## 2 Apr 2016

### Agler (2.3) Symbolic Logic: Syntax, Semantics, and Proof, "Syntax of PL," summary

by Corry Shores
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[The following is summary. Boldface (except for metavariables), underlining, and bracketed commentary are my own. I highly recommend Agler’s excellent book. It is one of the best introductions to logic I have come across.]

Summary of

David W. Agler

Symbolic Logic: Syntax, Semantics, and Proof

Ch.2: Language, Syntax, and Semantics

2.3 Syntax of PL

Brief Summary:
Propositions in the language of propositional logic (PL) must be formed according to specific syntactical rules, and when they are, we consider them to be “well-formed formulas” or “wffs”. We form wffs by using uppercase Roman numerals, which can be modified, combined, and organized using logical operators ∨, →, ↔, ¬, ∧ or scope indicators like parentheses. The propositions within the language of PL are in our object language, and the English sentences we use to discuss these  object language propositions is part of our metalanguage. In our metalanguage we might use metavariables, in this case, boldface Roman letters, to stand for any propositions in the object language. Using such metavariables and a metalanguage, we can state the rules for properly constructed wffs. Briefly, if a formula is an atomic formula, it is a wff. Or, if a proposition takes a negation symbol to its left, it is a wff. Or, if two formula have the symbols ∨, →, ↔, or ∧ placed between them, it is a wff. Nothing else, however, can be considered wffs. The literal negation of a proposition is the negation of the whole proposition; this means it could be a double negation in some cases, and in others the negation will go outside parentheses enclosing a complex proposition.

Summary

2.3 Syntax of PL

There are right and wrong ways to structure sentences in English. For example, “John is tall” and “Vic is short” are grammatical sentences, but “John is tall, Vic is short, and” is not. We will now examine well-formed formulas in PL.

2.3.1 Metalanguage, Object Language, Use, and Mention

The language we are talking about is the object language, and the language we are using to talk about the object language is the metalanguage. So far we have been using the metalanguage of English to talk about the object language of PL.

When we use a term in such a way that we are speaking about its referent, we say that this is a use of the term. For example, when we say, “John is looking rather sluggish today,” our predicate refers to John himself. But if we use a term in such a way that we are speaking about that term itself as the term that it is, then we say that we mention the term. For example, “ ‘John’ has four letters in it.” As you can see, we often put “scare-quotes” around the term to indicate that it is a mention and not a use of the term (38-39).

Agler notes that there are two ways that we indicate when a term is mentioned rather than used. The first, as we said, is with quotation marks, and Agler says we use the single quote marks (39).

The second way is “by putting the expressions on display” as for example when we give a new line just for the expression itself (39).

2.3.2 Metavariables, Literal Negation, and the Main Operator

We will need to make general statements for broad categories of things in PL. Agler gives an example of a way that will not work.

Suppose that you want to say that for any proposition in PL (atomic or complex), if you put a ‘¬’ in front of it, you will get a negation. Now let’s suppose that you try to do this as follows:

If ‘P’ is a proposition, then ‘¬P’ is a negation.

If this is your formulation of such a rule, you will fall miserably short of your goal.
(39)

The problem with this option is that it implies we are only talking about atomic proposition P and not any possible proposition whatsoever in PL (39).

Instead we need to use metalinguistic variables (also called metavariables).

A metavariable is a variable in the metalanguage of PL (not an actual part of the language of PL) that is used to talk about expressions in PL. In other words, metavariables are variables in the metalanguage that allow us to make general statements about the object language like the one we are currently aiming at.
(40)

Agler will use boldface on the uppercase Roman letters to indicate a metavariable (in other texts, I think, sometimes Greek letters are used for this purpose.)

Agler then notes two things about metavariables: a) “metavariables are part of the metalanguage of PL. This means that they will be used to talk about PL and are not part of PL itself” (40). b) “metavariables are variables for expressions in the object language” (40).

We now reformulate our generalization, this time using metavariables:

If ‘P’ is a proposition in PL, then ‘¬P’ is a negation.

Agler notes that metavariable “ P’ is not part of the vocabulary of PL. Instead, it is used to refer to any proposition in PL (e.g.,‘A,’‘¬A,’‘A→ B,’‘¬A→B,’etc.)” (40, underlining mine).

2.3.3 The Language and Syntax of PL

Agler now gives a more rigorous characterization of the language of PL. It consists of

1) Uppercase Roman (unbolded) letters. They may or may not have subscripted numerals:

A1, A2, A3, B, C,. . ., Z

2) Five truth functional operators, namely:

∨, →, ↔, ¬, ∧

3) Three scope indicators:

(), [], {}

(41)

We form other complex propositions by combining these three elements, and we must do so in accordance with the grammar of PL. When a proposition is formed correctly syntactically speaking, we call it a well-formed formula or wff (pronounced ‘woof’). Agler then lists the following formation rules for constructing wffs:

1) Every propositional letter (e.g., ‘P,’ ‘Q,’ ‘R’) is a wff.
2) If ‘P’ is a wff, then ‘¬P’ is a wff.
3) If ‘P’ and ‘R’ are wffs, then ‘(PR)’ is a wff.
4) If ‘P’ and ‘R’ are wffs, then ‘(PR)’ is a wff.
5) If ‘P’ and ‘R’ are wffs, then ‘(PR)’ is a wff.
6) If ‘P’ and ‘R’ are wffs, then ‘(PR)’ is a wff.
7) Nothing else is a wff except what can be formed by repeated application of rules (1)–(6).
(Agler 41)

The first rule establishes that any uppercase Roman letter (that is not in bold) is an atomic proposition. The second through the sixth rules specify how we can build more complex formula from simpler ones. And the seventh rule states that only formula made by these rules (and their repeated instantiations) are well- formed (41).

Thus formula not made by these rules are not well-formed. Agler will now show how we evaluate whether or not a formula is a wff. We begin with

P→¬Q

First he establishes that all the atomic letters in the formula are wffs, thereby satisfying the first rule.  Then we find the rules for their more complex modifications and combinations. We can enumerate the steps to show the evaluation process.

1     P and Q are wffs.          Rule 1
2    ¬Q is a wff.                       Line 1 + rule 2
3    P→¬Q is a wff.               Line 1, 2 + rule 5
(Agler 42)

Agler then gives a more complex example: show that P→(R∨¬M) is a wff:

1     P, R, M are wffs.              Rule 1
2     ¬M is a wff.                        Line 1 + rule 2
3     R∨¬M is a wff.                 Line 1, 2 + rule 4
4     P→(R∨¬M) is a wff.      Line 1–3 + rule 5
(Agler 42)

2.3.4 Literal Negation and the Main Operator

Literal negation occurs when an entire proposition (including complex ones) is negated. In the formulation below, we see how metavariables can be useful for defining literal negation.

Literal negation: If ‘P’ is a proposition in PL, the literal negation of a proposition ‘P’ is the proposition of the following form ‘¬P.’
(42)

Thus we would have the following literal negations:

P                   >    ¬P
¬Q                >   ¬¬Q
W→R           >   ¬(W→R)
P∧(R∨S)    >   ¬(P∧(R∨S)) or ¬[P∧(R∨S)]

(Agler 43)

Another use for metavariables is for specifying what a proposition’s main operator is.

1) If ‘P’ is an atomic proposition, then ‘P’ has no truth-functional operators and so has no main operator.
2) If ‘Q’ is a proposition, and if ‘P’ is of the form ‘¬Q,’ then the main operator of ‘P’ is the negation that occurs before ‘Q.’
3) If ‘Q’ and ‘R’ are both propositions, and if ‘P’ is of the form ‘QR,’ ‘QR,’ ‘QR,’ or ‘QR,’ then the main operator of ‘P’ is the truth-functional operator that occurs between ‘Q’ and ‘R.’ These are ‘∧,’‘∨,’‘→,’ and ‘↔,’ respectively.
(Agler 43)

So consider (P∧Q)→¬Z. It has the form QR. Thus its main operator is →. (43)

Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.

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