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notation in semantics book

What do the semicolon and period mean, please?

For context, this is from Truswell's 'Events, Phrases and Questions' and this section in particular is talking about the causal ordering of events

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    I'm voting to close this question as off-topic because tis is a question about Mathematics, try Mathematics Mar 8 '18 at 10:57
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    It's a question in my semantics textbook, not mathematics... ?
    – Eloisa
    Mar 8 '18 at 11:18
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    It's a Mathematical notation used in a book about Linguistics. I believe that somewhere in the preamble to the book there should be a brief explanation to Math notation and the Set theory used within?
    – bytebuster
    Mar 8 '18 at 13:39
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The semicolon in ∀;y is surely a typo. It should just be ∀y, just like the ∀x that precedes it.

A period is often used to introduce the scope of a variable binding expression like ∀x, e.g. ∀x.φ. By convention, the scope stretches all the way to the end of the entire expression. So, for example, in ∀x.Px ∧ Qx, the occurrence of x in Qx is bound by ∀x. That is, it's not equivalent to Qx ∧ ∀x.Px (by convention).

Some authors instead delimit the scope with square brackets or parentheses, e.g. ∀x[φ] or ∀x(φ), while others don't use anything at all, e.g. ∀xφ. (In the latter case, complex expressions like Px ∧ Qx are defined recursively to contain brackets, e.g. (Px ∧ Qx), which by convention are dropped only when no confusion arises. For quantified expressions, this then yields ∀x(Px ∧ Qx), so that the scope of ∀x is automatically clear.)

The author that you quote seems to use a combination of a period and parentheses. This is basically a case of redundancy: assuming by convention that a scope introduced by a period stretches all the way to the right, it would be just a clear to write

∀x.∀y.x < y → ∃z.x < z ∧ z < y

with just periods and no parentheses, or

∀x(∀y(x < y → ∃z(x < z ∧ z < y)))

with just parentheses and no periods.

For historical reasons that I don't fully understand, using a period to introduce the scope of a λ-binder, e.g. λx.φ, is particularly common (though not universal).

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  • According to a mathematician I asked, the semicolon is used to mark either a 1-2-1 mapping, or distinguish between parameters and variables. Are either convention used in linguistic notation at all?
    – Eloisa
    Mar 8 '18 at 14:29
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    I just checked relevant passage on Amazon (p. 51 of the book), and there is no semicolon. (Find the book on Amazon, click the book cover, then search inside the book for "is dense iff".) I'm guessing that you're using an electronic copy, which has been corrupted somehow. (This would also explain why your formula has a caret, ^, instead of an actual logical 'and' symbol, ∧.) Mar 8 '18 at 15:20

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