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The phrase "a third type of argument" is correct and usable in written English.
You can use it when discussing different categories or classifications of arguments in a debate, essay, or analysis.
Example: "In addition to the first two types of arguments, we must also consider a third type of argument that addresses counterclaims."
Alternatives: "an additional category of argument" or "a different kind of argument".
Exact(1)
But there is a third type of argument: we have the woodland only because previous generations passed it on to us and we have a similar obligation to pass it on to future generations.
Similar(59)
A second type of argument for ethical relativism is due to the Scottish philosopher David Hume (1711 76), who claimed that moral beliefs are based on "sentiment," or emotion, rather than on reason.
A second type of argument for expected utility theory relies on so-called representation theorems.
A second type of argument used against atomism involved showing that that the notion of conceptually minimal parts was philosophically and scientifically wanting (e.g., Avicenna 1983, III.4, 189 190).
The third type of argument involves a generalization of Chomsky's poverty of the stimulus argument for universal grammar.
This third type of argument for universalism has also been used to argue directly for the existence of temporal parts.
A third type of inductive justification recognizes the argument from analogy as a distinctive form, but treats its past successes as evidence for future success.
A second type of Leibnizean argument for God's real possibility returns to the thesis that God is the most perfect being, and adds that perfections are positive and simple, unanalyzable qualities.
Carruthers presents a clear outline of the first type of argument "the biological argument for massive modularity": "(1) Biological systems are designed systems, constructed incrementally.
The second type of argument is one side of a perennial debate in the philosophy of cognitive science.
The second type of argument makes no appeal to biological considerations whatsoever (although many evolutionary psychologists give these arguments a biological twist).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com