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ryang
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1. Here's a comparison of the various translations of $$∀x{∈}F\; P(x).$$

• For all $$x$$ in $$F,\ldots$$” sometimes sounds like property $$P$$ might belong to $$F$$ as a whole rather than to its individual members: “for all members of the family, they have a house” (1 house in total? or 5?). Contrast with “for each member of the family, they have a house” (definitely 5 houses in total).

• For every $$x$$ in $$F,\ldots$$”, despite ‘every’ too having a collective sense, says that property $$P$$ is common to the members of $$F.$$

• For each $$x$$ in $$F,\ldots$$” directly attributes property $$P$$ to individual members of $$F.$$

• For any $$x$$ in $$F,\ldots$$” doesn't strongly communicate that property $$P$$ belongs to each and every member of $$F;$$ nevertheless, “for any $$x$$” typically means “given an arbitrary $$x$$”, which, logically, is synonymous withfor every $$x$$” and “for each $$x$$”.

2. The phrase ‘not any’ actually means ‘not some’ instead of ‘not every’.

For example, observe that the sentences within each group below are equivalent to one another:

• She does $$\color\red{\textbf{not}}$$ have $$\color\red{\textbf{any}}$$ disease.

She does $$\color\red{\textbf{not}}$$ have $$\color\red{\textbf{some}}$$ disease.

$$\color\red{\boldsymbol{\lnot \exists}} x\, Dx$$

$$\forall x\, \lnot Dx$$

For every disease, she does not have it.

• She does not have every disease.

$$\lnot \forall x\, Dx$$

$$\exists x\, \lnot Dx$$

For some disease, she does not have it.

3. Consider this logical relationship: \begin{align} &&\forall x\;\big(Px\to Q\big)\tag{\color{brown}{1a}}\\ &\equiv{}&\big(\exists x\,Px\big)\to Q\tag{\color{brown}{1e}}\\&\large\not\equiv{}&\big(\forall x\,Px\big)\to Q\tag{\color{blue}{2a}}\\&\equiv{}&\exists x\;\big(Px\to Q\big).\tag{\color{blue}{2e}}\end{align}

For each/every/any $$x,$$ if $$x$$ has property $$P,$$ then $$Q$$ is true. $$\color{brown}{(1a)}$$
If some $$x$$ has property $$P,$$ then $$Q$$ is true.
$$\boxed{\textbf{Any }x}$$ has property $$P$$ implies that $$Q$$ is true. $$\quad\quad(\Large❔)$$
$$\boxed{\textbf{If for any }x},\,x$$ has property $$P,$$ then $$Q$$ is true.$$\quad\quad(\Large❔)$$
$$\boxed{\textbf{If any }x}$$ has property $$P,$$ then $$Q$$ is true. $$\quad\quad\quad\quad(\Large❔)$$
$$\color{brown}{(1e)}$$
If each/every $$x$$ has property $$P,$$ then $$Q$$ is true. $$\color{blue}{(2a)}$$
For some $$x,$$ if $$x$$ has property $$P,$$ then $$Q$$ is true. $$\color{blue}{(2e)}$$

When the word ‘any’ is embedded in a conditional's antecedent, the idiomatic meaning is typically—though not always—sentence $$\color{brown}{(1e)}$$ instead of sentence $$\color{blue}{(2a)};$$ that is, ‘if any’ frequently means ‘if some’ instead of ‘if every’; sometimes, its intended meaning is ambiguous.

For example, Wikipedia's definition of set disjointedness “A collection of two or more sets is disjoint $$\boxed{\text{if any}} \,$$two distinct sets of the collection are disjoint” requires external clarification whether three sets are disjoint

• if some pair of distinct sets is disjoint

or

• if every pair of distinct sets is disjoint;

it turns out from reading between the lines two sections down that Wikipedia intends the ‘if every’ reading, even as the ‘if some’ reading is probably more immediate. No wonder authors disagree between these two contrasting definitions!

Furthermore, compare

• $$\boxed{\text{If}}\,$$the dog understands $$\boxed{\text{any}}\,$$command, then it is a genius.

with

• $$\boxed{\text{If}}\,$$the dog understands $$\boxed{\text{any}}\,$$command, then it has been trained.

Summary: ‘for each’ and ‘for every’ are excellent translations of the universal quantifier; the word ‘any’ sometimes corresponds to ‘∃’ instead of ‘∀’, and in technical writing should be used judiciously.

ryang
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