# propositional logic , don't know the answer

Is the assertion "This statement is false" a proposition?

I think that it is a proposition because this("This Statement is false") may have truth values. the statement may be true or maybe false. therefore its proposition. but i don't know if i am correct. please correct me , if i'm wrong.

• In propositional logic a sentence like "This statement is false" must be symbolized simply as $p$, so it may have a truth value. In first-order logic, in order to "translate" it, you must use a predicate like: $false(x)$, but this kind of predicates, expressing semantical properties of the language, cannot be used "in" the language to "speak of" the language itself, in order to avoid the Liar paradox (that is the "circular" assertion you are using). A predicate $false(x)$ must be used in the meta-language for asserting the semantical properties of the language you are studying. – Mauro ALLEGRANZA Mar 3 '14 at 9:38
• thanks for giving the answer, – user3129893 Mar 3 '14 at 9:41
• but i really don't understand your answer because today is my first day on discrete mathematics , therefore not familiar to predicates , and other terminologies . So , would you provide the answer so that i can understand the explanation. – user3129893 Mar 3 '14 at 9:42
• In propositional logic you can model a statement like "the glass is empty" simply as $p$, because prop logic "can see" only the "structure" of a statement by way of the truth-functional connectives, like "not", "and", "or". So your statement is simply $p$, but in this way you of course loose the bits of information "inside it". The logical structure that prop logic "can see" in a "complex" statement like : "today it rains and my umbrella is broken" must be symbolized as : $p \land q$. – Mauro ALLEGRANZA Mar 3 '14 at 9:52
• Basically, the question is about philosophy as well as logic. Strictly speaking, it has nothing to do with propositional logic. – Mauro ALLEGRANZA Mar 3 '14 at 13:48

In modern logic, the predicate $True(\quad)$ is of "difficult usage", because in using it in a formal language we will encouter the problems connected with Self-Reference.

Regarding "standard" logical approach to the Liar, see :

4.2.1 Tarski's hierarchy of languages

Traditionally, the main avenue for resolving the paradox within classical logic is Tarski's hierarchy of languages and metalanguages. Tarski concluded from the paradox that no language could contain its own truth predicate (in his terminology, no language can be ‘semantically closed’).

Instead, Tarski proposed that the truth predicate for a language is to be found only in an expanded metalanguage. For instance, one starts with an interpreted language $L_0$ that contains no truth predicate. One then ‘steps up’ to an expanded language $L_1$, which contains a truth predicate, but one that only applies to sentences of $L_0$. With this restriction, it is easy enough to define a truth predicate which completely accurately states the truth values of every sentence in $L_0$ and yields no paradox. Of course, this process does not stop. If we want to describe truth in $L_1$, we need to step up to $L_2$ to get a truth predicate for $L_1$. And so on. The process goes on indefinitely. At each stage, a new classical interpreted language is produced, which expresses truth for languages below it.

Why is there no Liar paradox in this sort of hierarchy of languages? Because the restriction that no truth predicate can apply to sentences of its own language is enforced as a syntactic one. Any sentence $\varphi$ equivalent to $\lnot Tr(\ulcorner \varphi \urcorner)$ is not syntactically well-formed. There is no Liar paradox because there is no Liar sentence. See the entries on Tarski and Tarski's truth definitions for more on Tarski's views of truth.

I am currently going through a book on discrete mathematics (Rosen), and this was one of the questions in the first chapter. Of course, it was an even-numbered question, so I had to tap the internet to gain more information.

I think Mauro's answer is the most complete, and I'll try to add my thoughts as a pile-on to hopefully clarify what I'm thinking as well (and to potentially gain some feedback from the crowd).

Bottom line, the statement "This sentence is false," is NOT a proposition. The reason for this is because there is no "external reference" (for lack of more precise language) to which the statement makes reference. Or put another way, it is because there is no "variable" which can take the place of a 'word' in the sentence and give it meaning. For example, the statement "This sentence is false: 'this sentence is false'" IS a proposition, because there is a reference that can 'take the place of a variable (with a value of T or F)' that makes the entire statement T or F. In this case, the variable is the portion of the sentence following the “:” in single quotes. Take as another example the statement/proposition "P(x): x is a four-legged dog." In this example, there is some entity ‘X’ which assumes a role in the sentential function that either makes P(x) a true or false statement. P(Fido): Fido is a four-legged dog, is TRUE, whereas P(Telephone): Telephone is a four-legged dog, is FALSE, for obvious reasons. So here, P(x) is an example of a sentence with “space for a variable,” because it refers to something outside itself.

Now, returning to my earlier statement that “This sentence is false” is NOT a proposition. Let’s try to impose the ‘sentential function’ structure on it. For example, take P(x) to be “This sentence is false,” where ‘sentence’ is the variable (I’m getting a little sloppy here, but I think you can follow the point): Now, P(x) will become “This ‘This sentence is false’ is false.”…hmmm, it looks like the ‘sentence’ variable is still there (but wait, didn’t we just replace it?)…let’s do it again. P(x) after one more “iteration” becomes “This ‘This ‘This sentence is false’ is false’ is false.” See how it becomes “recursive” without a base case defined? A quick thought experiment continues this line of reasoning ad infinitum. Hence our problem, and why P(x) in this case is not a proposition; it makes no reference to anything outside itself.

So the actual “why,” I think, has to do with set and class theory – but I’m not as versed as I’d like to be. Briefly, statements that refer to themselves form what are known as proper classes (I think...), which are disallowed in mathematics and logic, because their entry effectively makes any statement true, which is “hardly a very desirable state of affairs,” according to Thomas Hungerford. The idea underlying all this is that we must place restrictions on what we say since it is possible in “ordinary” language to say things which have no meaning.

-Luke

It seems that this sentence ought to be a proposition, because it has the grammatical form of other meaningful sentences, but it is self referential, means little if anything, and is paradoxical. There are many other perfectly grammatical sentences which do not qualify as propositions in classical two-valued logic, because for one reason or other, they cannot be unambiguously classified as true or false.

A very informal analysis by a non-expert on such matters, so your instructor may very well disagree...

Consider the alternative, "This statement is true." No contradictions will arise from it, but it is clearly quite meaningless. It can have no truth-value. If a proposition is required to have meaning or truth-value, then this meaningless statement is not a proposition.

Now, if "This statement is true" is meaningless, then it stands to reason that "This statement is false" must also be meaningless. If it is meaningless with the word "true", it cannot suddenly have meaning when "true" is changed to "false". And if propositions must have meaning or truth-value, then it, too, must not be a proposition.

• If "This sentence is true" is true, there doesn't seem to be any contradiction, so there's not the same problem giving it a truth value as "this sentence is false". – Carl Mummert Mar 6 '14 at 4:22
• @CarlMummert My point is that, with both statements, there is the problem of meaninglessness or a lack of any truth-value. It is just easier to see with "This statement is true." – Dan Christensen Mar 7 '14 at 17:06
• @CarlMummert Intuitively, I would say that "This statement is true" is meaningless. It has no truth-value. – Dan Christensen Mar 7 '14 at 17:29
• Now that I think about it, there is not even a contradiction if the sentence is false (you said as much already, if I had re-read the answer). So I don't see any reason to think the sentence "This sentence is true" is meaningless. Is "This sentence has five words" also meaningless? – Carl Mummert Mar 7 '14 at 17:34
• @CarlMummert "This sentence has five words" is true and has meaning. "This sentence has only two words" has meaning, but it is false. The way I look at it, anyway. – Dan Christensen Mar 7 '14 at 17:39

Statements such as "this statement is false" and "Window is half open" are those which cannot have fixed truth values. Hence these are not propositions.
Reference:
http://mathworld.wolfram.com/Proposition.html

P.S. By not fixed truth values I mean its debatable. The glass is half full ;)

• why this (this statement is false) can't be proposition , even though it has truth value. suppose two person are talking each other one say a statement "your statement is false " and this may be false and may not be. therefore it could be proposition. Yes , window is half open is not proposition because this is not a declarative sentence and therefore doesn't has the truth values. – user3129893 Mar 3 '14 at 9:47
• @user3129893 - Why do you think that "Window is half open" is not a "correct" proposition ? In my office now my window is half open; so I can in a perfectly correct and intelligible way utter the statement "Window is half open" and this statement is true. – Mauro ALLEGRANZA Mar 3 '14 at 9:57
• ok , thanks for that. if i say that "X has umbrella over his head" then would it be proposition ?i think yes because it has truth value (true). and also don't forget to pass your view over the question("the statement is false"). – user3129893 Mar 3 '14 at 10:04
• @user3129893 : "This statement is false" refers to itself, and not like the case you gave of two person talking. Hence it is contradicting itself. Hence it is not a proposition. – Swapniel Mar 3 '14 at 10:36
• thanks for the comment but strictly speaking , i am still not getting the concept why its not a proposition. – user3129893 Mar 3 '14 at 11:02