TinyProf's Grammar Lesson: Can "Using" Be an Unattached Participle? 🎓

---

1. What We're Solving

You're asking whether "using" can legitimately function as an unattached (dangling) participle in mathematical writing — the way words like considering, assuming, and granted do — without being grammatically incorrect.

This sits right at the intersection of grammar rules and real-world language conventions.

---

2. The Approach

To answer this, we need to understand three things in sequence:

• 1. What a dangling participle normally is (and why it's usually wrong)
• 2. What makes certain participles exceptions — called absolute or sentential participles
• 3. Whether "using" qualifies for that same exception

Grammar isn't purely rule-based — usage, convention, and community acceptance all play a role.

---

3. Step-by-Step Explanation

🔹 Step 1: Understand the Basic Rule About Participles

Normally, a participial phrase must attach to a clear subject in the sentence. The subject performing the participle's action should be the grammatical subject of the main clause.

> ❌ Running down the street, the rain soaked me.

This is a classic dangling participle — the rain wasn't running. It's wrong because "running" has no logical subject in the sentence.

---

🔹 Step 2: Recognize That Some Participles Are Exceptions

English has a category of participles that grammarians call:

• Absolute participles, or
• Sentential (discourse-level) participles

These participles don't attach to the subject of the main clause. Instead, they comment on the entire statement — they modify the act of saying something, not a noun in the sentence.

Classic accepted examples include:

| Word | Example | |------|---------| | Considering | Considering the evidence, the verdict seems fair. | | Assuming | Assuming X > 0, we can divide both sides. | | Speaking | Speaking of which, did you hear the news? | | Granted | Granted, this is a simplification. | | Given | Given the constraints, this is optimal. |

✅ These are widely accepted even by strict grammarians because they've become idiomatic discourse markers — they signal how the reader should interpret the following statement.

---

🔹 Step 3: Ask — Does "Using" Behave the Same Way?

Now apply this test to your mathematical examples:

> "Then, using this argument, if U is a set of 5 vertices, then U contains a clique."

Ask yourself: Who is "using" the argument?

• It's not U (the grammatical subject)
• It's not the clique
• It's implicitly the mathematician/author — or more abstractly, the proof itself

This means "using" is functioning sententially — it tells the reader "with this tool in hand, observe that..."

Compare to assuming:

> "Assuming inequality (3), we get A > B > C."

The logical structure is almost identical. Both words introduce an intellectual instrument or premise that frames what follows.

---

🔹 Step 4: Check What Authorities and Convention Say

Here's the nuanced answer:

Strictly speaking, "using" is more vulnerable to the dangling participle criticism than "assuming" or "given" — because it hasn't been as fully grammaticalized into an idiom.

However, in mathematical and academic writing, "using" has developed a recognized discourse function:

• It signals: "by means of this tool/result, the following holds"
• It's understood by the entire community of readers
• Reputable mathematical journals and textbooks use it routinely

Linguists would say it has achieved pragmatic legitimacy within that discourse community — even if a traditional grammar textbook might flag it.

---

🔹 Step 5: Compare Your Two Examples

| Example | Assessment | |---------|------------| | "Then, using this argument, if U is a set..." | Slightly awkward — "using" has no clear agent. Could be improved. | | "Note that A>B>C (using inequality (3))." | More natural — parenthetical use feels like a citation/justification marker. |

The parenthetical form (your second example) is the stronger usage because it functions almost like a footnote or reference tag — very natural in math writing.

---

4. The Answer

✅ Yes, "using" CAN function as an unattached participle in mathematical writing, but with important nuance:

• It is not yet as fully accepted in general English as considering, assuming, or given
• In mathematical discourse specifically, it has become a conventionalized discourse marker and is widely understood and accepted
• The parenthetical form — Note that X (using lemma 3) — is especially natural
• For formal writing outside mathematics, you might prefer: "By using this argument..." or "Using this argument, we see that..." (with a clear we as the agent)

---

5. Memory Tip 💡

> "Dangling is bad; idiomatic is fine!"

Ask yourself: Has this participle become a community-accepted signal word in this field?

If mathematicians everywhere write "using lemma 3" and every reader understands it instantly — that's the language doing its job. Grammar serves communication, not the other way around! 😊