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I am trying to distinguish argument, inference, deduction and proof. First, let's look at the distinction between argument and inference (if there is one). This online source states:

An argument is a set of two or more propositions related to each other in such a way that all but one of them (the premises) are supposed to provide support for the remaining one (the conclusion). The transition or movement from premises to conclusion, the logical connection between them, is the inference upon which the argument relies.

While the definition of "argument" is pretty concrete (a set whose elements are a number of premises and a conclusion), the definition of "inference" is less rigorous, referring to "movement" from premises to conclusion.

  1. Is this "movement" different from the argument itself? Or can the set of the premises and conclusion also be the inference?

  2. Also, where do the terms "deduction" and "proof" fit?

Please consider my made-up premise P and conclusion C below:

P: "The number x satisfies 4x+8=32."

C: "x=6."

Also, consider the implications I1 and I2 below:

I1: "If 4x+8=32, then 4x=24"

I2: "If 4x=24, then x=6"

  1. In the above example, I would like to identify specifically

    • What is the argument?
    • What is the inference?
    • What is the deduction?
    • What is the proof?

According to the source article, since an argument is a set of premises together with the conclusion, then the argument would have to be {P,C}.

  1. Then what would you identify as the inference? Is the inference also the set of statements {P,C}?

  2. Are the terms 'argument" and 'inference' synonymous?

  3. What would you call {P,I1,I2,C}? It cannot be the argument, since it contains more statements than just the premise and conclusion; it also contains the "steps" leading from P to C. Is it an inference, deduction, or a proof (or more than one of the above)?

  4. What would you call the tuple (P,I1,I2,C), where order matters?

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One issue is that different authors use "argument" and "inference" in ways different from each other, and from the colloquial meaning. For example, your source interprets "argument" as just the list of premises and the conclusion, whereas in the colloquial sense it is the sequence of intermediate logically elementary steps that lead from premises to the conclusion that is called an argument. The word deduction is usually used to express that sequence, and in formal deductions "logically elementary" means that each step is a premise, an axiom (if a list of them is additionally assumed), or a direct consequence of previous steps by one of accepted inference rules (modus ponens, etc.).

Another issue is that there are two approaches to logical consequence used in arguments, the deductive described above, and semantic. In the semantic approach, the conclusion follows from premises not by force of adopted inference rules but by relying on interpretation of variables. B follows from A if under all interpretations (consistent with axioms if those are present) B is true whenever A is true. Then no steps are needed between the premises and the conclusion; there is no deduction, which is why authors that have this approach in mind collapse "argument" to premises+conclusion. The process of passing from premises to the conclusion is then called inference. It is valid if truth values align in all interpretations, roughly speaking deduction is replaced by inspecting a truth table.

Proof is what is used to support argument's validity. In mathematics and formal logic "rigorous proof" is usually identified with a formal deduction, but outside of it may be used more loosely. Presenting a truth table that validates semantic inference counts as "proof". Informal deduction, where some steps are not logically tight but plausible, counts as "proof" especially in philosophy, and even strong evidence supporting an inductive generalization counts as "experimental proof".

In your first quote with P and C, the whole thing is the argument (in semantic sense), there is no deduction (but one can be provided assuming the axioms of arithmetic) or proof (in any sense), and passage from P to C is the inference. In I1 and I2, you have implications, which are treated as single statements in logic, so technically there are no arguments there, or we can treat them as conclusions with empty set of premises. It is more natural however to split implications into premises and conclusion, and treat them as arguments that way.

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