Philosophy
Philosophy Language Semantic Theory Of Truth
Quick fact
Alfred Tarski showed that for languages like arithmetic, truth can be defined purely in terms of satisfaction—a relation between sentences and objects—without using any undefined notion of truth, thus avoiding the liar paradox.
Why this is interesting
Have you ever said 'This sentence is false'? It's a paradox that makes truth seem impossible to pin down. Yet logicians found a way to define truth without falling into that trap—by learning to talk about language itself.
Read the full explanation
Understanding Philosophy Language Semantic Theory Of Truth
Imagine you're playing a game where you describe a room. You have a list of sentences like 'The lamp is on' or 'The chair is brown.' The truth of these sentences depends on the actual state of the room: they are true if they match what you see. Tarski applied this idea to formal languages. Instead of using an everyday notion of truth, he proposed that truth is defined by a rule: a sentence is true if it is satisfied by every object (or by the actual world). For example, 'x is a cat' is true for each particular cat. To define truth for a whole language, you need two languages: the 'object language' (the one you're talking about) and the 'metalanguage' (the language you use to talk about the object language). This distinction prevents paradoxes because you never apply the truth predicate to a sentence from the same language that contains that predicate in an uncontrolled way.
A deeper explanation
Tarski's mechanism works by first defining 'satisfaction' for open sentences (sentences with free variables). For instance, 'x is odd' is satisfied by objects like 1, 3, 5, etc. Then, a closed sentence (no free variables) is true if it is satisfied by all objects (or by the actual domain). This is done recursively: the truth of 'A and B' depends on the truth of A and B; the truth of 'not A' depends on the falsity of A. Crucially, Tarski introduced 'Convention T', which requires that for every sentence S in the object language, the metalanguage can prove: 'S' is true if and only if S. This captures the intuition that truth means corresponding to reality. By keeping the truth predicate in the metalanguage, Tarski avoids the liar paradox: 'This sentence is false' cannot be formed in the object language because it would require a self-referential truth predicate. This theory matters because it gives a rigorous, consistent definition of truth for formal systems, which became a cornerstone of model theory and influenced how philosophers think about truth. It also supports a correspondence intuition while showing that truth can be treated as a logical property, leading to deflationary views that see truth as a mere device for disquotation.