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T6.3 4 of 6 The Definition of a PL Sentence Back on page one, we said that a sentence of PL is a particular kind of formula, one for which each variable has a quantifier. For example, (*) (%x)(Tx&Lx) is a sentence, but (**) (Tx&(%x)Lx) is not because the first instance of 'x' is without a quantifier in (**).) Now, we need to make this idea of "having a quantifier" precise. The scope of a quantifier in a PL formula P is the subformula of P for which that quantifier is main connective. So, in (*) the scope of the quantifier is the whole formula while in (**) the scope is only the second conjunct. An instance of a variable x in a formula is bound if and only if it is within the scope of an x-quantifier. Otherwise we say it is free. In (*) all instances of 'x' are bound, while in (**) the first instance is free. This is what makes (*) but not (**) a sentence of PL. A formula of PL is a sentence if and only if it has no free variables. Now, which of the following are sentences?
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