# How do you prove that a logic system is sound?

I am aware of the fact that a logic system must be sound, in order to be useful. However, I am not sure, about how, after setting up or coming up with the basic logic axioms that make up my system, I would prove that the logic system is sound.

In general, how much work is required to do so, how do I go about doing so, and where can I find some references online and in non format of how to do this.

Can someone please give me some examples?

What are the easier and most complex logic systems to prove that they are sound.

Thanks.

• It really shouldn't be that hard to search this online, use SEP for example. In any case, proof of soundness generally follows from induction. Aug 16, 2022 at 2:22
• See soundess first: Most proofs of soundness are trivial... If the system allows Hilbert-style deduction, it requires only verifying the validity of the axioms and one rule of inference, namely modus ponens... However, for systems such as PA its soundness has to ground to something else additionally, and still today there're some (ultrafinitism) mathematicians doubt its absolute consistency. Also for multivalued logics soundness proofs are generally complicated as discussed in a recent post... Aug 16, 2022 at 3:52
• " where can I find some references online and in non format of how to do this." Start from the simple case of propositional logic: the proof amounts to showing (e.g. using truth tables) that the axioms are valid (always true) ans that the rules (like Modus Ponens) preserve truth. Aug 16, 2022 at 8:29

I am assuming you are talking about a proof of soundness (and perhaps completeness) of a formal proof system. The short answer to how you do it is by induction on the length or complexity of the proof.

First, let's get clear about what we mean here. A proof system is sound with respect to a semantic theory for a language if all the sentences it proves are logical truths (or validities, or valid sentences, or perhaps tautologies) of that language. Completeness is the converse: a proof system is complete if all the logical truths of the language are provable. Strong soundness goes beyond this by requiring that for any set of sentences Γ in the language, if a sentence α is provable from Γ, then α is the semantic consequence of Γ. Likewise, strong completeness is the converse.

Proving soundness will depend on what your proof system looks like. In the case of a natural deduction system for classical propositional logic, it is simply a matter of showing that the basic rules are all soundness-preserving. If the syntax for the sentences of your language is defined recursively then you can show by induction on the size or complexity of a proof that there is no way to prove α from Γ unless α holds under all the models or interpretations or structures under which Γ holds.

With simple logics, proving soundness is fairly straightforward. If you are using a many-valued logic, or an infinitary logic, the proof becomes rather more complex.

Some examples: this shows a sketch of a proof of soundness for classical first-order logic; this shows a sketch of a proof of soundness for intuitionistic propositional logic.

How do you prove that a logic system is sound?

I am aware of the fact that a logic system must be sound, in order to be useful.

And there is your answer. I understand “soundness” to mean “that which aligns with observable reality“, or “that which is useful”. You design and conduct tests to show that your system produces results that agree with the things that you and others can see and use: in other words, that your system is practical.

In general, [1] how much work is required to do so, [2] how do I go about doing so, and [3] where can I find some references online and in non format of how to do this.

[1] Well, a lot of work is necessary, quite frankly. You are building a new system of thought. Such efforts take serious time. This is not at all sarcastic, or intended to be discouraging: expect to spend a lifetime on this project.

[2] I recommend doing so by studying the work of history’s system builders, such as Plato, Aristotle, Leibniz, Kant, and Marx. What did they do to assure that their ruminations squared with observations? What did they get wrong? Or right?

[3] I would start with Project Gutenberg. The project has done the heavy lifting when it comes to finding and making public original historical documents. And never forget SEP— the Stanford Encyclopedia of Philosophy.

Good luck to you!

• Hi. I would add Hegel as a system builder. Aug 19, 2022 at 2:16
• @psitae. Good thought. Aug 19, 2022 at 4:46

To prove means to show a particular result after a period of time. Since your question demands proof of logic system the area becomes too broad. Also time for a proof is not specified. Anyway 'to prove' often implies to use logic. If somebody proves logic system is sound that must mean logic system is sound; but only on the basis of logic (since he is proving it by means of logic).

But what does it mean if he proves that ‘logic system is not sound’? That means, the logic he uses to prove (or convince others) is not completely reliable.

Now what do you say? Is logic system the supreme?