Overview
The Problem Situation
The Tentative Solution
Critical Discussion
Correspondence
Content Logic
Class Logic
Logic of Arithmetic
Logic of Physics
Conclusion
Footnotes
Bibliography
This section focuses on arithmetic theorems. First, it studies the relationship between an arithmetic theorem and an argument formulated in natural language. Second, it examines a proof of an arithmetic theorem in terms of content logic.
The new quantitative logical form simply gives the counted quantity of things. Usually, expressions like “the number of” or “seven” indicate this form in everyday language. The example consists of three pieces of quantitative content.
Socrates put one pearl of wisdom on the table.
Karl put three pearls of wisdom on the table.
Therefore, there are now four pearls of wisdom on the table.
There is a connection between this argument and the following axiom of arithmetic.
If x, y are numbers, there is a number z with x + y = z.
Since this theorem does not say anything about our argument, especially not that it is valid, the problem ahead involves examining this connection. More specifically, the problem is to reconstruct the logically true implicative content stating the deducibility of the conclusion from the premises. The simplest solution connects premises and the conclusion as antecedent and consequent, respectively.
If Socrates put one pearl of wisdom on the table and Karl put three pearls of wisdom on the table, then there are four pearls of wisdom on the table right now.
Is this piece of content logically true? Its logical form is ‘if p and q, then r.’ So, it is certainly not true from the perspective of content logic. Does it suffice to add the arithmetic theorem to the antecedent to make it true?
If Socrates put one pearl of wisdom on the table and Karl put three pearls of wisdom on the table, and for numbers x and y, there is the sum x + y, then there are four pearls of wisdom on the table right now.
Unfortunately, this is not enough. If pearls of wisdom, like drops of water, were to merge in close proximity, then the consequent would be false and the antecedent true. Therefore, the antecedent must state that pearls of wisdom behave in an additive way or, in other words, that arithmetic applies to them. The arithmetic theorem is not enough because it simply states that the number 1 + 3 = 4 exists, and not that the number of pearls on the table equals four. The logically true implication assumes, therefore, this form as follows.
If Socrates put one pearl of wisdom on the table and Karl put three pearls of wisdom on the table, and pearls of wisdom act like solid objects, so that their quantities add up, and for numbers x and y, there is the sum x + y = z, then there are four pearls of wisdom on the table right now.
This logically true implicative content can serve as one premise in an argument based on modus ponens. The argument’s conclusion would be the consequent of the implication. The implication has three kinds of premises conjoined in the antecedent. First, there are contents describing the specific situation at hand—the initial conditions; second, contents characterize the part of the cosmos in a general way and are called theories; and third, there are mathematical theorems. 23 These contents are independent of each other. For example, consider that the true conclusion states there are eight pearls of wisdom on the table, and the initial conditions are true. An alternative theory could assume that pearls of wisdom somehow multiply in close proximity with a factor of two applied to the sum of pearls in the area. The last section resumes the discussion of these matters. The focus here is the use of arithmetic theorems.
To sum up, arithmetic theorems appear as premises in arguments consisting of contents in the quantitative logical form. A different question is whether this logical form provides the proper means to describe the cosmos at all. Progress in human science consists in part of new logical forms and theorems covering the argumentation with contents in these forms. Once a new logical form is available to scientists they can explore the new range of possible theories.
Yet another problem is determining the role theorems of content logic play in proofs of arithmetic theorems. The following proof 24 begins with a theorem of content logic called “reductio ad absurdum” and deploys the arithmetic theorem ‘if x < y, then y ≮ x.’
‘if (if p, then not p), then not p’ is true. (E8)
‘if p, then not p’ is true → ‘not p’ is true. (“p” = “x < x.”)
‘if x < x, then x ≮ x’ is true → ‘x ≮ x’ is true.
‘if x < y, then y ≮ x’ is true. (“y” = “x.”)
‘if x < x, then x ≮ x’ is true.
THEREFORE, ‘x ≮ x’ is true.
A logically true piece of metametalevel content states the deducibility of the conclusion from the premises explicitly.
IF ‘‘x < x’ is true → ‘x ≮ x’ is true’ IS TRUE AND (IF ‘‘x < x’ is true → ‘x ≮ x’ is true’ IS TRUE,
THEN ‘‘x ≮ x’ is true’ IS TRUE), THEN ‘‘x ≮ x’ is true’ IS TRUE.
At first glance, this proof differs only slightly from a proof in syntactical terms. 25 The main difference is that metalevel theorems of content logic serve as premises in proofs of mathematical contents, and that one piece of metametalevel content, modus ponens, is necessary to make claims to deducibility explicit.
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]