First-order logic
Main article: First-order logic
Whereas universal algebra provides the semantics for a signature, logic provides the syntax. With terms, identities and quasi-identities, even universal algebra has some limited syntactic tools; first-order logic is the result of making quantification explicit and adding negation into the picture.
A first-order formula is built out of atomic formulas such as R(f(x,y),z) or y = x + 1 by means of the Boolean connectives {\displaystyle \neg ,\land ,\lor ,\rightarrow } and prefixing of quantifiers {\displaystyle \forall v}or {\displaystyle \exists v}. A sentence is a formula in which each occurrence of a variable is in the scope of a corresponding quantifier. Examples for formulas are П† (or П†(x) to mark the fact that at most x is an unbound variable in П†) and П€ defined as follows:
1. (П† в€Ё П€), (П† ∧ П€), (В¬П†), (П† в†' П€) 2. (в€ѓxП†), (в€ЂxП†)
(Note that the equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the Пѓsmr-structure {\displaystyle {\mathcal {N}}}of the natural numbers, for example, an element n satisfies the formula П† if and only if n is a prime number. The formula П€ similarly defines irreducibility. Tarski gave a rigorous definition, sometimes called "Tarski's definition of truth", for the satisfaction relation {\displaystyle \models }, so that one easily proves:
Thus в€Ђy (rxy) в€Ё в€ѓy (rxy) means (в€Ђy (rxy)) в€Ё (в€ѓy (rxy)) .
A set T of sentences is called a (first-order) theory. A theory is satisfiable if it has a model M |= T, i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set T. Consistency of a theory is usually defined in a syntactical way, but in first-order logic by the completeness theorem there is no need to distinguish between satisfiability and consistency. Therefore, model theorists often use "consistent" as a synonym for "satisfiable".
A theory is called categorical if it determines a structure up to isomorphism, but it turns out that this definition is not useful, due to serious restrictions in the expressivity of first-order logic. The Löwenheim-Skolem theorem implies that for every theory T[2] which has an infinite model and for every infinite cardinal numberκ, there is a model {\displaystyle {\mathcal {M}}\models T} such that the number of elements of {\displaystyle {\mathcal {M}}} is exactly κ. Therefore, only finitary structures can be described by a categorical theory.
Lack of expressivity (when compared to higher logics such as second-order logic) has its advantages, though. For model theorists, the Löwenheim-Skolem theorem is an important practical tool rather than the source of Skolem's paradox. In a certain sense made precise by Lindström's theorem, first-order logic is the most expressive logic for which both the Löwenheim-Skolem theorem and the compactness theorem hold.
As a corollary (i.e., its contrapositive), the compactness theorem says that every unsatisfiable first-order theory has a finite unsatisfiable subset. This theorem is of central importance in infinite model theory, where the words "by compactness" are commonplace. One way to prove it is by means of ultraproducts. An alternative proof uses the completeness theorem, which is otherwise reduced to a marginal role in most of modern model theory.
From: Model Theory, Wikipedia
Also see, Introduction to the CTMU:
That this structure is completely self-distributed implies that it is locally indistinguishable for subsystems s; it could only be discerned against its absence, and it is nowhere absent in S. Spacetime is thus transparent from within, its syntactic structure invisible to its contents on the classical (macroscopic) level. Localized systems generally express and utilize only a part of this syntax on any given scale, as determined by their specific structures. I.e., where there exists a hological incoversion endomorphism D:SГ {rГЋS} carrying the whole structure of S into every internal point and region of S, objects (quantum-geometrodynamically) embedded in S take their recognition and state-transformation syntaxes directly from the ambient spatiotemporal background up to isomorphism. Objects thus utilize only those aspects of D(S) of which they are structural and functional representations.
From: http://www.ctmu.org/Articles/IntroCTMU.htm
Here we see a relationship between Quantum Mechanics and Logic.
Logic is Quantum. Quantum, logic. Recall that reality is logic(al). Hence, so to is Quantum reality. First order logic formulas are built by atomic formulas. Which themselves combine to form first order theories. A first order theory that is satisfiable has a model M |= T. We derive Quantum logic from this and the CTMU derives a reality that is Quantum-geometrodynamically embedded. Where spatiotemporal containment defines the location of objects within time and space. As the objects move through time and space, they take their state-recognition and state-transformation syntaxes directly from this ambient spatiotemporal background. Now, what does this have to do with first order logic? Well, if you read the very first paragraph of what I posted from the Wikipedia article, it says that Universal Algebra provides the semantics for the signature of a formal language, whereas logic provides the syntax. So we have reality being logically Quantum-geometrodynamically embedded within itself using syntax.
Does this make sense?
Nicholas I. Hosein, 2016.
