[The following is summary. Boldface, 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.]
David W. Agler
Symbolic Logic: Syntax, Semantics, and Proof
Ch.2: Language, Syntax, and Semantics
2.2 The Symbols of PL and Truth-Functional Operators
The language of Propositional Logic (PL) is composed of (a) upper-case Roman letters (sometimes numerically subscripted) for propositions, (b) the truth-functional operators ∨, →, ↔, ¬, and ∧, and (c) scope indicators, namely, parentheses, brackets, and braces. A conjunction ∧ is true only if both conjuncts are true, and it is false otherwise. Negation ¬ inverts the truth value of the proposition. When determining the true value of a complex formula, we determine the operated values beginning with the operator with the least scope and work toward the one with the greatest scope, called the “main operator”. It can be determined by finding the operator that operates directly or indirectly on all other sentence parts.
Agler will continue detailing the language of propositional logic (PL). He begins by listing the basic symbols of this language:
1 Uppercase Roman (unbolded) letters with or without subscripted integers (‘A1,’ ‘A2,’ ‘A3,’ ‘B,’ ‘C,’ ..., ‘Z’) for atomic propositions
2 Truth-functional operators (∨, →, ↔, ¬, ∧)
3 Parentheses, braces, and brackets to indicate the scope of truth-functional operators
We use capitalized Roman letters to abbreviate atomic propositions. So “John is grumpy” could be abbreviated as: J. We can use subscripts if we have many sentences we are working with.
We then can add truth functional operators to these symbols to make them more complex (30).
Agler introduces the conjunction operator:
In the language of PL, where ‘P’ is a proposition and ‘Q’ is a proposition, a proposition of the form
is called a conjunction. The ‘∧’ symbol is a truth-functional operator called the caret. Each of the two propositions that compose the conjunctions are called the proposi- | tion’s conjuncts.
Agler then defines the truth function for the caret symbol/conjunction operator as follows:
Conjunction = df. If the truth-value input of both of the propositions is true, then the complex proposition is true. If the truth-value input of either of the proposition is false, then the complex proposition is false.
Agler provides the following table for conjunction’s truth-functional input-output schema:
As we can see, the conjunction is only true when both conjuncts are true.
For the most part, the best translation of the caret in English is “and” (31).
However, there are other ways it can be found written in English.
Both John and Liz are happy. (H∧L)
Although Liz is happy, John is grumpy. (L∧G)
Liz is happy, but John is grumpy. (L∧G)
In PL, the negation of ‘P’ is written: ¬P. Agler defines its truth function as:
Negation = df. If the truth-value input of the proposition is true, then the complex proposition involving ‘¬’ is false. If the truth-value input of the proposition (atomic or complex) is false, then the complex proposition involving ‘¬’ is true.
And here is the truth table.
the negation function changes the truth value of the proposition it operates upon. If ‘M’ is true, then ‘¬M’ is false. And if ‘M’ is false, then ‘¬M’ is true. To put this in plain English, if a proposition is true, adding ‘¬’ to it changes it to false. If the proposition is false, adding ‘¬’ to it changes it to true.
Agler says that the best way to translate negation into English is to use “not” or “it is not the case that”. For example:
Liz is not happy. (¬L)
It is not the case that John is grumpy. (¬G)
It is false that Mary is a zombie. (¬Z)
Agler notes that when converting English sentences to PL, we should preserve as much of the underlying structure as we possibly can. So while it is possible to translate “John is not tall” as J, we should really translate it as ¬J. This allows us to deal with the truth-functional operations at work.
2.2.3 Scope Indicators and the Main Operator
We use parentheses
to indicate the scope of the operators’ operations. We will call these symbols scope indicators.
In mathematics we often need them to tell us which order to perform the mathematical operations in a formula.
We consider first negation. If we have the negation of an atomic formula, like
, the negation operates on M. This is because “In the absence of parentheses, ‘¬’ simply operates on the propositional letter to its immediate right” (34). If we want the negation to operate on an entire complex formulation, then we use parenthesis, as in
The negation operates only on the M.
Since the negation applies to more content in ¬(M∧J) than in the other cases, we say it has a wider scope.
In complex formulations, there is an operator with the widest scope, called the main operator.
Main operator: The main operator of a proposition is the truth-functional operator with the widest or largest scope.
We can use the main operator for classifying the proposition. So consider again
Here the main operator is negation, and it operates on the conjunction. Thus we classify it as a negated conjunction. Or consider:
Here the conjunction is the main operator, and one of the things it conjoins is a negated term. Thus we can call it a conjunction with a negated conjunct (34).
Scope indicators also tell us the order of the truth-functional operators.
Agler explains the order in this way: “in determining the truth value of a complex proposition, we move from the truth-functional operators with the least scope to the truth-functional operators with the most scope” (35).
We consider again
First we do the conjunction, since it has the least scope, and then we do the negation, since it has the next largest scope (35).
We can see how this works when we assign values to M and J in this proposition. We can write those values like this.
We then begin with the truth-functional operators with the least scope. Again, that is the conjunction, and here since one conjunct is false, the whole conjunction is false.
The truth-functional operator with the next-least scope is the negation. Since the whole conjunction is false, that makes its negation true. And thus the whole complex proposition ¬(M∧J) is true.
Instead see how it works for
. We begin with the same truth value assignments for M and J:
This time, the negation has the least scope. Since M is true, that means its negation is false.
And that makes one of the conjuncts false, and thus the whole conjunction is false. Unlike for the prior case, here the entire formula ¬M∧J is false.
Compare the two:
So as we can see, the order is very important when determining the truth values.
The scope indicators also influence how we translate the symbols into English. We will suppose that M = “Mary is a zombie” and J = “John is running.” We would then get:
¬(M∧J) : “It is not both the case that Mary is a zombie and John is running.”
¬M∧J : “Mary is not a zombie, and John is running.”
In the first case, the negation has in its scope the conjunction as a whole. Thus the phrase “it is not the case that” comes before the compounded clauses containing these two propositional parts.
Agler uses a convention where parentheses are used firstly, then brackets secondly, and then braces thirdly. We do not use any scope indicators for just atomic propositions, for just negations of atomic propositions, for two propositions combined with an operator, or when two are combined and either or both have a negation.
However, whenever we negate a compound operator, we need them.
And we also need them if we have three or more propositions.
Since the scope of ¬’s operation is just the one proposition (be it atomic or complex) to its right, we call it a unitary operator. But since the scope of ∧, ∨, →, ↔ are the two propositions (be them atomic or complex) to both their right and left, we call them binary operators. And we also call them connectives, because they connect these propositions.
We consider this example:
The operator with the greatest scope is the conjunction, “because it contains all other operators in its scope” (37). Thus it is also the main operator. The disjunction has the next greatest scope, then the biconditional, then the conditional, then the negation (37).
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.