Overview
The Problem Situation
The Tentative Solution
Critical Discussion
Correspondence
Content Logic
Class Logic
Logic of Arithmetic
Logic of Physics
Conclusion
Footnotes
Bibliography
To extend the logical means to class logic, this section introduces one new logical form along with two relations between classes, studies the proof of a theorem, and, finally, examines an individual argument based on a theorem of class logic.
Content in the new subclass form states that the subclass relation holds between two classes. A sentence creating content in this logical form must contain names of classes and a symbol for the subclass relation. If “K,” “L,” and “M” are constant names of classes, and “⊂” indicates the subclass relation, the sentences “K ⊂ L,” “L ⊂ M,” and “K ⊂ M” can substitute “p,” “q,” and “r” in axioms and theorems of content logic. Although ‘if p and q, then r’ is not a theorem of content logic, the substitution “p” = “K ⊂ L,” “q” = “L ⊂ M,” and “r” = “K ⊂ M” leads to a true theorem of class logic.
A If K ⊂ L and L ⊂ M, then K ⊂ M.
The following proof is based on two axioms of class logic. 22 Both axioms and the theorem use an operation called “union,” which yields a class when applied to two classes. Its symbol is “∪,” so that “K ∪ L” names the union of the classes “K” and “L.” The conclusion of the proof is theorem T.
B ‘K ∪ L ⊂ M, iff K ⊂ M and L ⊂ M’ is true.
C ‘K ⊂ K’ is true.
T ‘K ∪ K ⊂ K’ is true.
The leading questions of the following analysis are: Which theorems of content logic are necessary to prove theorem T? How does the proof apply these theorems?
B ‘K ∪ L ⊂ M, iff K ⊂ M and L ⊂ M’ is true. (“L” = “K,” “M” = “K.”)
B' ‘K ∪ K ⊂ K, iff K ⊂ K and K ⊂ K’ is true.
B' is in the logical form of a position at the metalevel. The named object-level content is in the form of an equivalence. According to the equivalences introduced in the previous section, this metalevel position is equivalent to B''.
B'' ‘K ∪ K ⊂ K’ is true ↔ ‘K ⊂ K and K ⊂ K’ is true.
The left-hand side of B'' is identical to T. The next step is to prove the right-hand side of B'' with the help of C and a theorem of content logic.
TC ‘if p, then (p and p)’ is true. (“p” = “K ⊂ K.”)
TC' ‘if K ⊂ K, then (K ⊂ K and K ⊂ K)’ is true.
This position corresponds to the following implication.
TC'' ‘K ⊂ K’ is true → ‘K ⊂ K and K ⊂ K’ is true.
The antecedent of this implication is the second axiom C. Applying TC'' implicitly leads to the right-hand side of B''.
C ‘K ⊂ K’ is true.
THEREFORE, ‘K ⊂ K and K ⊂ K’ is true.
Applying TC'' explicitly results in a different argument.
C ‘K ⊂ K’ is true.
TC'' ‘K ⊂ K’ is true → ‘K ⊂ K and K ⊂ K’ is true.
THEREFORE, ‘K ⊂ K and K ⊂ K’ is true.
The piece of metametalevel content stating the deducibility is in the form of modus ponens.
IF ‘P’ IS TRUE AND (IF ‘P’ IS TRUE, THEN ‘Q’ IS TRUE), THEN ‘Q’ IS TRUE.
“P” = “‘K ⊂ K’ is true,” “Q” = “‘K ⊂ K and K ⊂ K’ is true.”
IF ‘‘K ⊂ K’ is true’ IS TRUE AND IF ‘‘K ⊂ K’ is true’ IS TRUE, THEN ‘‘K ⊂ K and K ⊂ K’ is true’ IS TRUE, THEN ‘‘K ⊂ K and K ⊂ K’ is true’ IS TRUE.
The last step of the proof is based on a different theorem of content logic.
‘if q and (p, iff q), then p’ is true.
“p” = “K ∪ K ⊂ K,” “q” = “K ⊂ K and K ⊂ K.”
‘if K ⊂ K and K ⊂ K and ( K ∪ K ⊂ K, iff K ⊂ K and K ⊂ K), then K ∪ K ⊂ K’ is true.
‘K ⊂ K and K ⊂ K and (K ∪ K ⊂ K, iff K ⊂ K and K ⊂ K)’ is true → ‘K ∪ K ⊂ K’ is true.
The antecedent of this implication is the conjunction of the first conclusion—‘K ⊂ K and K ⊂ K’—with B'; the consequent is theorem T. The substitution “P” = “‘K ⊂ K and K ⊂ K and (K ∪ K ⊂ K, iff K ⊂ K and K ⊂ K)’ is true,” “Q” = “‘K ∪ K ⊂ K’ is true” in metametalevel modus ponens leads to a metametalevel content stating deducibility of T from suitable premises.
In sum, when proofs in class calculus use theorems of content logic explicitly, they use modus ponens in its metametalevel form implicitly. The remaining part of this section examines the relationship between theorems of class calculus and arguments.
All politicians are humans.
All humans are philosophers.
Thus, all politicians are philosophers.
This argument claims truth for its premises and deducibility of its conclusion from the premises implicitly. The general theorem involved is A.
A if K ⊂ L and L ⊂ M, then K ⊂ M.
Substituting constants for variables in A leads to the specific theorem A'.
A' if all politicians are humans and all humans are philosophers, then all politicians are philosophers.
Applying the equivalences previously introduced results in a piece of metalevel content claiming that a piece of object-level content in the positional logical form follows from a piece of object-level content in the conjunctive logical form.
A'' ‘All politicians are humans and all humans are philosophers’ is true →
‘All politicians are philosophers’ is true.
A'' and the explicit version of the example argument form a new argument.
‘All politicians are humans and all humans are philosophers’ is true.
‘All politicians are humans and all humans are philosophers’ is true →
‘All politicians are philosophers’ is true.
THEREFORE, ‘All politicians are philosophers’ is true.
The above premises and their conclusion are metalevel contents that state the truth and deducibility of object-level contents explicitly. The following metametalevel content in the logical form of modus ponens expresses the deducibility of the conclusion from the premises explicitly.
IF ‘P’ IS TRUE AND (IF ‘P’ IS TRUE, THEN ‘Q’ IS TRUE), THEN ‘Q’ IS TRUE.
“P” = “‘All politicians are humans and all humans are philosophers’ is true,”
“Q” = “‘All politicians are philosophers’ is true.”
IF ‘‘All politicians are humans and all humans are philosophers’ is true’ IS TRUE
AND (IF ‘‘All politicians are humans and all humans are philosophers’ is true’ IS TRUE, THEN ‘‘All politicians are philosophers’ is true’ IS TRUE), THEN ‘‘All politicians are philosophers’ is true’ IS TRUE.
In summary, one piece of metametalevel content in the logical form of modus ponens suffices to analyze proofs and applications of theorems of class calculus.
Overview
The Problem Situation
The Tentative Solution
Critical Discussion
Correspondence
Content Logic
Class Logic
Logic of Arithmetic
Logic of Physics
Conclusion
Footnotes
Bibliography
[Download a pdf version: Contents, Sentences, and Possibilities]