Syntax vs semantics in logic and metalogic

Question

Trying to better understand syntax vs semantics (structure vs meaning) in logic and metalogic.

I am using this as my reference point (and the example(s) below on this link): https://en.wikipedia.org/wiki/Propositional_calculus#Generic_description_of_a_propositional_calculus

e.g. let's say we have the connectives/operators:

$\Omega = \Omega_0 \cup \Omega_1 \cup \Omega_2 = \{T, F\} \cup \{\lnot\} \cup \{\to\}$

And the following rule of inference (modus ponens):

$P, P \to Q \vdash Q$

And the following axioms (Jan Łukasiewicz system, picking these arbitrarily):

I. $(p \to (q \to p))$

II. $((p \to (q \to r)) \to ((p \to q) \to (p \to r)))$

III. $((\lnot p \to \lnot q) \to (q \to p))$

Does this fully describe the entire logical system, or only in terms of syntax? Do the semantics follow or do we have to impose those on top of all this? Is imposing semantics a metalogical level concept?

To my current understanding, we'd use truth tables to define various operators and such to give semantic meaning but are these optional / more of a convenience, or are they necessary?

Are truth tables still metalogical?

What about other definitions, like introducing operators such as $\land$, $\lor$, etc, are these still metalogical or would they need to be embedded in $\Omega$ up front?

For example, how do we use all this to describe how $\lnot$ and $\to$ work? i.e. showing that:

$\lnot T = F$

$\lnot F = T$

$F \to F = T$

$F \to T = T$

$T \to F = F$

$T \to T = T$

Or law of excluded middle:

$p \lor \lnot p = T$ (after defining $\land$ and $\lor$, etc)

Do we explicitly define these with truth tables, or derive them from the axioms and inferences somehow? How do the axioms and rules of inference play with each other here and allow us to ultimately go from something involving $\vdash$ to something with $=$ or $\to$?

Trying to understand how we start off with a system and start unrolling a logical system as we might normally use it.

You can see the post Why do people separate syntax and semantics in mathematical logic ? — Mauro ALLEGRANZA, Sep 04 '18 at 13:29
See Formal language : we need symbols (the alphabet), rules for generating expression (i.e. grammatically correct strings of symbol), usually axioms (maybe none), and rules of inference (at least one: with the usual axiomatic approach we have tipically modis ponens; in other treatment, like Natural Deduction that are rules-based, there are no axioms and a couple of rules for every basic connective). — Mauro ALLEGRANZA, Sep 04 '18 at 13:32
About truth tables, see the post : Truth tables in propositional calculus: semantic or syntactic in nature ? — Mauro ALLEGRANZA, Sep 04 '18 at 13:34
See also the post : Axiom Systems and Formal Systems for some details and reference. — Mauro ALLEGRANZA, Sep 04 '18 at 13:35
And the post : Are axioms assumed to be true in a formal system ? — Mauro ALLEGRANZA, Sep 04 '18 at 13:37
Finally, you can use a good textbook, like e.g. van Dalen, Logic and Structure. — Mauro ALLEGRANZA, Sep 04 '18 at 13:44
For example, you can start with only $\bot$ and $\to$ and define $\lnot$ as $\varphi \to \bot$ and then $\lor$ and $\land$ and $\leftrightarrow$ as usual. We define them "syntacticallly", i.e. in the metalogic world we introduce $\varphi \lor \psi$ as an abbreviation for the formula $\lnot \varphi \to \psi$. But also $\lnot$ is an abbreviation, and thus we have that $\varphi \lor \psi$ will be defined as : $(\varphi \to \bot) \to \psi$. — Mauro ALLEGRANZA, Sep 04 '18 at 13:50
If we work in the classical framework, the above definition will match withe the truth table for the connectives ($\lnot \varphi \to \psi$ will be FALSE only when both $\varphi$ and $\psi$ are FALSE). — Mauro ALLEGRANZA, Sep 04 '18 at 13:52
Reading through those links, I'm not sure any of these address these questions directly. Most point out a syntax / semantic difference which I already accept but what I don't understand is how it's executed and categorized. — user525966, Sep 04 '18 at 14:09
I understand that we can define operators syntactically in terms of other operators — user525966, Sep 04 '18 at 14:10
Sure... they are answer to question on the site and not textbook: they address some more or less specific points. For a basic treatment with all the needed definitions and explanations, you have to read at least one textbook. — Mauro ALLEGRANZA, Sep 04 '18 at 14:11
What lets us treat T/F as 0-arity operators? Convention? They seem a bit different from everything else — user525966, Sep 04 '18 at 14:11
We have $\text T$ and $\text F$ as metalogical symbols to denote TRUE and FALSE, the "usual" meaning of proposition according to propositional calculus. If we have their counterpart in syntax, it is better to use two difefret symbols : $\top, \bot$: they are $0$-ary connectives or propositional constants. We can also define $\top$ as $\bot \to \bot$. — Mauro ALLEGRANZA, Sep 04 '18 at 14:15
Having done this, we interpret the two symbols $\top$ and $\bot$ as $\text T$ and $\text F$ respectively. — Mauro ALLEGRANZA, Sep 04 '18 at 14:16
But then technically we are using 2-arity connectives with 0-arity connectives as inputs, yes? Everything is technically connectives or do the connectives "return" a particular data type, i.e. a wff? True and false here each considered a wff? — user525966, Sep 04 '18 at 14:25
See discussion : What is the “correct” reading of $\bot$ ? — Mauro ALLEGRANZA, Sep 04 '18 at 14:31
So we can define all these relationships syntactically but don't they have to be in the form of our axioms? — user525966, Sep 04 '18 at 14:33
And how do we formally denote that these axioms are "true" (or rather that T symbol), does this need to be expressed explicitly in the syntax or is it something intrinsic to the axiom set $I$ of a propositional calculus? Or is this too an informal semantic assignment? — user525966, Sep 04 '18 at 14:51
We define in the metalogic world the concept of interpretation of a formula and then we define as a tautology a formula that is always true (as $\lnot p \lor p$). Then we check that axioms are tautologies and that modus ponens preserves truth. This is the core of the soundness property of the calculus. — Mauro ALLEGRANZA, Sep 04 '18 at 14:56
Checking out that book, it claims no need of set theory but then immediately introduces a bunch of set theory and undefined syntax / arrows on page 7, lol — user525966, Sep 04 '18 at 15:54
How do we show every theorem is a tautology and every tautology is a theorem? — user525966, Sep 04 '18 at 17:45
Regarding "set theory machinery" in the exposition of math logic, see the post : Set theoretic concepts in first order logic. You have to note that math log is part of modern mathematics, and modern mathematics uses set theory langiage as a "common language". — Mauro ALLEGRANZA, Sep 04 '18 at 18:39
See also the post : Why are ⊢ and ⊨ symbols from metalanguage? — Mauro ALLEGRANZA, Sep 04 '18 at 18:42
For soundness, see Argue that if a sentence has a proof, then it is a tautology and see [Model-theory and Proof-theory in Propositional Logic](Model-theory and Proof-theory in Propositional Logic). — Mauro ALLEGRANZA, Sep 04 '18 at 18:46
I'm still lost as to whether our definitions are bound by our axiomatic syntax and inferences. — user525966, Sep 04 '18 at 19:26
I also don't understand why we are allowed to use set theory to define logic when logic defines how set theory works. Feels circular. — user525966, Sep 04 '18 at 19:58
Regarding "circularity", see the post : Does mathematics become circular at the bottom? What is at the bottom of mathematics ? as well as the post When does the set enter set theory ? — Mauro ALLEGRANZA, Sep 05 '18 at 06:27
See also the post : How to avoid perceived circularity when defining a formal language ? — Mauro ALLEGRANZA, Sep 05 '18 at 07:00
On language vs metalanguage, see the post : Object language and meta-language. — Mauro ALLEGRANZA, Sep 07 '18 at 07:21

Syntax vs semantics in logic and metalogic

0 Answers0