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I've seen in the book "Logic for everyone" by Jason Decker that he writes the Propositions on the header of the tables as meta-Propositions. E.g. for the formula

A→C

π’œ π’ž π’œβ†’π’ž
T T T
T F F
F T T
F F T

When π’œ is the meta- sentence corresponding to A and π’ž corresponding to C. (Those words are mine, the author calls them metalinguistic variables)

I seen in other books that there is no use of meta-sentences when they write tables (for example in Kleene, Mathematical Logic).

I'm confused. What is the right way?

I have a tendency to think that a speech about the truth value of a Proposition is a metalinguistical reasoning, so the proposition as part of my metalinguistical reasoning is mentioned so I need to write π’œ

Additionally I don't know how to interpret the third header without quasi quotations.

But I can imagine that the frame, i.e. the table is itself a device in which the headers are a kind of quasi quotations and so it's correct to insert A and C without quotations or changing the font.

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  • Hi, my previous answer might have been wrong! I actually now don’t think it’s a question of quasi quoting, though it very much is still confusing! Commented Jul 2 at 10:33
  • Thanks so much @Sofie Ross . You writes inside the table Is cool(Blueberry) as it is written in the object language. Why? Commented Jul 2 at 10:34
  • So yes, our logic is the object language, the thing we’re using to explore symbolic logical connectives. I find it helps to use examples like β€œIsWhite(snow)” as individual propositions in the object language because they’re self contained little packets of meaning but still look artificial enough that we’re not going to mistake them for metalanguage meaningful sentences in English like β€œSnow is white” or β€œBlueberries are cool” :) Commented Jul 2 at 10:38
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    Obviously meta-. The truth table is not an expression of the formal language. And it can be applied to single formulas, that are expressions of the formal language, and to schema, that are recipes in the meta- used to produce formulas of the formal language. Commented Jul 2 at 11:09
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    @MauroALLEGRANZA the author calls them metalinguistic variables. I added it in the text. I wanted to show that some authors use the P, Q as they are written in the object language and he doesn't. Commented Jul 8 at 11:27

4 Answers 4

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Decker touches on this in pp20-22, but the root of his problem and your confusion comes down to the use of quotation marks where they would add clarity to the literal understanding of what he is trying to say.

Your column 3 I believe should be the quasi quote complex:

βŒœπ’œβ†’π’žβŒ

Metalanguage variables are variables that range over object language terms, such that you could substitute in object language strings (either an atomic logical sentence or a complex one) and talk about more complex abstract relationships between different parts of the object language.

Decker uses these variables to try to explain how the logical connectives of your object language work in terms of the semantic evaluation of their compositional parts.

In your example table above, the idea is that this is a truth table scheme for how we can approach sentences of the form βŒœπ’œβ†’π’žβŒ that works whatever instance of an object language string you put in there. This is the first table Decker typically uses when introducing logical connectives, because he’s saying something about general relationships between all the object languages strings using that connective and their fragments.

You can also construct truth table instances for specific object language compositions. For example:

IsBlue(blueberry) IsCool(blueberry) IsBlue(blueberry) β†’ IsCool(blueberry)
T T T
T F F
F T T
F F T

This is what we’d get by substituting a specific object language string β€œIsBlue(blueberry)β€œ in your metalanguage variable π’œ and β€œIsCool(blueberry)” for π’ž. This talks about a particular semantic evaluation of the complex sentence β€œIsBlue(blueberry) β†’ IsCool(blueberry)” which we can refer to because that sentence is of the form βŒœπ’œβ†’π’žβŒ.

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The added reference to Kleene's Mathematical Logic (1967) textbook allows me to add some more organized comments.

Page 3: the author introduces the term object language: the (formal) language to be studied by the logic discipline, and the observer's language ("usually called the 'metalanguage'"): the usual mathematical language=natural language+symbols, used by the logic discipline to discuss the properties of the object language.

Compare with a Latin grammer written in English: tthe "object language" is Latin while the "meta-" is English. When in the tex we find the exercise: "Translate the phrase 'Alea iacta est' ", we have an instruction to the reader expressed in the metalanguage regarding a sentence in the object language.

Page 4-5: the logic discipline will deal with certain (declarative) sentences of the object language that will be called formulas built from "some unambiguously constituted sentences [...] called prime formulas or atoms denoted by capital Roman letters as 'P', 'Q', 'R'...".

Starting from atoms, we can build composite formulas (or molecules) using the usual propositional connectives. Thus, formulas of the object language will be expressions like P (an atom is a formula), P β†’ Q, etc.

Having said that, the author gives a description (an informal specification) of the syntax of the formal language:

If each of A and B is a given formula, then A β†’ B, A & B, ∼A, etc are (composite) formulas.

The statement above is a statement in the metalanguage that gives us instructions how to build and parse formulas of the object language. A and B are metalinguistic variables (aka: schematic letters) that refer to formulas of the object language.

Written in "procedural langugae" the specifications say:

let A a formula whatever of the object language and let B a formula whatever (not necessarily distinct from the previous one) of the object language, then A β†’ B is a formula.

Thus, P β†’ Q and P β†’ P and (P & Q) β†’ R are formulas, while P β†’ & Q is not.

Page 9: truth tables are described using A, B, i.e. with meta-variables.

This means that e.g. the rightmost one (that for ∼) applies to formula ∼P and to formula ∼(P β†’ Q) as well.

See page 8: "[The truth] tables relate the truth value of each molecule to the truth value(s) of its immediate component(s)." It is not necessary that the immediate components are atoms.

A truth table is not an expression of the formal language. And, in general, quite all the text of a mathematical logic textbook is meta-, because it states facts (syntax, semantics) about the formal language.

In conclusion, there is no real difference (except for typographical aspects) with Decker's text; I've no acces to the text, but if Decker uses A,B,C as atoms, then π’œ, π’ž are clearly a metalinguistic variables.

Thus, wrt

"π’œ is the meta-sentence corresponding to A..."

the answer is: no. π’œ, π’ž,... stands for formulas and not only for atoms (sentential letters).

Regarding Quasi-quotation (aka: Quine corners), their use is a little bit tricky and IMO it is not necessary at the elementary level.

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Keep in mind that logic is symbolic. Whenever you see a proposition like 'A→C', 'A' and 'C' are merely symbols that can be assigned a truth-value. Those symbols might represent anything linguistic: elemental facts, simple propositions, complex statements, entire arguments, or anything at all that can reduce to a binary true/false assessment. There isn't really a distinction between sentences and meta-sentences, because everything is 'meta' by the nature of being symbolic. Decker is (apparently) trying to drive home the insight that this isn't restricted to simple statements, which is a common enough confusion among students.

This is a pedagogical tool, not something essential to logic.

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  • "There isn't really a distinction between sentences and meta-sentences, because everything is 'meta' by the nature of being symbolic." I'm not downvoting, but metalanguage has a distinct purpose that helps resolve ambiguity in an object language. In this way, a meta/object language distinction embody a very important act: interpretation. Thus, you can't have a compiler without a metalanguage. You can't have a type theory without a metalanguage. Commented Jul 8 at 16:32
  • @JD: I don't disagree. I'm just suggesting that logic is itself already a metalanguage: a way of structuring the processing of arbitrary language propositions. Commented Jul 8 at 18:20
  • Ah. I see. "logic is itself already a metalanguage" - I think it important to remember that logic is a mental process, and not a language of any sort. But I think it right to say that logical primitives qua processes are pre-linguistic. Quine drew a distinction between fragments of language called schema used in a metalanguage, and then the formal definition of logical primitives grounded in those schema in the object language. I would also caution that... Commented Jul 9 at 13:44
  • "Those symbols might represent anything LINGUISTIC: elemental facts, simple propositions..." Mauro's post cites en.wikipedia.org/wiki/Quasi-quotation which says "It does not make sense to quantify into quasi-quoted contexts using variables that range over things other than character strings". Here then, there's a distinction between symbols used as variables in general and symbols used specifically for linguistic variables such as those in a metalanguage to refer to the elements of the object language. But I'm just being picky. Commented Jul 9 at 13:46
  • @JD: Picky, yes, but rightly so. I'll edit that in. Commented Jul 9 at 16:42
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It looks like for Decker, truth tables are about formulas, whereas to other authors they are about truth values. In other words, Decker is saying, "If you have a formula of the form π’œβ†’π’ž where π’œ and π’ž are any formulas, then you interpret it according to this table: if the formula π’œ evaluates to true and the formula π’ž evaluates to true, then the formula π’œβ†’π’ž evaluates to true. (etc. for the other rows)."

Other authors treat it more like a mathematical expression. They don't have to talk about formulas because they are relying on their readers understanding how math works. When you write a multiplication table in mathematics, you don't say, "If you have a expression π’œxπ’ž where π’œ and π’ž are expressions, then if π’œ evaluates to 1 and π’ž evaluates to 1, then π’œxπ’ž evaluates to 1."

No, in math they just say "1 times 1 is 1", and they rely on you understanding that this applies to any formula, no matter how complex, that reduces to a multiplication and both sides evaluate to 1. Similarly, in a more standard logic text, they say T->T=T and rely on the reader to understand that this can be applied to any implication, no matter how complex, so long as the two sides both evaluate to T.

Another way to say this is that Decker seems to be talking about how to reduce formulas, whereas most logic texts would instead talk about how material implication operates on truth values.

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  • thanks so much for your answer. I'll try to understand it. Commented Jul 2 at 19:30

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