Truth: what type of things can be true; what makes them so?
Position: Logical subatomism
Truth is a central subject in philosophy. Any theory of truth should answer at least two fundamental questions
What kind of thing can be true or false? (the nature of truth-bearers)
What makes something true? (the nature of truth-makers)
Although its consequences range far beyond that field, a theory of truth belongs, at its core, to semantics, since truth-bearers tend to be semantic objects (like utterances, sentence-types, propositions…) or at least semantics-adjacent ones (beliefs, facts…). For instance, we have already seen that in Frege’s theory, the True and the False serve as the reference of propositions.
Theories of truth come in many flavours, but the metaphysically realist worldview I’ve been advocating is usually associated with one strand of thought. Painting with a broad stroke, we can describe it as the correspondence theory of truth. It can already be found, in a vague form, in the teachings of the Ancient Greeks. Aristotle proposed that “to say ‘it is’ about what is, or ‘it is not’ about what is not, is true”. To put it more clearly: what makes a statement true or false is the way that things are (or are not): every truth corresponds to an objective fact, or a state of affairs. In modern terms: a statement is true when the corresponding fact obtains. So if it is true that Sean is a physicist, it is by virtue of the corresponding fact (that Sean is a physicist).
The correspondence theory thus feeds into a broadly “objective” vision of truth. The reliance on facts (presumably abstract objects) encourages us to commit to their existence. And the bridge between semantics and metaphysic allows to import the logical laws which often are the underpinning of the latter. With classical propositional logic, this leads to a bivalent, non-contradictory truth.
This can lead astute readers to harbor concerns about the sheer adequacy of the concept. It is not clear at all that such commitments can help us build a coherent view of day-to-day assertions like "It is true that ’Penny Lane’ is a good song". This type of statement is called normative, and will be presented in the next section.
The most influential version of the correspondence theory is Tarski’s. The Polish logician dispensed with facts and states of affairs to ground truth in reference, satisfaction and logic. The theory follows from two simple principles.
The truth definition states that a sentence is true if and only if the corresponding formula is satisfied. This merits some development. The way this is often formalized is: \[\Vdash \phi \leftrightarrow \phi\] That can be misleading: Tarski insisted on the difference between the right-hand-side \(\phi\), which belongs to the object language and tells us something about the world, and the left-hand-side \(\phi\), which is a statement in the metalanguage. So even as we’re using the same symbol, or the same string of characters, on either side, we must remember they represent two distinct objects that just happen to be in correspondence.
A concrete example of the truth definition is that “snow is white” is true when snow is white. Due to the way we express this in a natural language (in this instance, English), this sounds like a tautology, but it’s not. A less natural but more precise version would be: the metalinguistic sentence “snow is white” is true when the object ‘snow’ satisfies the property ‘being white’. Every formula from the object language can be expressed as a sentence in the metalanguage, but the reverse is not true. That is actually a key point to consider. If the two languages were just mirror of each other, then the theory could be accused of redundancy. That position was defended by Frege. But what Tarski established rigorously is that any language that includes its own concept of truth as a predicate will inevitably face paradoxes. Confining truth to the metalanguage allows to dispel them.
Beyond the truth definition, the second element of Tarski’s model is the truth schema. It basically helps find the truth value of complex metalinguistic expressions by importing logical connectives and quantifiers from the object language’s syntax. For instance: \[”A \textrm{ and } B \textrm{ is true}” \leftrightarrow A \wedge B\], and so on. By using this schema recursively, the valuation of any metalinguistic sentence can in principle be reduced to multiple applications of the truth definition.
Marian David has called this approach “logical subatomism” as a way to highlight the filiation with Russell and Wittgenstein’s logical atomism. They designed a form of truth schema that allowed to walk back from complex expressions to atomic ones but they still promoted correspondence with facts and states of affairs. By relating the truth of atomics sentences to the satisfaction relation between two objects, Tarski could be said to have “split the atom”.
The system we have been building is very well-suited to Tarski’s theory of truth. It has been shown in Zalta’s work that \([\lambda \phi] \leftrightarrow \phi\) is a theorem. We can identify \([\lambda \phi]\) to truth-bearing propositions, that is to elements from the metalanguage, and \(\phi\) to the truth-making formula from the object language, landing exactly on Tarski’s general truth definition. The dual copula means that we have two kinds of atomic satisfaction relations: exemplification and encoding. As to the recursive truth schema, it can also be proved from the axioms.