Priest (1.1) An Introduction to Non-Classical Logic, ‘Introduction [to 1 Classical Logic and the Material Conditional]’, summary

 

by Corry Shores

 

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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other distracting mistakes, because I have not finished proofreading.]

 

 

 

Summary of

 

Graham Priest

 

An Introduction to Non-Classical Logic: From If to Is

 

1. Classical Logic and the Material Conditional

 

1.1. Introduction

 

 

 

Brief summary:

The main purpose of logic is to provide an account of validity, which determines what follows from what. The account is given in a metalanguage for a formal, object language. There are two types of validity: {1} Semantic validity (symbolized ⊨) which preserves truth: every interpretation that makes the premises true also makes the conclusion true. {2} Proof-theoretic validity (symbolized ⊢)  which is determined by means of a procedure operating on a symbolization of the inference. Most contemporary logicians think that semantic validity is more fundamental than proof-theoretic, but it is good nonetheless to provide a proof-theoretic notion of validity to correspond with a semantic notion. A proof-theory is sound when “every proof-theoretically valid inference is semantically valid (so that ⊢ entails ⊨)” and it is complete when “every semantically valid inference is proof-theoretically valid (so that ⊨ entails ⊢)” (4).

 

 

 

Summary

 

1.1.1

[First chapter half: basics of propositional logic.]

 

The first part of the chapter reviews classical propositional logic, along with semantic tableaux and basic terminology and notational conventions that will be used in the second part (3).

 

 

1.1.2

[Second half: conditional.]

 

The second half will examine the classical conditional along with its shortcomings.

 

 

1.1.3

[Logic is a matter of validity (which determines what follows from what), defined for a formal, object language.]

 

The main aim of logic is to give an account of validity, which is often defined for a formal language, considered the object language.

The point of logic is to give an account of the notion of validity: what follows from what. Standardly, validity is defined for inferences couched in a formal language, a language with a well-defined vocabulary and grammar, the object language. The relationship of the symbols of the formal language to the words of the vernacular, English in this case, is always an important issue.

(3)

 

 

 

1.1.4

[The account of validity is made in a metalanguage.]

 

But the account for an object language is made in a metalanguage that is often distinct from the object language. Here our metalanguage will be mathematical English (3).

 

 

1.1.5

[Two types of validity: {1} Semantic, which preserves truth: every interpretation that makes the premises true also makes the conclusion true. The metalinguistic symbol for semantic validity is: ⊨]

 

There are two sorts of validity: semantic and proof theoretic. Semantic validity is one where truth is preserved in the inference. [I am not sure I understand the notion of truth preservation. It might mean something like the following. We suppose we have sufficiently many true premises to make some inference on their basis. We also think that their truth in combination with certain logical structures or rules necessitates that the conclusion be true. So in a sense, in a semantically valid inference, the truth of the premises is preserved in the conclusion. Hence the definition that a semantically valid inference is one where it cannot be that the premises are true and the conclusion not true. In other words, a semantically valid inference is one where the truth of the premises guarantees the truth of the conclusion. Also, we will distinguish different logics by means of their notions of interpretation. Let me quote:]

It is also standard to define two notions of validity. The first is semantic. A valid inference is one that preserves truth, in a certain sense. Specifically, every interpretation (that is, crudely, a way of assigning truth values) that makes all the premises true makes the conclusion true. We use the metalinguistic symbol ‘⊨’ for this. What distinguishes different logics is the different notions of interpretation they employ.

(3)

 

 

1.1.6

[{2} Proof-theoretic validity, which is determined by means of a procedure operating on a symbolization of the inference. Its metalinguistic symbol is: ⊢]

 

Proof-theoretic validity is determined by means of a formal procedure performed on symbolic notation for the inference. Our method here employs tableaux. Priest says that the different tableaux procedures will distinguish the different types of logic (4).

The second notion of validity is proof-theoretic. Validity is defined in terms of some purely formal procedure (that is, one that makes reference only to the symbols of the inference). We use the metalinguistic symbol ‘⊢’ for this notion of validity. In our case, this procedure will (mainly) be one employing tableaux. What distinguish different logics here are the different tableau procedures employed.

(4)

 

 

1.1.7

[Semantic validity is more fundamental. A proof-theory is sound when “every proof-theoretically valid inference is semantically valid (so that ⊢ entails ⊨)” and it is complete when “every semantically valid inference is proof-theoretically valid (so that ⊨ entails ⊢)”]

 

Priest says that the consensus among contemporary logicians is that semantic validity is a more fundamental notion of validity than the proof-theoretic notion. But, he continues, it is useful to provide a proof-theoretic notion of validity corresponding to any semantic one, “in the sense that the two definitions always give the same answers”. A proof-theory is sound when all the proof-theoretically valid inferences [those determined as such by symbolic procedures] are also semantically valid [preserve the truth of the premises]. In that case, ⊢ entails ⊨. And if all semantically valid inferences are also proof-theoretically valid, then the proof-theory is said to be complete. In that case, ⊨ entails ⊢. [I am not sure why in both cases it is the proof-theory that is either sound or complete, and why not instead in the second case we do not say something like the semantic theory is complete, to make the formulation symmetrical.]

If every proof-theoretically valid inference is semantically valid (so that ⊢ entails ⊨) the proof-theory is said to be sound. If every semantically valid inference is proof-theoretically valid (so that ⊨ entails ⊢) the proof-theory is said to be complete.

(3)

 

 

 

 

 

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

.

Tarski (§9) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘9. Object-Language and Meta-Language’

by Corry Shores

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[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

9. Object-Language and Meta-Language

 

 

Brief Summary:

We are semantically defining truth using the (T) scheme:

(T) X is true if, and only if, p.

p is the sentence we are defining, while X is the name for that sentence, often given in quotation marks. For example,

“snow is white” if and only if snow is white.

The text in the quotation marks is in the language that the statement is talking about, and that language is called the object-language. The parts of that expression which are not in quotes and which are talking about the object-language are parts of what is called the meta-language.

Summary

Previously Tarski concluded that in order for our semantic theory to avoid the liar’s paradox, we cannot “employ semantically closed languages.” Instead, “we have to use two different languages in discussing the problem of the definition of truth and, more generally, any problems in the field of semantics” (341, both quotes from this section). The first language, called the object language, is the one we are talking about, while the second one, called the metalanguage, is doing that talking.

The first of these languages is the language which is “talked about” and which is the subject matter of the whole discussion; the definition of truth which we are seeking applies to the sentences of this language. The second is the language in which we “talk about” the first language, and in terms of which we wish, in particular, to construct the definition of | truth for the first language. We shall refer to the first language as “the object-language,” and to the second as “the meta-language.”
(341-342)

The terms ‘object-language’ and ‘meta-language’ have a relative sense, since if we in turn talk about some meta-language with another meta-language, then the first becomes the object-language.

It should be noticed that these terms “object-language” and “meta-language” have only a relative sense. If, for instance, we become interested in the notion of truth applying to sentences, not of our original object-language, but of its meta-language, the latter becomes automatically the object-language of our discussion; and in order to define truth for this language, we have to go to a new meta-language so to speak, to a meta-language of a higher level. In this way we arrive at a whole hierarchy of languages.
(342)

Tarski then explains how we obtain the vocabulary of the metalanguage. [It seems he is saying that the metalanguage is mostly what is contained in the (T) formulation. So it also includes everything as well in the object language, which will be stated in the (T) formulations. It is not explained yet what to do with the fact that ‘is true if and only if’ could easily be found in an object language like English or formal logic.]

The vocabulary of the meta-language is to a large extent determined by previously stated conditions under which a definition of truth will be considered materially adequate This definition, as we recall, has to imply all equivalences of the form (T):

      (T) X is true if, and only if, p.

The definition itself and all the equivalences implied by it are to be formulated in the meta-language. On the other hand, the symbol 'p' in (T) stands for an arbitrary sentence of our object-language. Hence it follows that every sentence which occurs in the object- language must also occur in the meta-language; in other words, the meta-language must contain the object-language as a part. This is at any rate necessary for the proof of the adequacy of the definition – even though the definition itself can sometimes be formulated in a less comprehensive meta-language which does not satisfy this requirement.

(The requirement in question can be somewhat modified, for it suffices to assume that the object-language can be translated into the meta-language; this necessitates a certain change in the interpretation of the symbol 'p' in (T). In all that follows we shall ignore the possibility of this modification.)
(342)

Because the (T) scheme needs names for sentences in the object-language, the meta-language should be rich enough to create names for all such sentences. As well, the meta-language needs basic logical terms like “if, and only if.”

Furthermore, the symbol 'X' in (T) represents the name of the sentence which 'p' stands for. We see therefore that the meta-language must be rich enough to provide possibilities of constructing a name for every sentence of the object-language.

In addition, the meta-language must obviously contain terms of a general logical character, such as the expression “if, and only if.”
(342)

The meta-language should not have undefined terms except ones in the object language, ones that refer to the object language expressions’ form, ones used to create names for those expressions, and logical ones.

It is desirable for the meta-language not to contain any undefined terms except such as are involved explicitly or implicitly in the remarks above, i.e.: terms of the object-language; terms referring to the form of the expressions of the object-language, and used in building names for these expressions; and terms of logic. In particular, we desire semantic terms (referring to the object-language) to be introduced into the meta-language only by definition. For, if this postulate is satisfied, the definition of truth, or of any other semantic concept, will fulfill what we intuitively expect from every definition; that is, it will explain the meaning of the term being defined in terms whose meaning appears to be completely clear and unequivocal. And, moreover, we have then a kind of guarantee that the use of semantic concepts will not involve us in any contradictions.
(343)

We do not need further requirements for the object and meta-languages. They should be like other known formalized languages.

We have no further requirements as to the formal structure of the object-language and the meta-language; we assume that it is similar to that of other formalized languages known at the present time. In particular, we assume that the usual formal rules of definition are observed in the meta-language.
(343)

 

Text:

Tarski, Alfred. “The Semantic Conception of Truth and the Foundations of Semantics”. In The Nature of Truth: Classic and Contemporary Perspectives. Michael P. Lynch, ed. Cambridge, Massachusetts / London: MIT, 2001, pp.331-363.

A hyperlinked online version can be found here:

http://www.ditext.com/tarski/tarski.html

The Lynch edited book writes this in the acknowledgments:

Alfred Tarski. “The Semantic Conception of Truth and the Foundations of Semantics.” Philosophy and Phenomenological Research 4 (1944). Copyright 1992 by the Estate of Alfred Tarski. Reprinted by permission of Jan Tarski.

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

Alfred Tarski (1944) The semantic conception of truth and the foundations of semantics (Reprinted as Chapter 4 of Martinich’s anthology). This is an abridged and updated version of his 1935 long paper Der Wahrheitsbegriff in den formalisierten Sprache (The concept of truth in formalized languages), itself a translation from his book in Polish of 1933.

And yet further bibliographical information from the German wiki page for Tarski

http://de.wikipedia.org/wiki/Alfred_Tarski:

Der Wahrheitsbegriff in den formalisierten Sprachen. In: Studia Philosophica. [Lemberg] 1 (1936), S. 261–405 (Vorabdruck datiert 1935).[4] Der Artikel ist eine deutsche Übersetzung der erstmals 1933 gedruckten polnischen Arbeit, die aber schon 1931 der Öffentlichkeit präsentiert wurde. Nachdruck in Karel Berka, Lothar Kreiser (Hrsg.): Logik-Texte. Kommentierte Auswahl zur Geschichte der modernen Logik. Akademie-Verlag, Berlin 1983, S. 445–546, in englischer Sprache in Tarski: Logic, Semantics and Metamathematics - papers from 1923 to 1938 by Alfred Tarski. Oxford 1956, 1983.

The German text can be found here:

http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf

 

Vergauwen, A Metalogical Theory of Reference, 1.3, §23

by Corry Shores
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[The following is summary. Paragraph headings are my own.]

Roger Vergauwen

A Metalogical Theory of Reference: Realism and Essentialism in Semantics

Chapter 1.3 Semantic closedness of Languages and the Paradox of the Liar

§23 Tarski's Truth Definition, Metalanguage & Object Language, and the Liar Paradox

Tarski's truth definition functioned well for formal languages. But he did not believe that the concept of truth could be applied to natural languages. In fact, he claimed it would inevitably produce confusions and contradictions.

We saw that a truth definition could be:

"snow is white" is true in English if and only if snow is white.

So truth definitions take the general form:

The sentence X is true (in L) if and only if p.

Notice the quotation marks in "snow is white." We are no longer considering what snow is and what white is. We are treating the whole proposition as one symbol or object.

In such a truth definition, there are two sorts of language being used.

a) object language: the language whose meaning we are explicating by means of truth definitions. We put the sentences of the object language in quotation marks. This turns the whole proposition into a single term, name, or object, as it were. It is treated as no more than a series of symbols. That particular series of symbols will be true under certain conditions.

b) metalanguage: those truth conditions are given in a language that has more expressive 'power' than the object language. It can say more, because it can talk about everything that is in the object language, and it can say things about the object language.

Consider that grass is green. The topic or subject that we are talking about here is grass. And, we see that "being green" is the predicament that the subject 'grass' is in. So 'is green' is the predicate for the subject, 'grass.'

In our truth definition, we say "the sentence X is true..."

Here, the subject of the truth definition is 'the sentence X.' But it is also in a predicament. It is predicated by "is true." So "is true" we call the truth predicate.

So consider our example

"snow is white" is true if and only if snow is white.

We see that the predicate is true is not a part of the quoted name for the object-language sentence. So the truth predicate must belong to the metalanguage. And we apply this predicate to the object sentence only under the given conditions. So if snow is not white, we would not predicate it with being true. But since snow is white, we say that "snow is white" is true.

Now consider that English is a possible object language. It itself contains the truth predicate "is true." In this case we cannot establish a strict division between object language and metalanguage. So we call such a language semantically universal.

One consequence of this is the liar paradox. Consider if someone says, "I am lying." If they are telling the truth, then they are lying. If they are lying, then they are telling the truth. Let's render such a statement into a different form. We'll create a statement that we call "sentence (1)." But that is just its name. Now we will say what Sentence (1) says:

(1):

Sentence (1) is false.

So Sentence (1) says of itself that it is false. So if we wanted to use a truth definition to give its meaning, we would have:

The sentence "sentence (1) is false" is true if and only if sentence (1) is false.

For (1) to be true, it must be false. But, if (1) is false, then it is true on account of the truth definition.

It appears then that we cannot use the object language as the metalanguage as well. For, this allows the object language to refer to itself, and then to contradict itself.

Tarski was pessimistic about using the truth definitions to formalize the syntax of natural language.

But there are reasons to think that it is still possible.

1) We think that the language of mathematics to be univocal and safe from paradox. However Gödel showed otherwise.

2) We can begin by having a limited version of the object language, then have a hierarchy of languages where one serves as a metalanguage for another one below it.

Later Vergauwen will show that it is possible to have a semantic theory based on Tarski's approach.

[Other Entries in this Series]

Vergauwen, Roger. A Metalogical Theory of Reference: Realism and Essentialism in Semantics. London: University Press of America, 1993.