# Taxonomy and structure of concepts of logic system (propositional)

I'd like to know a conventional and clear structure of the main basic concepts of system of logic, how they relate between each other and so forth.

Say Atom is a symbol P. Statement is an atom with natural language text. Axiom is a statement with a truth value. Proposition is either a set of statements OR Axioms, depending if it forms a theorem or something else? Argument is a set of propositions, where the last proposition is a conclusion. This is just a draft of the idea to show what I mean.

I'd like also to know, how synonymous terms / concepts should be defined, where does hypothese and other terms involve in the system. And would you just call whole system "Logic system" or what?

I read some articles from Google and Wiki for example, but I couldn't paint a clear simple picture of this.

My aim is to model this with some programming language at the end. I'm thinkin of propositional logic in first hand but sure other systems, how they relate might help to see bigger picture and make some definition decisions.

Id like to stress the importance of the topic since other people have asked similar questions:

https://www.quora.com/Is-there-a-taxonomy-of-logic-as-in-a-visual-representation-of-how-all-the-concepts-of-reasoning-and-logic-are-connected

See e.g. :

an atom (or atomic proposition) is either a proposition symbol (or proposition variable) : $p_i$ or a constant : $\top$ (the true) and $\bot$ (the false).

Is a symbol that is used as name for a sentence of natural language.

A formula (of propositional logic) is a "complex" string built up from atoms and (propositional) connectives :

$\lnot, \lor, \land, \to, \ldots$.

Thus, $\lnot p_i, p_i \lor p_j, \bot \to \bot$ are example of formulaee.

A calculus (or proof system) is a set of (zero or more) "designated" formuale (the axioms) and a set of (one or more) rules of inference, like e.g. modus ponens.

A derivation of a formula $\varphi$ (the conclusion) from a set of assumptions $\Gamma$, in symbol:

$\Gamma \vdash \varphi$

is a sequence of formulae such that every formula in the sequence is either an axiom, or a formula in $\Gamma$, or is derived from previous formulae in the sequence with the application of a rule of inference, and the last formula of the sequence is the conclusion $\varphi$.

An argument is a set of sentences with some premises (call them $\Gamma$) and a concusion: $\varphi$.

A valid argument is an argument such that the conclusion is logical consequence of the premises; in symbol:

$\Gamma \vDash \varphi$.

The usual semantics for propositional logic interprets a formula with a valuation function, i.e. a function that maps every atom (i.e. every member of the set $\text {Atom}$) into a truth-value:

$v : \text {Atom} \to \{ 0 , 1 \}$.

This function assigns a truth value to each atom (with the obvious proviso that $v(\top)=1$ and $v(\bot)=0$); with the usual truth tables for the propositional connectives we can compute the truth value of every propositional formula.

The axioms of propositional calculus are formulae that are tautologies, i.e. true for every possible valuation.

The rules of inference are sound, i.e. they "produce" true conclusions from true assumptions.

Thus, every propositional formula derivable by way of the inference rules from the axioms of the propositional calculus are tautologies: this propeerty is called soundness of the calculus.

It can be generalized to the following "stronger" form:

if $\varphi$ is derivable from the set $\Gamma$ of assumptions (i.e. $\Gamma \vdash \varphi$) then $\varphi$ is a logical consequence of the set $\Gamma$ (i.e. $\Gamma \vDash \varphi$).

The above concepts can be easily applied to more expressive languages, like the first order one.

• So should I take formulae = axiom = assumption = proposition = premise, synonyms for each other? Maybe this is the most confusing part. Also should I think sentence as a combination of a symbol (atom), natural language phrase and a truth value? For $⊤$ phrase would simply be "true" and value true/1 (depending on the chosen underlying programable variable assignment) – MarkokraM Feb 27 '17 at 14:20
• @MarkokraM - We have 1) the alphabet made of symbols. Then we have 2) expressions, i.e. (usually) finite strings of symbols. Some expressions are 3) formulae according to the rules of the syntax: with the usual rules of prop logic the expression $p \to q$ is a formula while $\lnot \to q$ is not. 4) An assumption is a formula that is the "starting point" of an argument: it is assumed to be true for the context of the argument. – Mauro ALLEGRANZA Feb 27 '17 at 14:24
• This answer helped me enough to move on. – MarkokraM Mar 8 '17 at 8:56