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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?

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    Am I mistaken, or is this question really about your own personal taste? You say that formal and symbolic logic is "too obscure" and you find it "unnecessary for philosophical usage", and then ask us if it is possible for philosophical logic be done without it being "too" symbolic. But how are we to judge what is "too obscure" or "too symbolic" in this context? Clearly, there are varying degrees of formalization found in various philosophical texts; what criteria would you like us to use to assess these? – Michael Dorfman Sep 30 '12 at 18:53
  • Plenty of philosophers don't use or even study any formal logic beyond basic undergraduate logic courses. So if the question is whether we can do good philosophy without symbolic logic, the answer is "of course we can." In fact, artificially formalizing arguments when you need not introduce formalisms is frowned upon in academic philosophy and can easily get papers rejected from publication. – transitionsynthesis Oct 2 at 2:38
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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:

  • While I agree with most of your points (+1), I feel compelled to point out that we have no evidence that humans are better theorem-solvers than Turing machines. We have come up with clever ways to solve individual problems (and classes thereof) that would naively be out of reach of a Turing machine, but there is no proof that a Turing machine could not implement the same clever approaches that we use, nor can we demonstrate that we are universal solvers (though it's tempting to assume we are!). – Rex Kerr Oct 3 '12 at 21:43
  • @RexKerr There are good reasons to believe that all (present, past and future) human minds together (including the tools they will invent) are better problem solvers than any given Turing machine using sound reasoning, at least with respect to their creativity. They won't use only sound reasoning, and they have the "game theoretic" privilege to always have the last word. – Thomas Klimpel Oct 20 '12 at 21:15
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    @ThomasKlimpel - Waving your hands about creativity is waving your hands. Do you have any theoretical basis for assuming that one cannot compute creativity? Or that various unproven alternatives cannot be explored in depth by an algorithm? (Chess programs already do the latter.) Does having the last word make one's truth-statements any more true? – Rex Kerr Oct 20 '12 at 21:39
  • Rex: I do not mean to agree with Lucas or Penrose (I don't), I just wanted to point out that an understanding of their argument requires a fairly deep understanding of logic. – Xodarap Oct 20 '12 at 21:42
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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.

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