James R Maxlow
james@maxlow.net
www.maxlow.net


The Truth Paradoxes

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The truth or falsehood of any given statement is determined by assessing if it accurately represents an external object's state. For example:

I am happy.

To determine the truth of this statement, one must turn to the object it represents, namely "I" and determine if the state of that object matches the state declared in the statement. If "I" really am "happy" then the statement is true. Conversely, if "I" am not "happy" then the statement is false. So, to determine truth, one examines the objects state and then compares it with the statement's declaration about that object. A statement is true if it accurately represents the state of an object, and is false if it does not accurately represent the state of an object.

We run into complications, though, when a statement declares a representation of an internal state. For example:

This statement is true.

The object whose state the statement represents is itself... there is no external object state to look to in order to evaluate the truth. Assume for the moment, though, that the statement as an object is true. If so, then the statement accurately represents the object's state, and so the statement is true. Assume then that the statement as an object is false. If so, then the statement does not accurately represent the object's state, and so the statement is false. In both cases, making the assumption automatically validates the assumption, and no other conclusion is possible, which is of course a trivially irrelevant logical conclusion. This suggests that the statement has no inherent truth or falsehood, because there are no means by which to evaluate their truths... in other words, can a statement really be true if we can never determine that it is true?

The other way a statement may declare a representation of an internal state is by denying its state, as in the following example:

This statement is false.

Again, the object whose state the statement represents is itself. Assume that the statement as an object is true. If so, then the statement does not accurately represent the object's state, and so the statement is false. Assume then that the statement as an object is false. If so, then the statement accurately represents the object's state, and so the statement is true. In both cases, making the assumption automatically invalidates the assumption, which is logically inconsistent - a paradox, if you will - where the application of logical operations can never lead to a valid conclusion. This also suggests that the statement has no inherent truth or falsehood, because any claim we make about its truth is incorrect... in other words, can we determine the truth of a statement if our determination is always wrong?

These complications can even exist when chaining statements together. Consider the non-complicated case:

The statement below is true.
The ball is red.

To determine the truth of the first statement, we must compare the declared state to the actual state of the object the statement represents. In this case, that object is the second statement, and the declared state is truth. So, we must determine if the second statement is true in order to determine if the first statement is true... thus, the chain. But the second statement is simple enough, as it refers to an object - the ball - and declares that the ball has a state of being red. Assume the ball is red. If so, then the second statement accurately represents the state of its object, and hence is true. Now we compare and find that yes, the first statement accurately represented the state of its object; therefore the first statement is also true. Assuming the ball is not red, then the second statement does not accurately represent the state of its object, and hence is false. Upon comparison, we see that the first statement does not accurately represent the state of its object; therefore the first statement is false. There are no complications in this chain, because both statements declare a representation of an external object.

The complications arise, as before, when statements declare representations of internal states. Consider this chain:

The statement below is true.
This statement is true.

Again we must determine the truth of the second statement to determine the truth of the first. However, we have shown earlier that the truth of a statement such as the second cannot be determined. It follows that the truth of the first, which depends on the second, can also not be determined. So, even though the first statement declared a representation of an external object, it was part of a chain with a statement that declared an internal state. In essence, one truthfully indeterminate link makes all other links in the chain truthfully indeterminate (for this reasoning could be extended to an indefinite number of linked statements.) Note that even changing the second statement to "This statement is false" provides no relief, as it was shown to have a truth that we always determined incorrectly.

The last type of cases considered is circular chains of statements:

The statement below is true.
The statement above is true.

To determine the truth of the first statement, whose object is the second, we must determine the truth of the second. But to determine the truth of the second statement, whose object is the first, we must determine the truth of the first. Here we see the circular linkage. Let us assume, though, that we know the second statement is true. If so, then the first statement accurately represents the state of its object, therefore the first is true. If we then wish to verify our assumption about the second statement, we can see that the second statement accurately represents the state of its object (that we just determined was true) and therefore the second is also true. As earlier, making our assumption validated our assumption, but nothing more. If we assume the second statement is false, then the first does not accurately represent the state of its object, and hence the first is false. In verifying our assumption about the second statement, we can see that the second statement does not accurately represent the state of its object (that we just determined was false) and there the second is also false. Again, our assumption validated our assumption, which logically irrelevant. In these cases, each statement declares a representation of an external object, but the circular linkage prevents us from actually determining anything about the truth of either.

Consider, then:

The statement below is false.
The statement above is false.

Assume the second is true. If so, the first does not accurately represent the state of its object, and hence the first is false. If we wish to verify our assumption about the second, we can see that it accurately represents the state of its object (that we just determined was false) and therefore our assumption is validated. Assume then that the second is false. If so, the first accurately represents the state of its object, and hence the first is true. If we wish to verify our assumption about the second, we can see that it does not accurately represent the state of its object (that we just determined was true) and hence our assumption is validated. This is a duplicate case of the one directly above, despite the change in declared states. Circular linkage prevents us from determining the truth of any link.

The last case to consider is a circularly linked chain with differing claimed states:

The statement below is true.
The statement above is false.

Assume the second is true. If so, the first accurately represents the state of the second, and hence the first is true. To verify our assumption about the second, we see that it does not accurately represent the state of its object (that we just determined was true) and hence the second is false. This invalidates our assumption. We then assume that the second must be false. If so, the first does not accurately represent the state of the second, and so the first is false. To verify our assumption about the second, we see that it accurately represents the state of its object (that we just determined was false) and hence the second is true. Our assumption is again invalidated. It is apparent in this case that any determination we make about the truthfulness of either statement is automatically incorrect.

The above cases and their terminal conditions are summarized as follows:

  • The truth of single statements referring to external object states can be determined correctly.
  • The truth of single statements referring to internal states cannot be determined (or are always determined incorrectly, if negation is involved.)
  • The truth of linearly linked statements, where each statement refers to an external object state, can be determined correctly.
  • The truth of linearly linked statements, where any statement refers to an internal state cannot be determined (or are always determined incorrectly, if negation is involved.)
  • The truth of circularly linked statements with identical declared states cannot be determined.
  • The truths of circularly linked statements with oppositely declared states are always determined incorrectly.

More succinctly:

  • Any single statement or every statement of a linear chain of statements has indeterminate truth if any reference to an internal state is made, and has determinate truth otherwise.
  • Every statement of a circular chain of statements has indeterminate truth.

It follows, then, that any statement has one of three truth values: it can be true, it can be false, or it can be indeterminate. For the sake of philosophy, let us offer choices for what we mean by indeterminate: it could mean non-existent, it could mean defined but forever unknowable or it could mean true and false simultaneously.


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