Two Birds, One Stone

In the course of a conversation, I mentioned denial entails belief. Surprisingly, the conversation stopped so that I could justify saying so.

I started off far too technically. I’ll do the opposite here because giving the gist up front means you’ve dropped off from reading either because interest is lost, it has at that point been justified enough, or the level of technicality has no more benefit.

Anyway, commonly, in many fields of study — Epistemology and Psychology, for example — belief is defined as an attitudinal disposition toward a state of affairs. So, denial would in itself be a disposition outright, while for instance, withholding judgment is not genuinely a disposition at all. But let’s go a little farther and kill a second bird, some 2,300+ years old in the process.

The nature of “denial entails belief” is axiomatic and therefore absolutely true because of all the meanings of the words. To show some axiom is wrong, it takes a counter-example. No amount of word-smithing will.

Such an example will meet the criteria of being 1) an assertion that 2) a person doesn’t withhold judgment on and 3) denies is true while simultaneously 4) not believing it is untrue, and where 5) all criteria are met.

I can indeed give such an example, but it only further illustrates “denial entails belief”, and I’d much rather see what examples may unfold.

An example that I have been given was the “Liar Paradox” in the version of “This statement is false.” The problem is, it’s not an assertion nor a proposition. In explaining this example, I can illustrate a common view of assertions, propositions, and exactly why denial entails to at least the belief the assertion denied is untrue. In this view, the age old paradox is solved as well.

To short-hand things, I’ll define the logical forms of propositions and assertions.

Propositions have the form:

[the proposed] + [true or false, “verity”]

Where “the proposed” must at least in theory have a case, be something than can be true or false. In my makeshift formal syntax, a proposition (P from here on) is:

[x] [verity claim]

Assertions are easy as they are merely a form of speech act, easily discerned because spoken or implied, they are always an affirmation of a “verity claim”, and always in the form of simply:

[it is true that]

Assertions from here on are shortened to A, but for the sake of clarity, I will redundantly use “that” in order to say that a valid assertion takes the form:

A that P

Using the full formula as-is, I’ll then use our example “This sentence is false” in diagram. Denial is just ~A; meaning that in the same way as A, D is [it is untrue that].

(Positive forms)

A = [it is true that] ( [x] [is true] )
D = [it is untrue that] ( [x] [is true] )

(Negative forms)

A = [it is true that] ( [x] [is false] )
D = [it is untrue that] ( [x] [is false] )

Here, it’s easy to see both A and D entail to at least one belief about any P; which is an affirming or negating disposition toward the verity of P. It has to be noted that A and D are only concerned with the verity claim in P and the only role of “the proposed” is to lead to some A, some D, or withholding judgment.

Our real example follows this diagramming:

P = [This sentence] [is false]

We have to note, as A.J. Ayer does in “Language, Truth, and Logic”, propositions are not English but bear a striking similarity to common language. One is that if I’m talking to you and tell you, “Take out the trash!”, it’s understood I mean “You, take out the trash”. The same is true of propositions. Unless stated otherwise, all propositions are assumed to be positive verity claims; asserting “The sky is blue” is the same as asserting “It is true, the sky is blue” (formally: [It is true that] [the sky is blue] [is true]).

It happens then that we only ever specify the verity claims of negative propositions; which is exactly what “This statement is false” is.

Immediately, you can see the solution. “The proposed” is not capable of being either true nor false. It simply states: “This statement”. This is why the Liar Paradox doesn’t stand as an example that counters the axiomatic, “denial entails belief”, and why it’s not a paradox but a statement like any other, such as “The red feather” or “Morning”.

Not to leave anything out, let’s suppose “the proposed” is actually the whole sentence and, like common English, we ought to assume that [it is true] is the verity claim. Besides begging the question “Why?”, it only shows the rule of paradox for self referential propositions is that “the proposed” must match the verity claim so that it always takes a positive propositional form. Otherwise, the proposition is incoherent and unanalyzable. For instance:

[This statement is false] [is true]

Is incoherent, but these are not:

[This statement is true] [is true]
[This statement is false] [is false]

At any rate, given the theories of belief, propositions, and assertions used here, it would take other theories, it seems to me, in order to suggest denial doesn’t entail belief or to adequately criticize this solution to a very, very old seemingly paradoxical proposition.

Just a thought.

Comments not only welcome but encouraged.

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