## The difference between argument, inference, deduction and proof?

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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?