negation results in a nonsense statement

This needs clearing up if a puzzle is to be unambiguous.

• I seem to recall that Smullyan referred to them as knights and knaves to avoid the connotations of "liar" and provided a rigorous definition. I can only add my advice to read everything by Raymond Smullyan you can get your hands on. Commented Sep 2, 2016 at 18:25
• Again, depends on what questions you can ask. If you can say "Tell me the number of an unsafe door" and they are all liars, they will lie and give you the safe door's number. Commented Sep 25, 2016 at 8:45
• @HughMeyers; i did it in 5 - ask each one for the safe door in the remaining set of 'unsafe' doors, but then my liar got cocky and said - 'but B IS a safe door'!
– JMP
Commented Sep 25, 2016 at 8:47
• For yes/no you can always go binary: "is the safe door one of these three?" Etc. Commented Sep 25, 2016 at 8:57
• For what it's worth, people mean different things when they use the word "lie". Say you ask me what color my front door is. I answer "green" because it was green when I left the house. Am I a liar if you secretly painted it white after I left? Similarly, if I had said "white" intending to deceive you would I be saved from being a liar by the "accidental" truth of my statement? I believe you will find opinions differ. Commented Sep 25, 2016 at 9:26

You're not the first to notice the complications around the issue of what exactly defines a lie for the purposes of these logic puzzles. After the following question was published in Scientific American:

Here's a recent twist on an old type of logic puzzle. A logician vacationing in the South Seas finds himself on an island inhabited by the two proverbial tribes of liars and truth-tellers. Members of one tribe always tell the truth, members of the other always lie. He comes to a fork in a road and has to ask a native bystander which branch he should take to reach a village. He has no way of telling whether the native is a truth-teller or a liar. The logician thinks a moment, then asks one question only. From the reply he knows which road to take. What question does he ask?

... the magazine received a letter from Willison Crichton and Donald E. Lamphiear of Ann Arbor, Michigan, which dissected the problem of different possible definitions of lies and liars. They divided liars into simple liars, honest liars, and artistic liars, each of whom would respond differently to classical logic puzzles. Here is their letter quoted for your interest, pleasure, and amusement - any good pedant will have a field day reading this! (All emphasis is mine.)

It is a sad commentary on the rise of logic that it leads to the decay of the art of lying. Even among liars, the life of reason seems to be gaining ground over the better life. We refer to puzzle number 4 in the February issue, and its solution. If we accept the proposed solution, we must believe that liars can always be made the dupes of their own principles, a situation, indeed, which is bound to arise whenever lying takes the form of slavish adherence to arbitrary rules.

For the anthropologist to say to the native, "If I were to ask you if this road leads to the village, would you say 'yes'?" expecting him to interpret the question as counter-factual conditional in meaning as well as form, presupposes a certain preciosity on the part of the native. If the anthropologist asks the question casually, the native is almost certain to mistake the odd phraseology for some civility of manner taught in Western democracies, and answer as if the question were simply, "Does this road lead to the village?" On the other hand, if he fixes him with a glittering eye in order to emphasize the logical intent of the question, he also reveals its purpose, arousing the native's suspicion that he is being tricked. The native, if he is worthy the name of liar, will pursue a method of counter-trickery, leaving the anthropologist misinformed. On this latter view, the proposed solution is inadequate, but even in terms of strictly formal lying, it is faulty because of its ambiguity.

The investigation of unambiguous solutions leads us to a more detailed analysis of the nature of lying. The traditional definition employed by logicians is that a liar is one who always says what is false. The ambiguity of this definition appears when toe try to predict that a liar will answer to a compound truth functional question, such as, "Is it true that if this is the way to town, you are a liar?" Will he evaluate the two components correctly in order to evaluate the function and reverse his evaluation in the telling, or will he follow the impartial policy of lying to himself as well as to others, reversing the evaluation of each component before computing the value of the function, and then reversing the computed value of the function? Here we distinguish the simple liar who always utters what is simply false from the honest liar who always utters the logical dual of the truth.

The question, "Is it true that if this is the way to town, you are a liar?" is a solution if our liars are honest liars. The honest liar and the truth-teller both answer "yes" if the indicated road is not the way to town, and "no" if it is. The simple liar, however, will answer "no" regardless of where the village is. By substituting equivalence for implication we obtain a solution which works for both simple and honest liars. The question becomes, "Is it true that this is the way to town if and only if you are a liar?" The answer is uniformly "no" if it is the way, and "yes" if it is not.

But no lying primitive savage could be expected to display the scrupulous consistency required by these conceptions, nor would any liar capable of such acumen be so easily outwitted. We must therefore consider the case of the artistic liar whose principle is always to deceive. Against such an opponent the anthropologist can only hope to maximize the probability of a favorable outcome. No logical question can be an infallible solution, for if the liar's principle is to deceive, he will counter with a strategy of deception which circumvents logic. Clearly the essential feature of the anthropologist's strategy must be its psychological soundness. Such a strategy is admissible since it is even more effective against the honest and the simple liar than against the more refractory artistic liar.

We therefore propose as the most general solution the following question or its moral equivalent, "Did you know that they are serving free beer in the village?" The truth-teller answers "no" and immediately sets off for the village, the anthropologist following. The simple or honest liar answers "yes" and sets off for the village. The artistic liar, making the polite assumption that the anthropologist is also devoted to trickery, chooses his strategy accordingly. Confronted with two contrary motives, he may pursue the chance of satisfying both of them by answering, "Ugh! I hate beer!" and starting for the village. This will not confuse a good anthropologist. But if the liar sees through the ruse, he will recognize the inadequacy of this response. He may then make the supreme sacrifice for the sake of art and start down the wrong road. He achieves a technical victory, but even so, the anthropologist may claim a moral victory, for the liar is punished by the gnawing suspicion that he has missed some free beer.

I found this letter published in one of Martin Gardner's excellent puzzle books, Hexaflexagons and Other Mathematical Diversions, whose text you can find as a PDF document here.

For the purposes of logic puzzles, we assume that our liars are "honest liars", who always utter the logical dual of the truth. Logic puzzles are written by logicians and intended to be exercises for the brain, not entirely realistic scenarios. As Deusovi said, questions is such puzzles tend to be yes/no questions, thus making it easier to define the logical dual of the truth.

• so different liars that use different rules to form their lies can arrive at different answers to the same compound question. but ultimately their own lie systems will betray them.
– JMP
Commented Sep 2, 2016 at 16:53

Typically, in these types of puzzles, all questions must be answerable with "yes" or "no". The liar then takes the opposite of the "correct" answer.

In any situation more complicated then that, you should probably request clarification from OP in the comments.

(Not an answer, but another thing to watch out for.)

In addition to the type of liars puzzle already discussed (where you ask a set of people true/false questions), there is another type where a set of people make statements that are either lies or truths, and you have to determine which are which.

This leads into a whole different problem called the paradox of material implication (which I first encountered while working Thirteen people telling truths or lies).

The basic version of the paradox comes when we assume two things:

• Iff a statement $X$ is a lie, then $\neg X$ must be true.
• When someone says, "if $A$, then $B$," this is equivalent to $A \Rightarrow B$

Now logically, $A \Rightarrow B$ is logically equivalent to $\neg A \lor B$. Therefore if a liar says "if $A$, then $B$," this means that $\neg(\neg A \lor B)$, which is logically equivalent to $A \land \neg B$.

This means, for example, that if a liar says "if it rains tomorrow, I will get wet," it must be true that it rains tomorrow and the liar does not get wet.

This is obviously ridiculous, since the liar's statement cannot control the weather, and common sense would say that the statement is a "correct" lie if the liar is planning on staying indoors tomorrow, regardless of whether it rains or not.

This means that one of the above assumptions is false; or more generally, that natural-language statements do not map to logical statements in a simple one-to-one manner.