For what it is worth I certainly did not know the answer to this question. I find that first sentence very difficult to comprehend and would find myself guessing at this one!

So anyone who struggled with this know that I am right there with you. I have never heard such a turn of phrase in English "She claims it is possible to deduce matters of fact from logic and, with just as little epigram, claims to derive ethical and economic truth as well."

If you missed this one please do not worry too much about it!
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why it is license??

License for two main reasons: first off, because with POE is the only word that fits the bill. Secondly,

The laws of logic, in her opinion, (ii)__________her proclamation that “existence exists,” which is very much like saying that the law of thermodynamics is hot.

License give her the authority to proclaim that “existence exists.

Hope this helps. Ask if ay is unclear to you.

regards
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why is not blank 1 warrant?................This line is from book "The Reactionary Mind: Conservatism from Edmund Burke to Sarah Palin"
By Corey Robin. Over there it is warrant. please explain.

Warrant means permission. authorization or a legal document.

Epigram is a remark expressing an idea.

Warrant is completely out of scope

I hope now is more clear.

Regards
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The word epigram has a literary connotation and it is the wrong word to put there. With just as little "fallacy" actually sounds good, because the kind of claims she makes does not seem very sensible or acceptable. How does one deduce matters of fact from logic? It seems to imply the exercise is in-sensible.

But warrant is also okay there, though the phrase "with just as little warrant, aims to derive..." sounds weird.

Also, deduce, logic and fallacy are all terms from logic and so my first preference is fallacy, and then warrant and last would be epigram.
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The clue for the sentence is little ( so epigram means a short pithy saying) so it matches here. License, is the correct word for the second blank.
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question is same level as official one?

The following official solution has been given for this exact question on page 126 of 543 of Manhattan official guide 8 for Text Completion and Sentence Equivalence (4th Edition):

Question 20 official answer: Warrant, license.

Official Explanation: It is clear from “claims” and the generally derisive tone of the sentence that the author does not think it is possible to deduce matters of fact from logic (that would be pretty silly, come to think about it). Thus, the “she” who is the subject of the text has little warrant (justification) for doing this, and just as little warrant for using logic to inappropriately derive other “truths.” A fallacy is a “mistaken belief, especially one based on unsound argument”—she has lots of fallacies, not the dearth of fallacy indicated by “just as little.” An epigram is a “witticism” or “quip,” an extra meaning not indicated by any clues. License is similar to warrant and serves the same function in the second sentence. The “laws of logic” do not occlude (close, cover, or obstruct) or galvanize (spur to action) “her proclamation.”

Having completely solved Manhattah 5lb, all 5 Manhattan mocks and many of their topic specific GRE guides, have found 12+ mistakes across all their materials. To err is human- therefore, request users to log onto the Manhattan online portal where they post a list of corrections across their materials.

Best wishes.

Blank (i): (B) warrant
Blank (ii): (F) galvanize

Explanation:
The sentence talks about how the person in question believes that it is possible to deduce matters of fact from logic, and that she also aims to derive ethical and economic truths from the same. The word "just as little" suggests that the person is confident in her ability to do so without much difficulty or hesitation. The word that would best fit in the first blank is "warrant", meaning to justify or give authority for something. The person believes that logic gives her the authority to deduce matters of fact and derive ethical and economic truths.

In the second part of the sentence, the speaker refers to the "proclamation that 'existence exists,'" which is compared to a physical law like the law of thermodynamics. The word that would best fit in the second blank is "galvanize", meaning to stimulate or excite into action. The person believes that the laws of logic support her proclamation and give it strength and validity, much like the law of thermodynamics is a well-established and accepted concept in the field of physics.

What do you guys think of my explanation, do you think it makes sense?