In writing an appendix to WIKAL I set myself the task of cataloguing all the ways in which the statements collected in WIKAL are wrong.
The statements collected in ‘What I Know About Logic’ are wrong in a lot of different ways, cataloguing all of these ways is a daunting task, and may result in the appendix turning into a case-specific handbook of rhetorical subterfuge, or a system for the assemblage and presentation of non-knowledge, or an experiment in notation, or an amalgamation of diary and bibliography genres.
For now though all I’ve done is provide a list of the assertions which function as crux points in the piece’s ‘argument’ along with the refutations, or negations pertinent to each of them. These refutations take the form of facts, the set of facts that WIKAL can be variously diagnosed as misrepresenting, misremembering, disfiguring or, if we have decided to be more forgiving in our assessment, paraphrasing. It should provide a picture of the fundamental tissue of inaccuracies running through the piece, leaving it for the reader to identify the lines of consequential reverberation they extend through the rest of ‘What I Know About Logic’, to extract the scaffolding of falsehoods erected as compensation or entrenchment or support around them. The facts (or the particular words in which they are expressed) are taken from Wikipedia entries (it may be worth noting that Wikipedia is itself a canonical and notorious example of the scandalous instability of the internet-as-knowledge-corpus).
It (FOL) was the development required to make modern mathematical discoveries expressible or tractable in the language of logic, it allowed the two disciplines to ‘speak to one another.’
First-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics.
The share the same symbols and operations, but certain rules governing the course of deduction in FOL don’t apply in SOL.
First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations… Second-order logic also includes quantification over sets, functions, and other variables.
it is only within the deductive environments furnished by both FOL and SOL that these systems can work at all.
There are many deductive systems for first-order logic which are both sound (i.e., all provable statements are true in all models) and complete (i.e. all statements which are true in all models are provable).
Before the creation of FOL logical arguments were hampered by an inability to take differences in quantity into account
First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as “Socrates is a man”, one can have expressions in the form “there exists x such that x is Socrates and x is a man”, where “there exists” is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations.
FOL is a synthesis of these two men’s achievements; Frege’s in the field of Logic, Cantor’s in Mathematics.
The foundations of first-order logic were developed independently by Gottlob Frege and Charles Sanders Peirce.
For an object to Belong to a set every component into which it can be broken must also belong to the set, it must be able to resist recombination as a different set no matter how it is broken apart. This fidelity isn’t required of components that are included in a set, Inclusion is a more relaxed affair, come and go as you please.[Belonging is a relation between a Set and the elements or objects it contains] Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, the notation o ∈ A [o belongs to A] is used… [Inclusion is a relation between a Set and the other Sets it contains, its Subsets] If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B [A is included in B].
It was an encompassing and expository book, entrenching the mathematisation of logic. In this book they also introduced their own contribution to this continuing program