If a person responded to an argument by saying, "This argument is highly illogical", would that be an appropriate response in any case involving logical argumentation?
in my shallow understanding, it certainly can, but the other side could ask to accusing side to pinpoint where is the illogical step in the argumentation? If the accusing side can't give a reason but only an accuse, then it's not an appropriate response?
Fuzzy logic was proposed as an attempt to provide a logic which works as you say "on a gradient": A logical statement can have a truth value anywhere in the [0,1] interval.
Completely true statements have a truth value of 1, completely false statements have a truth value of 0, and ambiguous statements have values values in between, depending on "How true" they are.
For example the statement "the weather is hot" is clearly true if the temperature outside is 50°C, and is clearly false if the temperature is -3° C, but what about 28°C or 23°C ?
Fuzzy logic would assign a gradient of truth values to the statement "the weather is hot". On can then assign a truth value of 1 to "the weather is hot" at 50°C, a truth value of 0.75 to "the weather is hot" at 28°C, a truth value of 0.5 to "the weather is hot" at 23°C, etc...
It should be noted that Fuzzy logic is a subset of a more general classe called Multivalued Logic, and that it was developed in the context of computer science and AI, as a tool to model vagueness in human speech, not in the context of philosophical logic.
Like all good questions, the answer is "it depends." One must define "logical" to arrive at an answer, and it gets defined in different contexts.
If the context is highly mathematical, it is likely that the definition of "logical" has to do with formal logic and proofs. Most formal systems such as First Order Logic are very binary in their approach to the validity of arguments. An argument is either logical or illogical in such an environment.
Some formal systems do permit gradients between logical and illogical. Alexander S King's answer goes into detail on one of them, Fuzzy Logic.
If we expand beyond the realm of simple mathematical contexts, other gradients do come forth. In conversational situations, it is common to provide only a high level sketch of an argument rather than digging into every little detail. The other individual then can pick which parts of the argument they feel they wish to see more detail. In such an environment, one could argue that a "logical" argument could be an argument that one feels can be turned into a rigorous mathematical style proof, and an "illogical" argument is one that one feels likely cannot be turned into such a proof.
Beyond that, one may explore other meanings of "rational." In Rational Choice Theory, the meaning of "rational" is chosen to be quite different:
The concept of rationality used in rational choice theory is different from the colloquial and most philosophical use of the word. Colloquially, "rational" behaviour typically means "sensible", "predictable", or "in a thoughtful, clear-headed manner." Rational choice theory uses a narrower definition of rationality. At its most basic level, behavior is rational if it is goal-oriented, reflective (evaluative), and consistent (across time and different choice situations). This contrasts with behavior that is random, impulsive, conditioned, or adopted by (unevaluative) imitation.
So, in all, the answer is "it depends," as always. However, in most environments there is some room for a gradient.