I tried to define a proposition and rediscovered programming
This post explores the concept of a proposition. Below is a definition coined by an LLM in a discussion I am engaged in regarding expressing the essence of atomic claims. It is treated as provisional. The following exercise is an investigation into assumptions of shared understanding between operator and agent.
Let's start with our working definition:
A proposition is the meaning expressed by a statement, interpreted in its context, that can be evaluated for truth or falsity. Quotations and paraphrases are representations of that meaning.
We are going to evaluate this definition through applying it to a scenario.
Here is an example claim taken from Sean Goedecke's article Human-AI partnerships are for alignment, not capability:
Lots of people think that we’re currently in a world of software engineering centaurs.
My informal reading of the proposition in this statement would be as:
we're currently in a world of software engineering centaurs.
i.e., that which follows "that".
"Lots of people think", itself interpretable as a proposition, would be the qualifier, or per the working definition above, the "context" that the proposition appears in, on a local sentence-specific level. The greater argument would be the "meta-context". In between that could be any number of layers of context abstractions.
In this case, the "meta-context" is easy to fetch:
Human-AI partnerships are for alignment, not capability.
This statement is both the URL and the title, and the essay is structured around it.
The context within which our claim is being made is a claim itself.
Moving on, to understand the nature of our claim regarding engineering centaurs, we need to define engineering centaurs, according to the intention of the source's author. To paraphrase from the article:
[Centaurs are] human-AI partnerships that are stronger than unassisted AIs or humans
To which we could add "according to some objective measurement."
So, at this point we have:
- a working definition of what a proposition is
- an example claim that can be associated with a proposition
- a proposition bound to the example claim
- a definition of dependent terms (i.e., centaur) necessary to understand to make the claim atomic
- an understanding of the context the claim appears within
This raises a question - does the proposition exist within the context held by its corresponding claim? According to the working definition, yes. But I propose that there is an uncertainty which has been implicitly smuggled into the definition, hinging on the following:
If a proposition is a verbatim fragment, i.e., if 'proposition' defines the part of the sentence following "that" which reads "we're currently in a world of software engineering centaurs", then a proposition is a semantic marker, and simply points to a portion of a claim. In this case, yes, the proposition is to be seen in the context of the claim.
If a proposition is a concept that exists separately from the portion of the claim itself, then no, the proposition is NOT to be seen in the context of the claim. It is to be seen as something that exists wholly outside of the claim, and that the source claim points to.
So, either the idea of a proposition is a pointer to a type of sentence fragment, and nothing more (type 1), or a proposition is a thing that can be pointed to by the portion of the claim associated with the proposition, and nothing more (type 2).
I am going to make an unscientific decision that, for the purposes of this post, a proposition is a thing that can be pointed to by a portion of the claim associated with the proposition (type 2).
Now let's revisit our original LLM-coined definition:
A proposition is the meaning expressed by a statement, interpreted in its context, that can be
evaluated for truth or falsity. Quotations and paraphrases are
representations of that meaning.
This seems to simultaneously support and undermine either type.
"The meaning expressed by a statement" separates clearly the meaning from the statement. So does "quotations and paraphrases are representations of that meaning". This is unambiguous.
"interpreted in its context" is potentially in conflict with our working requirement that a type 2 proposition is NOT to be seen in the context of the claim.
Something needs to give - either our working definition of a proposition needs to be modified, or the claims of the rule above need to be modified. This needs to be reconciled.
I am going to make an argument against our working definition, based on language within the definition itself. If 'meaning' is defined as a thing outside of the 'statement', then it is individuated conceptually while being dependently associated. If this is the case, it must have its own ability to have independent relationships. If this is the case, other claim statements should be able to point to it. If this is the case, it must exist independent of the context of all dependent claim statements, unless the contexts are discretely attached.
Coming back to our example:
The full claim statement
Lots of people think that we’re currently in a world of software engineering centaurs.
The context (another proposition)
Lots of people think
The claim statement
we’re currently in a world of software engineering centaurs.
The proposition
we’re currently in a world of software engineering centaurs.
At this point, the proposition, although separate from claim statement 1, is reflected verbatim, as there is no rule that would say otherwise.
Now imagine a new claim statement:
software engineering centaurs outperform AI engineers and human engineers according to the Centaur benchmark
This new claim statement can point to the same proposition, which is now a shared object, IF a judgement phase has allowed for either a human, agent, or centaur to determine that the semantic meaning of some portion of each phrase overlaps sufficiently to imply shared ownership of the proposition. Let's assume that process has occurred, and shared ownership has been agreed upon.
At this point, the proposition can either:
- retain its original wording
- be rewritten to better reflect the shared ownership
Option 1 would use sequence as a first class citizen in determining the ability to define. This is arbitrary.
Option 2 would better reflect the shared ownership model, but require a process of rewriting and then judging the change. Let's assume that, like the judgement phase above, this would be made explicit.
I favor option 2, so am going to take the liberty of choosing it for our system.
Here is, thanks to me, but could be thanks to agents or centaurs as well, our new modified proposition:
Software centaurs currently exist in the world.
In finding the claim overlap between the statements, we do not find room to express the claim statement 2 makes, that centaurs outperform AI and human engineers. The best we can express is that software centaurs exist at a point in time, which is not that far from a definition.
The claim that centaurs outperform AI and human engineers needs to be a new additional proposition, that does NOT redundantly include the previous proposition, that centaurs exist:
software engineering centaurs outperform AI engineers and human engineers according to the Centaur benchmark
The fact that this proposition is a verbatim copy of the claim is not a problem; what would be a problem would be to discretely state what is already expressed by a linked proposition, i.e., that centaurs exist.
So it appears that propositions arise emergently as they are extracted from statements, as new connections between statements and the propositions they link to make clear the need for the semantic evolution of propositions. Propositions appear to be living approximations across sources.
We now have a chance to revisit our working definition of a proposition, which could now be arguably expressed as:
The meaning expressed by a statement, understandable separately from the statement's context, that can be
evaluated for truth or falsity. Quotations and paraphrases are
representations of that meaning.
"Interpreted in its context" has been changed to "understandable separately from the statement's context."
The last remaining issue is "currently." This is defined by the context, i.e., when the claim was made. This is resolvable through substitution:
Software centaurs exist in the world as of 9/27/2026.
Centaurs outperform AI and human engineers as of 9/27/2026 according to the Centaur benchmark.
Damn. We have resolved a contextual dependency, in that "currently" can be translated into the date the claim was made to achieve the modular independence that our definition of proposition requires, but at the expense of the general applicability of the proposition. The value of a proposition is to express a core meaning along with tracing how that meaning might relate to other sources. Hardcoding the date into the proposition limits the time the proposition concerns, so we would need a date for every possible day the proposition might occur on. The date is really just a variable, and you need the ability to pass a date to it. We want to be able to connect a family of related dated claims while retaining the distinctions between their individual instances.
Here is a notation that could enable this:
Software centaurs <currently> exist in the world.
Where "<currently>", or any text surrounded by "<>"s, is a variable. This restores the general purpose functionality of the proposition statement irrespective of the date contained in the context.
It seems that rediscovering programming allows us to create dynamic self-contained propositions.
It also seems that propositions, according to this model, can exist in two states:
- Parameterized propositions: propositions with unresolved contextual dependencies represented by variables
- Bound propositions: resolved propositions whose contextual dependencies have been resolved and are suitable for evaluation for truth/falsity.
This terminology allows us to differentiate between depending on the original statement’s context and explicitly representing the information that context supplies.
The discussion of temporal qualification in the Stanford Encyclopedia of Philosophy's entry is relevant to this choice; it treats the relationship between statements and temporally qualified propositions as a substantive issue. SEP, §3.2
Extending this idea of parameterized and bound propositions into earlier claims:
Lots of people think
This statement was declared as context above and noted as itself being a proposition. But it is currently a "dangling proposition" if we might coin (another) phrase, its content is unspecified. However, as a parameterized proposition, it can be:
Lots of people think that <p>
For further consideration, as it turns out, resolving contextual dependencies while preserving distinctions between propositions may improve retrieval for questions that depend on those distinctions.