# Chapter 5: Formal semantics

In this chapter, we describe a formal semantics for SL and for QL. The word ‘semantics’ comes from the greek word for ‘mark’ and means ‘related to meaning.’ So a formal semantics will be a mathematical account of meaning in the formal language.

A formal, logical language is built from two kinds of elements: logical symbols and non-logical symbols. Connectives (like ‘&’) and quantiﬁers (like ‘∀’) are logical symbols, because their meaning is speciﬁed within the formal language. When writing a symbolization key, you are not allowed to change the meaning of the logical symbols. You cannot say, for instance, that the ‘¬’ symbol will mean ‘not’ in one argument and ‘perhaps’ in another. The ‘¬’ symbol always means logical negation. It is used to translate the English language word ‘not’, but it is a symbol of a formal language and is deﬁned by its truth conditions.

The sentence letters in SL are non-logical symbols, because their meaning is not deﬁned by the logical structure of SL. When we translate an argument from English to SL, for example, the sentence letter $$M$$ does not have its meaning ﬁxed in advance; instead, we provide a symbolization key that says how $$M$$ should be interpreted in that argument. In QL, the predicates and constants are non-logical symbols.

In translating from English to a formal language, we provided symbolization keys which were interpretations of all the non-logical symbols we used in the translation. An interpretation gives a meaning to all the non-logical elements of the language.

It is possible to provide diﬀerent interpretations that make no formal diﬀerence. In SL, for example, we might say that $$D$$ means ‘Today is Tuesday’; we might say instead that $$D$$ means ‘Today is the day after Monday.’ These are two diﬀerent interpretations, because they use diﬀerent English sentences for the meaning of $$D$$. Yet, formally, there is no diﬀerence between them. All that matters once we have symbolized these sentences is whether they are true or false. In order to characterize what makes a diﬀerence in the formal language, we need to know what makes sentences true or false. For this, we need a formal characterization of truth.

When we gave deﬁnitions for a sentence of SL and for a sentence of QL, we distinguished between the object language and the metalanguage. The object language is the language that we are talking about: either SL or QL. The metalanguage is the language that we use to talk about the object language: English, supplemented with some mathematical jargon. It will be important to keep this distinction in mind.