The most important element for expression of truth is trough an argument, with premises and conclusion. Argumentation requires to avoid fallacies and adhere to the truth. However logic if treated as a subject itself and not a part of something else grows bigger and deeper, that it leads to symbols which I find unnecessary for philosophical usage of asserting an existential claim or in other words it turns to mathematical than philosophical, take for example the concepts of propositional and other forms of modern logic with terms and functions to obscure to be used for philosophical assertions. So can the basic knowledge about argumentation, fallacies, and adherence to truth be sufficient for philosophical logic or philosophical purposes without too symbolic or mathematical concepts?
I assume by "basic knowledge" you mean "Aristotelian logic", namely syllogisms.
The answer is no, this is not sufficient. Inductive logic cannot be expressed in this manner, and (more damningly) Frege's introduction of quantifiers and variables showed that there were a large number of sentences which can't be expressed in standard Aristotelian form.
However, there are an infinite number of ways of writing logic (more formally, an infinite number of Universal Turing Machines which enumerate theorems), and so any particular encoding is arbitrary. If you don't like the current symbols, then by all means substitute your own. (Or if you prefer writing "for all" to the upside-down A, etc.)
Some examples where "complicated" logic is useful in Philosophy:
Both traditional (e.g., syllogistic or dialectical reasoning) and modern symbolic logic are based on equivalent rules of formal inference. What is essential to either case is not the content expressed in the premises or conclusions of the arguments, but the mediation of the premises according to certain abstract laws of thought, which can be expressed either formally or informally.
In a categorical syllogism, for example, the major premise “All men are mortal,” and minor premise “Socrates is a man” is what supplies the content rather than the form of the argument. In this case, the inference rule is categorical since it coordinates the middle term (man) with the major term (mortal) and minor term (Socrates) to yield the conclusion that "Socrates is mortal." In other words, a categorical syllogism is a formal pattern of reasoning that certifies the inference from universal to particular.
The very same rule can also be expressed in predicate logic using, for instance, the law of universal instantiation, as in the following: ∀xP(x); ∴ P(c). Stated in natural language, this argument would be rendered as: all (∀ = the universal quantifier) mortals (x = the subject or variable) are men (P = the predicate); therefore (∴ = the logical consequence or entailment) Socrates is mortal, as Socrates is an instance (c = an element of the domain) of men (P): or more economically ∀xP(x); ∴ P(Socrates).
The point here is that there are always formal laws of logic operative in philosophy. Whether these are expressed implicitly in a natural language, or explicitly using artificial formalisms is more a matter of taste and the particular kinds of problems one has to deal with. A complex empirical theory, for example, might require the precision and mathematical grip made possible by the latter, whereas a more creative or synthetic account may depend on the fluid or rhetorical properties of speech made possible by the former. With either approach, though, you are still accountable to the same logical laws; that is, the distinction is closer to vocabulary or mode of presentation, than of logical rigour or expressive power.