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Here, Terence Tao writes:

This change is almost trivial to enact (it is often little more than just taking the contrapositive of the original statement), but it does offer a slightly different “non-counterfactual” (or more precisely, “not necessarily counterfactual”) perspective on these arguments which may assist in understanding how they work.

What does he mean with the words marked in bold? All the hypotheses of the arguments he presented of the "non-counterfactual"-type are actually satisfiable, for example:

Let A be a set.

Let n be a natural number.

...

Why should one precisely say "not necessarily counterfactual" instead of "non-counterfactual"? What's wrong with just saying "non-counterfactual"?

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    I mentally replaced "counterfactual" with "false" when reading that, but I'm not entirely sure that was intended. – 2017-02-21
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    In that context, I'm pretty sure that Tao means "assuming a hypothesis which is in fact false" by "counterfactual". – 2017-02-21
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    Please Help: if you're going to keep posting questions about three words, or one sentence, you need to provide (in your post itself) more context. Without reading the post you link to (which no user should be expected to do), your question makes little sense. Please think more before you post, and when you do post, include more context in which your quote emerges. – 2017-02-21
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    @amWhy: Thanks for your comment. You say: "your question reveals your own misunderstanding of the material". Could you please point out where my misunderstanding lies? – 2017-02-21
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    OK, I clicked through to the link. In this case, I think he means that "A actually has some value which may or may not be a set" or "n has a value which may or may not be a natural number"; but in both cases we can use particular examples for A and n which do satisfy these requirements, whereas if you say "There is no set A such that..." then demonstrating with an example is more difficult. – 2017-02-21
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    @Kevin: Thanks. That's also what I thought. But I also guessed that it's quite unprobable that he would make such an remark because statements such as "n is not a natural number" for a non-number-object n hasn't much value. – 2017-02-21
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    @amWhy: Do you think Kevin and I interpreted Tao's sentence correctly? – 2017-02-21
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    This question is about Tao's opinions and can only have an opinion-based answer. – 2017-02-21
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    @RobArthan I disagree. The question is "what does he mean" not "is his opinion right." – 2017-02-21
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    The person who PleaseHelp should be consulting is Tao, not the users at MSE. – 2017-02-21
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    @amWhy: exactly! A debate here about degrees of counter-factuality would be absurd (particularly, as **in my opinion**, Tao is rather confused and/or confusing about the distinction between counter-factual arguments and constructive reasoning). – 2017-02-21
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    @amWhy Tao is the ideal person to answer this question, but there's no reason to assume critical readers couldn't figure out his meaning from his writing. Certainly if that's impossible, then Tao is probably a bad writer! – 2017-02-21
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    @StellaBiderman: he's a very fine writer. I would hate to be judged by your standard: one lapse of clarity or precision in a blog where ideas are being explored and the author becomes a bad writer? – 2017-02-21
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    You think I should go to the famous Tao to ask him that question? Really? Seriously? – 2017-02-21
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    @Please Help: yes! Why not? It's a blog. There's a form at the bottom for you to leave a reply. – 2017-02-21
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    He won't answer. – 2017-02-21
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    Please Help: So you just decide to inundate this site in search of what you think we might think about what Tao thinks let alone what he means? – 2017-02-21

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I thought about Kevin's comment a bit and I think I can answer my own question:

Consider for example:

Proposition 1 (No largest natural number). There does not exist a natural number N that is larger than all the other natural numbers.

In the first article on the no self-defeating object argument, Tao gives the following counterfactual proof:

Proof: Suppose for contradiction that there was such a largest natural number ${N}$. Then ${N+1}$ is also a natural number which is strictly larger than ${N}$, contradicting the hypothesis that ${N}$ is the largest natural number.

This argument is counterfactual, because there is no situation in which "$N$ is a largest natural number" is true.

Now look at the "non-counterfactual" version:

Proposition 1′. Given any natural number $N$, one can find another natural number $N'$ which is larger than $N$.

The only hypothesis here is "$N$ is a natural number". And this is satisfiable (for example by $N=1$). But technically, it is not valid in every situation. For example, if $N = \pi$, then the hypothesis is false, and thus the hypothesis would be counterfactual. That's why Tao says "not necessarily counterfactual".

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    Given any natural number $N$, there certainly does exist (so one can find) another natural number $N' = N+1$ such that $N'\gt N$. That is absolutely true for all $N\in \mathbb N$. And note, if $N$ is a natural number (hypothesis), then $N \neq \pi$, right? in your second example, we have the true statement $$\forall N \in \mathbb N, \exists N'\in \mathbb N(N'\gt N)$$ – 2017-02-21
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    What you are noting is, $\exists N\in \mathbb N \forall N'\in \mathbb N\; (N'\gt N)$ which is clearly false, but this is not equivalent to the statement I wrote immediately above. That is, Proposition 1 $\neq$ Proposition 1' – 2017-02-21