Is this a valid solution to the liar paradox?

Question

Example:
Is the statement "These letters are black." true or false?

If the letters are black, then the statement is true.
If the letters are not black, then the statement is false.

In this example we have a subject (These letters) and a proposed description of that subject (black). In the same way, we have the Liar Paradox:

Is the statement "This sentence is false." true or false?

If the sentence is true, then the statement is false.
If the sentence is false, then the statement is true.

Here we have a subject (This sentence) and a proposed description of that subject (false).

The questions is, how do we determine if the description is correct?

To start, we have to realize that in this context the statement itself cannot tell us that the sentence is false as it might appear to. This is because when asking "Is the statement 'This sentence is false.' true or false?" we are implying that the description of the subject may be either true or false. This means that the description (false) of the subject (This sentence) is solely a proposed description as stated above.

Therefore, the word "false" is not defining the truth value of the sentence as it might first appear, but only proposing a truth value for it. It is our job to determine whether this proposed truth value is correct or incorrect.

To do that, we need to understand what the proposed description (false) is actually proposing. What does it mean to describe something as "false"?

True and false are names for relationships between a subject and some proposed description of that subject. If the proposed description matches the subject, then the relationship is named true. If the proposed description does not match the subject, then the relationship is named false.

If true and false are only names given to relationships between a subject and a proposed description of that subject, then you need both a subject and a proposed description of that subject before you can determine which relationship exists.

In the statement 'This sentence is false,' we have a subject (This sentence) and a proposed relationship that subject has with some proposed description of it (false). The proposed description of it is not stated, therefore it is impossible to determine whether or not the proposed relationship is correct or incorrect.

This means that the statement "This sentence is false" cannot hold a truth value because there is nothing to attach one to; in the same way that you cannot attach a truth value to the statement "Water is not made of things." until you know what the term "things" is referring to.

To put it another way, "This sentence is false" refers to a proposed relationship between "This sentence" and conditions for truth and falsity. We are not shown which of these conditions exist, so the truth value of the proposed relationship cannot be determined.

To be clear, I did not contradict my terminology when stating that "false" is both a proposed description, and a proposed relationship between the subject and a proposed description of that subject. I am considering the word "description" as a variable for any proposition made about the subject.

Feedback and critiques on my reasoning would be greatly appreciated. I apologize for any elementary mistakes and unconventional terminology that may have been used. I have never taken a course in logic so I am relying almost entirely on intuition. I am also unable to read symbolic logic easily, so I would appreciate it if replies could avoid using it.

Answer 1

See Tarski's Truth Definitions: in a nutshell, T's definition amounts to adopting the intuitive principle concerning truth that, for any sentence p, we have the so-called T-biconditional:

‘p’ is true iff p

where ‘p’ is a "name" for p.

In your language, we have a "subject" p (i.e. a "piece of reality") and a "proposed description" (i.e. a linguistic entity) of it: ‘p’.

Tarski's theory "formalize" the intuitive understanding of truth; as you said:

True and false are names for relationships between a subject and some proposed description of that subject. If the proposed description matches the subject, then the relationship is named true.

As soon as we apply this theory to language itself, we are lead to the liar paradox if the language (has natural language does) has enough resources to talk about its own semantics.

Various solutions has been formulated:

• the distinction between object language and metalanguage

• the theory of type

• truth value gaps.

Answer 2

This is a topic that great mathematicians like Tarski have beat on quite a bit (I highly recommend looking up his definition of truth: "P" is true if and only if P). Part of the issue is that there is an intuitive desire to stat that sentences can have a truth value associated to them, because we do so intuitively all the time.

Your particular approach runs afowl of classical logic, because you have a thing whose truth value is not true nor false. The logic you recommend looks much like intuitionalist logic, which does not assume the law of the excluded middle. This permits a statement's truth value to be not true and also not false. It is a valid form of logic, just not the most popular one today. There are a lot of proofs that are much harder when you don't have the law of the excluded middle, so intuitionalist logic doesn't prove as many things.

Answer 3

A good attempt!

Heres how I do it:

1) this sentence is false

What sentence?

2) this sentence = "this sentence is false"

Sorry but line 2 IS FALSE!

PROOF:

By Leibniz Law we get from line 2

3) this sentence is true IFF "this sentence is false" is true

And the definition of truth allows simplification

4) this sentence is true IFF this sentence is false

The contradiction proves line 2 to be false (QED)

The sentence claimed by line 1 to be false can therefore NOT be the sentence in line 1 :)