Logical system is a deductive system together with additional (non-logical) axioms and a semantics. An example of a logical system is Peano arithmetic. Wikipedia
Can you please explain what is non-logical axioms and a semantics in this context?
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tryst with freedom
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Oleg Dats
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3A logical axiom is a formula that is true in every interpretation, like e.g. $\forall x (x=x)$. A non-logical axiom is a formula that is true in some interpretation. Example: the Peano axiom $\forall x\ (0\neq S(x))$ is true in $\mathbb N$ but not in every interpretation. – Mauro ALLEGRANZA Jan 19 '21 at 16:21
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1Having said that, no clear-cut difference – Mauro ALLEGRANZA Jan 19 '21 at 18:28
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@MauroALLEGRANZA What about semantics? Does the deductive system include semantics? – Oleg Dats Jan 20 '21 at 08:02
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As per discussions on previous similar posts we have many terms with similar meaning.
Maybe the first one is Formal language: an alphabet made of an initial (usually finite) stock of basic symbols and a set (usually finite) of rules to produce expressions (usually with finite length) that are called formulas.
With it we build a Formal system (also called a logical calculus, or a logical system) consisting of a formal language together with a deductive apparatus (also called a deductive system).
The deductive apparatus is a set (usually finite) of rules, called rules of inference, that we use to produce formulas from an initial stock of formulas: the axioms.
This is the syntactical part.
The logical system is used to derive logical truths (like e.g. $\vdash p \lor \lnot p$) and to derive consequences from assumptions: $\Gamma \vdash \varphi$.
We may call it the "pure" logical theory.
When the set of assumptions $\Gamma$ is the formalized version of the collection of axioms of some mathematical theory, like e.g. the firts-order arithmetic that formalizes the theory based on Peano axioms, we have an "applied" logical theory.
A formalized mathematical theory can be interpreted: for first-order arithmetic, the "natural" intepretation is the mathematical structure base on the set $\mathbb N$ of natural numbers.
The intuitive notion of interpretation, i.e. the way to assign meaning to symbols and formulas of a formal language, is made rigorous through the mathematical definition of Interpretation of a formal language.
This is the semantics.
Mauro ALLEGRANZA
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Great answer. Let me add that the terminology used in Wikipedia's page about formal system is not standard and slightly misleading, especially in its definitions of "logical system". – Taroccoesbrocco Jan 20 '21 at 08:45
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@MauroALLEGRANZA thank you for a great answer. Can you please recommend a good book with exercises to learn logic and everething you mentioned in your answer? – Oleg Dats Jan 20 '21 at 13:42
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@MauroALLEGRANZA is the order of the books is the order of how should I study them? – Oleg Dats Jan 20 '21 at 14:10
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@MauroALLEGRANZA I am really interested in Philosophy, Theory of knowledge, and Artificial Intelligence. Any recommendation for a book (Logic is core) that also mentiones these topics? – Oleg Dats Jan 20 '21 at 14:23
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1@OlegDats - some sugesetions: Carlo Cellucci, Rethinking Logic : Logic in Relation to Mathematics Evolution and Method (Springer, 2013), Carlo Cellucci, Rethinking Knowledge: The Heuristic View (Springer, 2017), Pavel Pudlak, Logical Foundations of Mathematics and Computational Complexity (Springer, 2013). – Mauro ALLEGRANZA Jan 20 '21 at 14:31
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1Harrie de Swart, Philosophical and Mathematical Logic (Springer, 2018). – Mauro ALLEGRANZA Jan 20 '21 at 14:32
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@Buraian - see Calculus: "calculation; computation. Any formal system in which symbolic expressions are manipulated according to fixed rules: lambda calculus, predicate calculus." – Mauro ALLEGRANZA Mar 09 '22 at 14:00
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Is all of mathematics then calculus? Because if one is to communicate they must use symbolic expa – tryst with freedom Mar 09 '22 at 15:17
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@Buraian - not very clear... In principle, mathematics is not equal to "symbolic"; we can currently read Euclid's Elements and there are no formulas in modern terms. But modern mathematics blow up since Renaissance due to the "invention" of symbolic thinking. – Mauro ALLEGRANZA Mar 09 '22 at 15:20
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I mean isn't language in itself a calculus by the type of definition you suggest – tryst with freedom Mar 09 '22 at 16:06