Contents, Sentences, and Possibilities

Overview The Problem Situation The Tentative Solution
Critical Discussion Correspondence Content Logic Class Logic Logic of Arithmetic Logic of Physics
Conclusion Footnotes Bibliography

Content Logic

This section elaborates upon the idea that logic is the theory of contents, their logical forms, levels, and relations. It adapts Tarski’s semantic theory of truth to the philosophy of content logic, introduces three basic and one derived logical form, defines the deducibility of contents, and analyzes the logical structure of arguments. Finally, it studies how theorems of content logic are proven.

The semantic theory of truth is, according to Alfred Tarski, formulated in a language that speaks about sentences and that to which they refer. 13 Any sentence can be part of a partial definition of truth. 14

“Socrates is the wisest man” is true, if and only if Socrates is the wisest man.

The sentence on the left-hand side of this equivalence uses the name of a sentence to assign truth to it. The right-hand side consists of the metalanguage occurrence of the object language sentence whose name appears on the left-hand side.

The philosophy of content logic modifies this equivalence and opens the door to an entirely new approach to propositional logic and logic in general. The modified equivalence replaces the name of a sentence with the name of a piece of content.

‘Socrates is the wisest man’ is true, if and only if Socrates is the wisest man.

How does the philosophy of content logic interpret this modified equivalence? The name on the left-hand side designates a piece of object-level content, so the sentence on the left-hand side as a whole creates a piece of metalevel content. Therefore, the right-hand side of the sentence creates a piece of metalevel content too. Yet, there is an obvious difference. Whoever utters a sentence like “Socrates is the wisest man” claims that a piece of object-level content is true. The right-hand side of the modified equivalence makes this contention implicitly, while the left-hand side makes it explicitly. Because contents cannot state their own truth and, thus, claims to truth always refer to a lower level of content, all sentences and contents retain implicit claims on some level. The demand that all claims become explicit results in an infinite regression. The philosophy of content logic takes the modified equivalence as a way to switch between explicit and implicit representations of the same content. Applying the modified equivalence is nothing else but realizing a possibility inherent in language constructs and their environments.

The general form of the modified equivalence requires variable names of contents. The single quotation mark notation can be used for this purpose too, and the first general equivalence takes on the following form.

E1 ‘p’ is true, if and only if p.

The interpretation of “‘p’” is straightforward as a variable name of object-level contents; but “p” on the right-hand side seems to be an expression that must be substituted by a sentence. Since sentences are not names, “p” is not a sentence variable, and the status of “p” in E1 is unresolved. According to the above interpretation exemplified by individual sentences and contents, the right-hand sides of the equivalences create metalevel contents. This interpretation extends to “p.” Therefore, “p” represents the metalevel content ‘‘p’ is true’ implicitly. In short, “p” is an implicit and “‘p’” an explicit variable name of object-level contents.

The theory of logical forms and levels is at the core of the philosophy of content logic. The content ‘Socrates is the wisest man’ is in the logical form of a position. ‘Socrates is not ignorant’ is in the logical form of a negation. The position and the negation are basic logical forms of content logic. Other parts of logic—e.g., class logic—provide a fine-grained view of contents, but content logic cannot resolve the logical forms of the position and the negation any further. Logical forms in content logic are analogous to sentential functions in sentence logic. 15 Sentential functions contain so-called free variables. Sentences are derived from sentential functions either by binding free variables with quantifiers or by substituting constants for variables. Contents and forms, on the other hand, cannot be manipulated directly. To derive individual contents from logical forms, sentences must be modified. When variables substitute constants, the corresponding individual piece of content becomes a mere form. A piece of content is in the logical form F, if its sentence derives from F’s sentence by substitution. The semantic relationship of satisfaction prevails among forms and things, rather than sentential functions and things. 16 Three logical levels are important in the following analysis, and three different notations mark these levels. 17 Lowercase words “and,” “or,” “not,” “iff,” “if” characterize the object level; the symbols “∧,” “∨,” “→,” “↔” mark the metalevel; and uppercase words “AND,” “OR,” and “IF” represent the metametalevel.

The following two equivalences introduce the positive and the negative logical forms. “True” and “false” appear as undefined expressions.

E2 ‘p’ is true ↔ ‘not p’ is false.
E3 ‘p’ is false ↔ ‘not p’ is true.

The next primitive logical form is the conjunction.

E4 ‘p and q’ is true ↔ (‘p’ is true ∧ ‘q’ is true).

The content of the right-hand side—‘‘p’ is true ∧ ‘q’ is true’—is in the logical form of a conjunction on the metalevel. It is the explicit claim that two pieces of object-level content ‘p’ and ‘q’ are true. The claim to its own truth remains implicit. If it were made explicit, it would be a piece of metametalevel content. The content of the left-hand side—‘‘p and q’ is true’—is in the logical form of a position on the metalevel. It is the explicit claim that the piece of object-level content ‘p and q’ in the logical form of the conjunction is true. The claim to its own truth remains implicit too. Since E1–4 state the logical equivalence of metalevel contents, they are not definitions of logical symbols. The philosophy of content logic deals with contents first and only secondarily with expressions and symbols. More precisely, E4 defines the object level conjunction in metalevel terms. Formalizing a language, on the other hand, amounts to assigning logical forms to sentences with rules that employ names of sentences, expressions, and logical forms. 18 The syntactical form of a sentence in a formalized language indicates, therefore, the logical form of its content.

E5 and E6 derive the object level implication from previously introduced logical forms. E7 and E8 define it in metalevel terms.

E5 ‘if p then q’ is true ↔ ‘not (p and not q)’ is true.
E6 ‘if p then q’ is true ↔ ‘p and not q’ is false.
E7 ‘if p then q’ is true ↔ (‘p’ is false ∨ ‘q’ is true).
E8 ‘if p then q’ is true ↔ (‘p’ is true → ‘q’ is true).

The above and similar equivalences define all the logical forms of content logic. Applying E1 and E8 to theorems of content logic poses a problem. A theorem known as “modus (ponendo) ponens” serves as an example.

if p and (if p then q), then q.

Substituting “Socrates is the wisest man” for “p” and “Karl is ignorant” for “q” leads to the following sentence.

if Socrates is the wisest man and (if Socrates is the wisest man, then Karl is ignorant), then Karl is ignorant.

Applying E1 renders the result explicit.

‘if Socrates is the wisest man and (if Socrates is the wisest man, then Karl is ignorant), then Karl is ignorant’ is true.

Applying E8 results in an implication that makes the problem acute.

‘Socrates is the wisest man and if Socrates is the wisest man, then Karl is ignorant’ is true → ‘Karl is ignorant’ is true.

The question is whether contents in the implicative logical form claim deducibility. The above metalevel content seems to claim that two individual pieces of content are deducible because its sentence uses the word “true.” Not all implicative contents express deducibility, though. Two independent axioms form a counterexample. ‘if (p and q) then (if p then q)’ is a theorem of content logic. The conjunction of two independent axioms is true, and because of the above theorem, their implication is true too. Thus, ‘if p then q’ cannot state deducibility. Only implications whose negations entail contradictions—logically true implications—state deducibility. 19 The following equivalence defines deducibility for object-level contents.

‘q’ is a logical consequence of ‘p’ ↔ ‘if p then q’ is logically true.

This equivalence explains why not all implications state deducibility. Content logic is too coarse to establish the logical truth of a piece of content in the implicative logical form ‘if p then q.’

The definition of deducibility interprets theorems of content logic as contents claiming deducibility without stating how arguments apply these theorems. The following analysis of example arguments based on content logic answers this question.

Karl is ignorant, because Socrates is the wisest man.

This argument contains two pieces of object-level content: ‘Socrates is the wisest man’ and ‘Karl is ignorant.’ Content logic cannot prove deducibility for these contents because its logical means are too weak. Rephrasing the argument changes the situation.

Socrates is the wisest man.
if Socrates is the wisest man, then Karl is ignorant.
Therefore, Karl is ignorant.

This form of the argument separates premise and conclusion and does not claim that a piece of content in the positional logical form follows from another position, but implicitly that the position ‘Karl is ignorant’ follows from the conjunction ‘Socrates is the wisest man, and if Socrates is the wisest man, then Karl is ignorant.’

The equivalence ‘‘p’ is true ↔ p’ transforms this argument into the explicit form.

‘Socrates is the wisest man’ is true.
‘if Socrates is the wisest man, then Karl is ignorant’ is true.
THEREFORE, ‘Karl is ignorant’ is true.

The use of “THEREFORE” indicates implicit claims that metalevel contents are true and deducible. A piece of metametalevel content presented in mixed notation states these claims explicitly.

IF ‘‘Socrates is the wisest man’ is true’ IS TRUE AND ‘‘if Socrates is the wisest man, then Karl is ignorant’ is true’ IS TRUE, THEN ‘‘Karl is ignorant’ is true’ IS TRUE.

Its logical form is: ‘IF ‘P’ IS TRUE AND ‘Q’ IS TRUE, THEN ‘R’ IS TRUE.’ Applying a metametalevel equivalence changes its logical form.

‘‘if p then q’ is true’ IS TRUE IFF IF ‘P’ IS TRUE, THEN ‘Q’ IS TRUE.
IF ‘‘Socrates is the wisest man’ is true’ IS TRUE AND (IF ‘‘Socrates is the wisest man’ is true’ IS TRUE, THEN ‘‘Karl is ignorant’ is true’ IS TRUE), THEN ‘‘Karl is ignorant’ is true’ IS TRUE.

This metametalevel content is now in the logical form of modus (ponendo) ponens.

IF ‘P’ IS TRUE AND (IF ‘P’ IS TRUE, THEN ‘Q’ IS TRUE), THEN ‘Q’ IS TRUE.

The substitution is “P” = “‘Socrates is the wisest man’ is true,” “Q” = “‘Karl is ignorant’ is true.”

Since modus ponens is true, the argument is valid. Moreover, the analysis—in terms of the philosophy of content logic—has replaced rules of inference with metametalevel theorems that are applied by substitution. One could assume that each argument has its own specific metametalevel content, stating deducibility of the conclusion from the premise. The next steps show that this is not necessarily the case. The use of propositional theorems as premises reduces the number of rules to one. 20 In content logic, it is the number of metametalevel contents that is reduced to one. First, applying the equivalences transforms the piece of metametalevel content into its metalevel form. The revised argument is still based on modus ponens. Second, performing the procedure with a different argument shows that just one piece of metametalevel content is necessary. The following alternating sequence of contents, equivalences and substitutions transforms the piece of metametalevel content in the logical form of modus ponens into a piece of metalevel content.

IF ‘‘Socrates is the wisest man’ is true’ IS TRUE AND (IF ‘‘Socrates is the wisest man’ is true’ IS TRUE, THEN ‘‘Karl is ignorant’ is true’ IS TRUE), THEN ‘‘Karl is ignorant’ is true’ IS TRUE.
‘P → Q’ IS TRUE IFF IF ‘P’ IS TRUE, THEN ‘Q’ IS TRUE.
“P” = “‘Socrates is the wisest man’ is true,” “Q” = “‘Karl is ignorant’ is true.”
IF ‘‘Socrates is the wisest man’ is true’ IS TRUE AND ‘‘Socrates is the wisest man’ is true → ‘Karl is ignorant’ is true’ IS TRUE, THEN ‘‘Karl is ignorant’ is true’ IS TRUE.
‘P ∧ Q’ IS TRUE IFF ‘P’ IS TRUE AND ‘Q’ IS TRUE.
“P” = “‘ Socrates is the wisest man’ is true,”
“Q” = “‘Socrates is the wisest man’ is true → ‘Karl is ignorant’ is true.”
IF ‘‘Socrates is the wisest man’ is true ∧ (‘Socrates is the wisest man’ is true → ‘Karl is ignorant’ is true)’ IS TRUE, THEN ‘‘Karl is ignorant’ is true’ IS TRUE.
‘P → Q’ IS TRUE IFF IF ‘P’ IS TRUE, THEN ‘Q’ IS TRUE.
“P” = “‘Socrates is the wisest man’ is true ∧ (‘Socrates is the wisest man’ is true → ‘Karl is ignorant’) is true,”
“Q” = “‘Karl is ignorant’ is true.”
‘‘Socrates is the wisest man’ is true ∧ (‘Socrates is the wisest man’ is true → ‘Karl is ignorant’ is true) → ‘Karl is ignorant’ is true’ IS TRUE.
P IFF ‘P’ IS TRUE.
“P” = “‘Socrates is the wisest man’ is true ∧ (‘Socrates is the wisest man’ is true → ‘Karl is ignorant’ is true) → ‘Karl is ignorant’ is true.”
‘Socrates is the wisest man’ is true ∧ (‘Socrates is the wisest man’ is true → ‘Karl is ignorant’ is true) → ‘Karl is ignorant’ is true.

A similar procedure transforms such metalevel content in the logical form of modus ponus into a metalevel position in either explicit or implicit form.

‘if Socrates is the wisest man and (if Socrates is the wisest man, then Karl is ignorant), then Karl is ignorant’ is true.
if Socrates is the wisest man and (if Socrates is the wisest man, then Karl is ignorant), then Karl is ignorant.

The revised version of the first argument has an additional premise in the implicative logical form that can be derived from the above metalevel position with E8.

‘Socrates is the wisest man and if Socrates is the wisest man, then Karl is ignorant’ is true.
‘Socrates is the wisest man and (if Socrates is the wisest man, then Karl is ignorant)’ is true → ‘Karl is ignorant’ is true.
THEREFORE, ‘Karl is ignorant’ is true.

A piece of metametalevel content in the logical form of modus ponens states the deducibility of the conclusion from the premise. This time, the substitution is “P” = “‘Socrates is the wisest man and if Socrates is the wisest man, then Karl is ignorant’ is true,” “Q” = “‘Karl is ignorant’ is true.”

IF ‘‘Socrates is the wisest man and if Socrates is the wisest man, then Karl is ignorant’ is true’ IS TRUE AND (IF ‘‘Socrates is the wisest man and if Socrates is the wisest man, then Karl is ignorant’ is true’ IS TRUE, THEN ‘‘Karl is ignorant’ is true’ IS TRUE), THEN ‘‘Karl is ignorant’ is true’ IS TRUE.

The first step seems futile; no obvious simplification has been realized. On the contrary, introducing another premise complicates the matter. Yet modus ponens on the metametalevel still states that the conclusion of the revised version of the argument follows from its premise. Applying the procedure to a second argument based on a different initial theorem shows that just one piece of metametalevel content is necessary.

‘Socrates is the wisest man and Karl is ignorant’ is true.
‘Socrates is the wisest man and Karl is ignorant’ is true → ‘Karl is ignorant’ is true.
THEREFORE, ‘Karl is ignorant’ is true.

The true implication stating deducibility of the conclusion from the premise resides, as shown above, on the metametalevel.

IF ‘‘Socrates is the wisest man and Karl is ignorant’ is true’ IS TRUE AND (IF ‘‘Socrates is the wisest man and Karl is ignorant’ is true’ IS TRUE, THEN ‘‘Karl is ignorant’ is true’ IS TRUE), THEN ‘‘Karl is ignorant’ is true’ IS TRUE.

This piece of metametalevel content is in the same logical form, modus ponens, as the analogous content of the first example. Without the additional premise, the metametalevel implication is in the logical form below.

IF ‘P’ IS TRUE AND ‘Q’ IS TRUE, THEN ‘Q’ IS TRUE.

The substitution is “P” = “‘Socrates is the wisest man’ is true,” “Q” = “‘Karl is ignorant’ is true.” Therefore, only one piece of metametalevel content, modus ponens, is necessary to apply theorems of propositional logic.

So far, the analysis has focused on individual arguments. Proofs of content logic theorems are nothing else but arguments. The example discussed here consists of two axioms and one theorem. 21

A1 ‘if p then if q then p’ is true.
A2 ‘if (if p then (if p then q)) then (if p then q)’ is true.
T1 ‘if p then p’ is true.

The first substitution is “p” for “q” in A1 and A2.

A1' ‘if p then if p then p’ is true.
A2' ‘if (if p then (if p then p)) then p’ is true.

A piece of metametalevel content in the logical form of modus ponus states the deducibility of T1 from A1' and A2'. The substitution is “P” = “‘if p then if p then p’ is true,” “Q” = “‘if p then p’ is true.”

IF ‘‘if p then if p then p’ is true’ IS TRUE AND (IF ‘‘if p then if p then p’ is true’ IS TRUE, THEN ‘‘if p then p’ is true’ IS TRUE), THEN ‘‘if p then p’ is true’ IS TRUE.

If this claim to deducibility is made implicit, the proof of T1 assumes the form below.

A1' ‘if p then if p then p’ is true.
A2' ‘if p then if p then p’ is true → ‘if p then p’ is true.
T1' THEREFORE, ‘if p then p’ is true.

The above definition of deducibility uses the notion of logically true contents, defined as contents whose negations implied contradictions. All theorems of content logic are logically true in this sense. To demonstrate how these proofs appear in terms of content logic, first, theorem T1 is proven, and then modus ponens.

‘if p then p’ is false. (E3, E5)
‘p and not p’ is true.
THEREFORE, ‘if p then p’ is logically true.
‘if p and (if p then q), then q’ is false. (E3, E5)
‘p and (if p then q) and not q’ is true. (E4, E6)
‘p’ is true ∧ ‘p and not q’ is false ∧ ‘not q’ is true. (unspecified)
‘q’ is true ∧ ‘not q’ is true. (E4)
‘q and not q’ is true.
THEREFORE, ‘if p and (if p then q), then q’ is logically true.

In sum, the philosophy of content logic replaces rules of inference with metametalevel theorems. Simply put, there are no rules of inference. The use of metalevel equivalents reduces the number of necessary metametalevel contents to one. Analyzing arguments in terms of logical forms, levels, and substitutions suffices to describe both the application of theorems and their proofs. Metametalevel and metalevel equivalences are necessary to convert contents to different levels and forms. So far, the discussion of logical matters has remained within the range of content logic.

Overview The Problem Situation The Tentative Solution
Critical Discussion Correspondence Content Logic Class Logic Logic of Arithmetic Logic of Physics
Conclusion Footnotes Bibliography

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