1. Operands and operators. The “operands” of arithmetic are 1, 0, etc., and the “operators” are +, -, *, /, etc. Algebra, then, which builds on top of arithmetic, introduces the “operands” x, y, etc.
2. The operands as distinguished into constants and variables. For example: 1 and 0 are “constants,” and x and y are “variables.” In arithmetic and algebra, the constants are numbers (e.g., 1, 0). The variables, then, are like blanks to be filled in with those constants (which, again, are numbers in the present context). x + 1 = 2 is like _ + 1 = 2. What constant/number, if put into that blank, would make a true proposition? The answer is of course 1. Thus, x = 1. That is: _ = 1. But why aren’t blanks actually used? The reason is that x is like a _ that must be filled in with the same constant/number everywhere—well, everywhere in the circumscribed sphere of that x’s usage. For example, consider: 2x + 3x = 10. That’s like 2_ + 3_ = 10 with the constraint that both _ must be filled in with the same constant/number. By contrast: 2x + 3y = 10 is like 2_ + 3_ without that constraint.
3. Relations. The “relations” of arithmetic and algebra are =, >, <, etc. Without the relations, no proposition can be made (a proposition being anything that’s either true or false). For example: 1 + 1 isn’t a proposition, for it can neither be true nor false. But both 1 + 1 = 2 and 1 + 1 = 3 are propositions (with the former happening to be true and the latter happening to be false).
4. x^2 + 6 = 5x is such that x = 2 or 3. Consider, though, that x + y = y + x is such that x can be any number and so can y. Thus: For some number(s) x, x^2 + 6 = 5x. And for all numbers x and y, x + y = y + x.
5. A pair of propositions in English analogous to the foregoing: (1) “For some American people x, x’s parents are from America.” (2) “For all Japanese people x, x’s parents are from Japan.”
6. Another pair: (1) “For some species of birds x, the prototypical x can fly.” (2) “For all species of fish x, the prototypical x can swim.”
7. I’ll also need to define the vocabulary 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, along with the grammatical system inherent in putting, say, 1 (⚫︎) before 2 (⚫︎⚫︎), and getting 12 (⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎). That should come before bringing up the operands, the operators, and the relations, along with bringing up quantifiers etc.

In the logical language, the joint-attentional frame variant [establish], which is symbolized both linearly and diagrammatically as a square and is one of the four joint-attentional frame variants, will work as follows:

1. In “bicycle [establish] electric,” the joint-attentional frame is established on “bicycle” (whether referentially or categorically) and then “electric” is said about that frame. To translate that into English (if interpreted based on the extralinguistic context of the utterance as referentially definite and singular): “The bicycle is electric.”
2. In “electric [establish] bicycle,” the reverse is true. The joint-attentional frame is established on “electric (thing)” and then “bicycle” is said about that frame. To translate that into English in the foregoing way: “The electric (thing) is (a) bicycle.”
3. In “bicycle electric [establish]” or “electric bicycle [establish]”—those two phrases being logically identical to each other—the joint-attentional frame is established on what’s both a “bicycle” and “electric.”

To compare that to how English and Japanese work:

1. X口Y is like “X is Y” and XはY
2. Y口X is like “Y is X” and YはX
3. XY口 and YX口 are like “X Y is” and “Y X is,” XYは and YXは

For now, though, let’s analyze only X口Y and YX口. For example, in English and Japanese:

1. X口Y in English is like “X is Y,” e.g. “(the) cat is black”
2. X口Y in Japanese is like XはY, e.g. 日本人は時間を守る
3. YX口 in English is like “Y X is,” e.g. “(the) black cat is”
4. YX口 in Japanese is like YXは, e.g. 時間を守る日本人は

In both English and Japanese, then, the grammar uses the position of X and Y with respect to “is” or は (the reversal of the order of X and Y in the above examples being incidental for the present purpose) in order to differentiate between—to return to the camera analogy, although that analogy is, strictly or technically speaking, ideal only in the very limited scope of the speaker talking about something referential in the visual modality—”using a label in order to point the camera” and “labeling what the camera is pointing at.”

That is, both English and Japanese use word order as a grammatical tool for the purpose of differentiating between (1) establishing a joint-attentional frame and (2) saying something about that frame.

But how should the logical language handle the non-口 cases, i.e. the other three joint-attentional frame variants? And how do English and Japanese handle those cases?

Consider for example:

1. “John slapped Jane.”

It’s possible to use emphasis in order to distinguish as follows:

1. “John slapped Jane.” (That is, it wasn’t a different person who slapped Jane.)
2. “John slapped Jane.” (That is, it wasn’t a different action that John took with regard to Jane.)
3. “John slapped Jane.” (That is, it wasn’t a different person who got slapped by John.)

For background: Whether we’re analyzing English, Japanese, or any other language possible to evolve naturally among human beings, there’s always the distinction—again, fundamental to all natural human language—between (1) how joint attention is established on X and (2) what’s said about X.

Interestingly: When using one of the copular verbs in English (e.g., “is,” “are”), there are only two places (or slots) for the arguments: before the copular verb and after the copular verb. That is—to bring up, appositionally, the traditional terminology—there’s the copular verb along with its “subject” and its “predicate.” And those two places/slots correspond to (1) how joint attention is established on X and (2) what’s said about X.

Consider for example:

1. “That man is the owner.”
2. “The owner is that man.”

When not using any of the copular verbs in English, however, and instead using one of the non-copular verbs: Even when there are, like with any of the copular verbs, the two places/slots of [before] and [after] for the arguments, with no other places/slots, those two places/slots don’t correspond to (1) how joint attention is established on X and (2) what’s said about X. Case in point: If we reverse “that man is the owner” to “the owner is that man,” that’s a different semantic change that if we reverse “John slapped Jane” to “Jane slapped John.” The question becomes, then: The semantic change that we get from the swapping of [before] and [after] with copular verbs in English, how do we get that same semantic change, again in English, with non-copular verbs?

As shown at the beginning of the present essay, emphasis is one of the tools that can help: “John slapped Jane” is different than “John slapped Jane.” There’s also another tool that can help (in a perhaps more logically rigorous way): “It was John who slapped Jane” is different than “it was Jane who was slapped by John.”

For copular verbs, then: There’s only one distinction that I’m concerned with in the present essay: that between (1) the place/slot/argument that establishes joint attention on X and (2) the place/slot/argument that says something about X. But for non-copular verbs: There are two distinctions: not only that between the foregoing, but also that between, e.g., the agent and the patient for the verb “slap.”

What pattern may there be of semantic difference between basic nouns and basic verbs? At first glance, relative permanence seems related. Consider for example: “The duck flew away.” The duck was almost definitely already in existence before it flew away, and it was almost definitely still in existence after it flew away. Only in a story with magic could it be otherwise. That is (again, unless there’s magic, e.g. a wizard casting a spell that causes a duck to appear in existence only long enough to fly away before then disappearing from existence): For any given thing in space that the basic noun “duck” is a true label or description of, the basic verb “fly” is a true label or description of that thing at fewer moments in time. Temporally speaking, there’s a whole-part relationship: The label/description “duck” is always true of that thing, meaning that being a duck is something true of every moment in time of its existence, and the label/description “fly” is sometimes true.

However, that proposed distinction doesn’t work because not only is the relationship between nouns like “duck” and verbs like “fly” a temporal whole-part relationship, but the relationship between nouns like “duck” and adjectives like “small” is also like that. That is: A duck is still a duck whether it starts or stops flying, whether it starts or stops being small. The nominal quality of being a duck is relatively permanent, and both the verbal and adjectival qualities of flying and being small are relatively impermanent.

With all of that said, my proposed distinction is as follows (without using the terms “verb,” “noun,” and “adjective”):

1. There are the always-true labels (e.g., “duck”).
2. There are the sometimes-true labels that go back and forth between being true and false more freely (e.g., “fly”).
3. And there are the sometimes-true labels that go back and forth between being true and false less freely (e.g., “small”).