Overview
The Problem Situation
The Tentative Solution
Critical Discussion
Correspondence
Content Logic
Class Logic
Logic of Arithmetic
Logic of Physics
Conclusion
Footnotes
Bibliography
Common sense makes a distinction between sentences and contents. The following two fictitious statements regarding the same book illustrate this distinction.
Karl: I found it difficult to follow the plot because my Spanish is not very good.
Socrates: I found it difficult to follow the plot because there were so many characters involved in the plot.
Karl’s insufficient knowledge of the language makes it difficult for him to grasp the content of the book. Socrates’s difficulty involved, supposedly, the content itself.
The problems posed by this distinction require attention. What is the relationship between contents and sentences? Do sentences designate contents, or are sentences and contents identical? What is the ontological status of contents? Are contents figments of human imagination, or are they objective structures that exist independently of anyone’s imagination? What is the relationship between contents and facts? Do sentences or contents correspond to facts, if they are true? Wherein lies this correspondence?
The identity thesis contains unacceptable consequences. Usually, double quotation marks are used to ascribe truth and falsity to sentences.
“Socrates is the wisest man” is true.
The above sentence, as a whole, assigns truth to a sentence. How can a sentence be true? The question about the correspondence between sentences and facts, in the sense of their similarity, seems impossible to answer. Moreover, if the identity thesis were true, two different sentences could not have the same contents.
There are unacceptable consequences of the identity thesis in another area as well. Theorems in propositional logic contain variables. The theorem—if (p and if p, then q), then q—contains the variables “p” and “q.” As a rule, constant names substitute variable names. In the mathematical theorem “a + b = b + a,” “a” and “b” are variable names of natural numbers. Substituting “4” for “a” and “5” for “b” leads to a true theorem of the calculus of natural numbers. Applied to the propositional theorem, however, this procedure does not lead to a true theorem. Substituting names of sentences, ““Socrates is the wisest man”” for “p” and ““Karl is ignorant”” for “q” leads to a result that is inadequate on formal grounds: If (“Socrates is the wisest man” and if “Socrates is the wisest man,” then “Karl is ignorant”), then “Karl is ignorant.” The mistake, one could assume, is to substitute names of sentences instead of sentences themselves. Substituting sentences seems straightforward but makes sentences names. As, for the identity thesis, sentences and contents are identical, sentences cannot name contents. What do sentences name? The identity thesis leaves this question unanswered.
Another possibility is to interpret “p” and “q” as variable names and sentences as constant names of contents. This assumption allows for the substitution of sentences for “p” and “q” and accepts contents as separate entities. Although this approach counters the above criticism, it leads to another insurmountable difficulty. Consider the following equivalence. 2
P is true, if and only if p.
The letter “p” on the right-hand side of this equivalence is, according to Alfred Tarski, the metalanguage occurrence of an object language sentence, whose metalanguage name must be substituted for the variable “P.” If sentences were names of contents, the sentence itself would be the substitute for “P” to attribute truth or falsity to its content. Thus, the assumption that sentences are names of contents is incompatible with Alfred Tarski’s interpretation of the above equivalence.
Gottlob Frege explicitly rejects the idea that sentences designate contents, but upholds the argument that sentences designate things. He transfers this idea from words to sentences and assumes that every sentence designates a truth-value and is associated with a sense that he identifies as its content. 3 Tarski’s theory of truth and Frege’s approach are irreconcilable because one uses sentences as names, and the other, names of sentences instead. Furthermore, Frege denies the correspondence theory of truth and assumes that truth is a property that might be beyond definition and full understanding. 4
Another tentative solution assumes that true sentences are names of facts or existing states of affairs. 5 Tarski’s semantic theory of truth focuses on sentences, and both approaches are reconcilable. The drawback is that it does not account for the relationship between contents and sentences. If sentences somehow express contents, this approach leaves them unnamed. Moreover, if sentences are names of facts, propositional theorems employing variables become theorems about facts, and propositional logic altogether vanishes.
These brief notes outline a problem situation that, allegedly, neither of the mentioned philosophical views solves. The solution to this problem situation surmounts these difficulties without causing new intractable problems and provides new tentative solutions to problems covered by its predecessors. 6
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]