Every definition is a proposition where the predicate and subject are interchangeable. Every axiom is a proposition where the predicate and subject are not interchangeable. Both are assumptions considered to be undeniable or incredibly agreeable. Every other proposition is part of the reasoning following the definitions and axioms-but not undeniable or incredibly agreeable.
Every proposition will consist of two to four parts: the existential import, the claim, the proof, and a remark. Every existential import is the necessity of the existence of a predicate or subject within a proof. Every claim is what the author intends to prove. Every proof is the demonstration of the truth of a claim. Every remark is additional information about the proposition contextualizing the content. A proposition does not need an existential import or a remark; the other two elements will always be present.
Despite the formatting, similar to the geometrical style, not all claims will be proven deductively; the texts will use inductive, analogical, and abductive arguments. The text will point out when a proof does not use deductive reasoning. The text will point out connectives within a rationale due to natural language's vagueness.
Every proof will use classical truth-functional propositional and traditional logic to demonstrate the truth of a proposition deductively. Inductive arguments will only take the form of statistical syllogisms. Furthermore, definitions can switch their predicate and subject within the proof without being mentioned. The text may shorten axioms and definitions without being noted. The text may not swap the definition's predicate and subject when shortened. The text may elongate an original definition. The text will not explicitly state greater length.
Proposition 1.1. SOME DOGS ARE HUSKIES.
Given some huskies are existing.
Proof. Every husky is a mammal of Canidae that has been domesticated for thousands of years [Axiom 1.1]. Every mammal of Canidae that has been domesticated for thousands of years is a dog [Def. 1.1].
Remark. The proposition's title is the claim-which is trying to be demonstrated. The first sentence after the proposition number and title is the existential import; the reasoning given necessitates the existence of huskies. The subsequent paragraph is denoted as the proof-what demonstrates the claim. After the proof, a remark is denoted to explain or contextualize a proposition.
Proposition 1.2. EVERY HUSKY IS A DOG.
Proof. Every mammal of Canidae that has been domesticated for thousands of years is a dog [Def. 1.1]. Every husky is a mammal of Canidae that has been domesticated for thousands of years [X.X].
Remark. Within this example, the existential import is missing since the reasoning does not necessitate the existence of some husky. The axioms [Axiom 1.1], definition [Def. 1.1], and proposition [X.X] cited are a way to make the proof more accessible. It is a way for the reader to check the homework, so to speak, of the author's demonstration.
Every S, M, and P are terms. Every H is itself a proposition.
Every S is M. Every S2 is M and P. Therefore, every S is P.
Every S is M. Candidate explanations of every S is M are: H1,...,Hn. If Hi explains every S is M better than H1,...,Hn, then Hi. Hi explains every S is M better than H1,...,Hn. Therefore, Hi.
If H, then -H. Therefore, -H.
If -H, then H. Therefore, H.