When two people are talking to each other, each utterance is such that there’s a speaker and a listener. Furthermore, there’s everybody who’s neither the speaker nor the listener.

When a first-person pronoun is used (e.g., “I,” “me”), the speaker is referring to themselves. But it’s also possible for the speaker to refer to something close to themselves (e.g., “this,” “these”) or somewhere close to themselves (e.g., “here”).

In deixis, there’s:

1. The speaker
2. The listener
3. Neither the speaker nor the listener
4. The location in space of the speaker, the listener, or neither the speaker nor the listener
5. The location in time of the utterance

Thus, it’s possible to refer to:

1. The speaker of the utterance
2. The listener
3. Neither the speaker nor the listener
4. Something near the speaker of the utterance
5. Something near the listener
6. Something near neither the speaker nor the listener
7. Somewhere near the speaker of the utterance
8. Somewhere near the listener
9. Somewhere near neither the speaker nor the listener
10. The past with respect to the utterance
11. The present with respect to the utterance
12. The future with respect to the utterance

There’s also:

1. Male vs. female
2. Singular vs. plural

My interest in using notation in logic (e.g., ~ for “not,” strictly defined) is in part a result of often finding it useful to keep track of how I would notate the logical skeleton of what’s fleshed out as natural prose. For example, in natural prose the word “not” doesn’t always mean ~. In natural prose, there’s no 1-to-1 correspondence between the linguistic form and the logical substance. With the symbol ~ strictly defined, you can, whenever doing so would be useful, ask yourself: “Is the present usage of the word ‘not’ equivalent to ~?”

When I’m reading or writing, my internal experience is often such that I visualize ~ and other symbols as furigana. That is, I often ask myself whether I could justifiably put a certain logical symbol above a given word or phrase.

When writing, that technique helps you get the best of both worlds of the artificial and the natural: the artificiality of scientific writing and the naturalness of artistic writing. It helps you keep track of the logic without there being any need for you to artificially limit or regiment how you use natural language.

1. There are the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. 1 is ⚫︎, 2 is ⚫︎⚫︎, etc., and 0 is nothing. When put together: For the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, xy = x * (9 + 1) + y. For example: 23 = 2 * (9 + 1) + 3. 45 = 4 * (9 + 1) + 5. And 67 = 6 * (9 + 1) + 7.
2. Importantly, the xy in that algebraic equation isn’t another way of writing x * y. In the present context, xy means writing the symbol for the number x in front of the symbol for the number y, with the possible numbers being from ⚫︎ to ⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎, e.g. writing the symbol 2 in front of the symbol 3.
3. Put differently: For the digits (the term “digit” meaning a certain kind of symbol) 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, the digit x written in front of the digit y = the number associated with the digit x * ⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎⚫︎ + the number associated with the digit y.

1. In arithmetical notation, there are the seemingly fundamental symbols +, -, *, and /. Interestingly, though: The symbols + and – are ambiguous between (1) +1 vs. -1 read as “positive one” vs. “negative one” and (2) 1 + 1 vs. 1 – 1 read as “one plus one” vs. “one minus one.” That is: + is ambiguous between “positive” and “plus,” and – is ambiguous between “negative” and “minus.”
2. Semantically speaking, the difference between those two conceptual pairs is that a number being positive or negative is a static state, and a number being added (with a + read “plus”) or subtracted (with a – read “minus”) is a dynamic state. For example, let’s say that you’re looking at your bank account. If you have a positive bank balance (e.g., +50 thousand dollars, with the + usually being left off), then the bank owes you $50,000. And if you have a negative bank balance (e.g., -50 thousand dollars), then you owe the bank that amount of money. That’s about the static state of your bank account. But if you add or subtract money from your bank account—i.e., if you make a deposit or withdrawal—then you change the bank balance in a positive or negative direction (whether in doing so you go far enough as to change whether you’re “in the black” or “in the red”). That’s about the dynamic state of your bank account.
3. Thus: In the logical language, there will be: (1) a symbol for the static state of being a positive number, (2) a symbol for the static state of being a negative number, (3) a symbol for the dynamic state of a number moving, or being made to move, in a positive direction, and (4) a symbol for the dynamic state of a number moving, or being made to move, in a negative direction.
4. What about the symbols * and /, though? Interestingly, there’s no analogous ambiguity with those symbols. * is straightforwardly just multiplication, and / is straightforwardly just division.
5. Addition and subtraction are counterpart operations in arithmetic in that what one does, the other undoes—the term “operation” here implying a dynamic state. Multiplication and division too are counterpart operations in the same sense. For example: Take 3 + 4 = 7. The start is 3, the operation is + 4, and the end is 7. Next, take that end as the start and undo what was done: 7 – 4 = 3. Analogously, consider the “doing” of 3 * 4 = 12 and the counterpart “undoing” of 12 / 4 = 3.
6. To generalize: (1) For all numbers x, y, and z, x + y = z implies z – y = x. And (2) for all numbers x, y, and z, x * y = z implies z / y = x.
7. It’s inelegant that the traditional notation is such that ab is the same as a * b but 22 isn’t the same as 2 * 2.

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.”

A deictic utterance is such that the question of what the referent(s) are of the utterance isn’t possible to answer without knowing:

1. Who said the words to whom
2. And/or where the speaker was, and/or where the listener was
3. And/or when the words were said

For example: “I ate here yesterday.” Without knowing who said the words, there’s no way to know who “I” is. Without knowing where the words were said, there’s no way to know where “here” is. And without knowing when the words were said, there’s no way to know when “yesterday” was.

Put differently, there’s:

1. The speaker of the utterance
2. The listener of the utterance
3. The location in space of the speaker
4. The location in space of the listener
5. The location in time of the speaker and the listener

If we can refer to the speaker (i.e., the so-called “1st person”), and we can refer to the listener (i.e., the so-called “2nd person”), then we can also refer to he who’s neither the speaker nor the listener (i.e., the so-called “3rd person”), for that’s just a negation of the foregoing. And if we can refer to the location in space of the speaker, then we can refer to the thing that’s near that location. Same for the 2nd and 3rd person.

Thus, there’s:

1. The speaker (e.g., “I,” “me”)
2. The listener (e.g., “you”)
3. Neither the speaker nor the listener (e.g., “he,” “she”)
4. Something near the speaker (e.g., これ)
5. Something near the listener (e.g., それ)
6. Something near neither the speaker nor the listener (e.g., あれ)
7. Where the speaker is (e.g., ここ)
8. Where the listener is (e.g., そこ)
9. Somewhere away from both the speaker and listener (e.g., あそこ)
10. The past with respect to the utterance
11. The present with respect to the utterance
12. The future with respect to the utterance

The first conjugational distinction:

1. Something permanent about the whole thing (e.g., 彼はサッカー選手です)
2. Something permanent about part of the whole thing
3. Something temporary about the whole thing (e.g., 彼はサッカー選手として活躍しています)
4. Something temporary about part of the whole thing

Put differently: There’s the temporal aspect, which can either be “whole-whole” or “whole-part,” and there’s also the spatial aspect, which can also either be “whole-whole” or “whole-part.” To be temporally whole-whole is to be permanent in the delimited context of the utterance, and to be temporally whole-part is to be temporary in the delimited context of the utterance. For example: (a) “He’s the goalkeeper” (his whole temporal existence being as the goalkeeper in the delimited context of the utterance, i.e. the present game of soccer). (b) “He’s reading a book” (only part of his whole temporal existence being reading the book, in that it’s natural to imagine, say, asking him a question, which would make him take a break from reading). Furthermore, to be spatially whole-whole is for the whole of the spatial existence of the subject to be such that the predicate describes it, and to be spatially whole-part is for only part of the whole of the spatial existence of the subject to be such that the predicate describes it. For example: (a) “That’s a house” (every spatial point making up the referent of the subject being a house). (b) “That house has a chimney” (only some of the spatial points making up the referent of the subject being the house’s chimney).

For clarity, and to recap, the first conjugational system again, put appositionally:

1. Temporal whole-whole (i.e., permanent), spatial whole-whole
2. Temporal whole-whole (i.e., permanent), spatial whole-part (e.g., 彼は鼻が高いです)
3. Temporal whole-part (i.e., temporary), spatial whole-whole
4. Temporal whole-part (i.e., temporary, spatial whole-part

The second conjugational distinction:

1. Past
2. Present
3. Future

The third (and final) conjugational distinction:

1. Static
2. Dynamic
3. Static, agent-oriented
4. Dynamic, agent-oriented
5. Static, patient-oriented
6. Dynamic, patient-oriented

For a verb to be static is for the verb to describe an unchanging state, and for a verb to be dynamic is for the verb to describe a change of state. For example: (a) “The mouse is dead.” (b) “The mouse died.” Furthermore: For a verb to be agent-oriented is for the verb to describe what the agent did (which had an effect on the patient). And for a verb to be patient-oriented is for the verb to describe what the effect on the patient was. For example: (c) “The man shot the woman” (shooting being what the agent did, with the effect left unspecified). (d) “The man killed the woman” (being killed being what happened to the patient, with the cause left unspecified).

Interestingly, it’s possible in English to put the agent-oriented and the patient-oriented together. For example: “I sang her to sleep.” Here the agent sang to the patient, which had an effect; the effect was sleep. Analogously: “He shot her dead.”

I should also be clear that whether agent-oriented or patient-oriented, what’s static or dynamic is the patient’s state. Either the patient’s state is kept the same, if static, or made to change, if dynamic. For example: In “I sang her to sleep,” the verb “sang” is dynamic and agent-oriented, for singing is what the agent did that changed the state of the patient, and the verb “sleep” is dynamic and patient-oriented, for sleeping is what the state was that the patient changed to.

To recap:

1. Static (e.g., これは赤い)
2. Dynamic (e.g., これは赤くなった)
3. Static, agent-oriented
4. Dynamic, agent-oriented (e.g., “the man shot the woman”)
5. Static, patient-oriented
6. Dynamic, patient-oriented (e.g., “the man killed the woman”)