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3. The above metalanguage can be viewed as a MetaEnglish because it is based on English. , we will always use MetaEnglish as our metalanguage. 4. What Tarski called metalanguage is close to what we call metalanguage but not quite the same. The difference is that Tarski allows metalanguage to contain the symbols of original object language written without quotation marks. So for him (but not for us), if the language is Formal, then the following is a metasentence: “∀x∃ys(x, y)” if and only if ∀x∃ys(x, y) Allowing the above to be a metasentence helped Tarski define truth in a language (the Tarski T scheme); we will not do this here.

Occurrences of these variables by t, s, ... to get a formula P xy... A more suggestive (but less precise) notation is as follows. We write P (x) instead of P and then we ts... write P (t) instead of P xt . ) instead of P xy... ), etc. notation from now on. Similarly if u is a term containing x and t is another term then one may replace all occurrences of x in u by t to get a term which we may denote by u xt ; if we write u(x) instead of u then we can write u(t) instead of u xt . And similarly we may ts replace two variables x, y in a term u by two terms t, s to get a term u xy , etc.

One gives a similar metadefinition for witnesses c , c , c , ... ), etc. 8. Explicitly if P (x, y) is a formula with free variables x, y the witnesses c , c for ∃x∀yP (x, y) are given by: c c = c∀yP (x,y) , = cP (c∀yP (x,y) ,y) . 9. If we deal with languages with witnesses we will always tacitly assume that all translations are compatible (in the obvious sense) with the witness assignments. Compatibility can be typically achieved as follows: if one is given a translation of a language L0 into a language L0 then this translation can be extended uniquely to a translation, compatible with witness assignments, of the witness closure L of L0 into the witness closure L of L0 .