Agler (Intro, 1.1-1.2) Symbolic Logic: Syntax, Semantics, and Proof, summary

by Corry Shores
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[The following is summary. Bracketed commentary is my own. I highly recommend Agler’s book.] 

David W. Agler

Symbolic Logic: Syntax, Semantics, and Proof

Introduction

and

Ch.1: Propositions, Arguments, and Logical Properties

Sections 1.1-1.2: Propositions – Arguments

Summary

Brief Summary:

Logic studies the structures and mechanics of correct reasoning that takes an argumentative form. An argument is made of premises that lead to some conclusion. An argument may be (a) inductive, meaning that the conclusion is based on probabilities discerned from the premises, (b) abductive, meaning that the conclusion provides the best explanation (or theory) to account for the premises, or (c) deductively valid, meaning that the conclusion follows by logical necessity from the premises. 

I.1 What is Symbolic Logic?

Symbolic logic is a branch of logic that represents how we ought to reason by using a formal language consisting of abstract symbols. These abstract symbols, their method of combination, and their meanings (interpretations) provide a precise, widely applicable, and highly efficient language in which to reason.
(1)

I.2 Why Study Logic?

Logic has a variety of applications, including learning circuit design, learning other formal languages (as with computer programming), developing skills in rule-following behavior, developing problem- solving and analytic skills, and enhancing a study of philosophy.

I.3 How Do I Study Logic?

You learn logic through study and practice.

I.4 How is the Book Structured?

Part I provides basic concepts like argument and validity. Part II describes propositional logic. Part III examines predicate logic, including formal proofs and semantics.

Chapter 1:

Propositions, Argument, and Logical Properties

“Logic is a science that aims to identify principles for good and bad reasoning” (5). It is prescriptive: it tells us how we should reason. The two main concepts of logic are propositions and arguments (5).

Propositions are the bearers of truth and falsity, while an argument is a series of propositions separated by those that are premises (or assumptions) and those that are conclusions.
(5)

1.1 Propositions

Agler defines proposition this way:

Proposition: A proposition is a sentence (or something expressed by a sentence) that is capable of being true or false.
(6)

Certain sentences may or may not be propositions, because it can depend on their use in context (6). But for the most part, commands, questions, and exclamations do not constitute propositions (7).

1.2 Arguments

1.2.1 What is an Argument?

Here is Agler’s definition of an argument:

Argument: An argument is a series of propositions in which a certain proposition (a conclusion) is represented as following from a set of premises or assumptions.
(7)

Some arguments start with assumptions, which are “propositions that are not claimed to be true but instead are supposed to be true for the purpose of argument” as with reductio ad absurdum arguments (8).

1.2.2 Identifying Arguments

Agler explains that if we are looking for arguments in everyday discourse, we can look for these two important features:

1) Arguments often have “argument indicators” that tell us there are inferential steps, like “therefore,” “in conclusion”, and

2) Often there is a conclusion that one is claiming follows from the premises or assumptions.
(Agler 8)

1.2.3 Argument Indicators

Agler lists the following argument indicators: Therefore, I infer that, Since, So, It follows that, Hence, In conclusion, For the reason that, Thus, Consequently, Inasmuch as, We deduce that, It implies, Ergo, We can conclude that (9).

1.2.4 Types of Arguments

There are different ways that the conclusion will follow from the premisses. In other words, there are different general types of inference.

1) Inductive argument: “The premises provide some degree of support for the conclusion but certainly do not guarantee the truth of the conclusion” (10). For example, on the table is a brown bag full of beans. We pull out one bean at a time. The first hundred times we pull out a black bean. We then conclude that the next bean will be black.

2) Abductive argument: Here “the truth of the premises does not guarantee the truth of the conclusion or even provide direct support for the conclusion.” Nonetheless, “the conclusion would (perhaps best) explain the premises” (11a). For example, imagine now that there is a closed brown bag on the table. Next to it are a bunch of black beans. We then conclude from these two facts that the beans came from the brown bag. [This sort of reasoning is like proposing the best theory to explain a set of facts.]

3) Deductively valid argument: Here “the truth of the premises guarantees the truth of the conclusion. That is, it is necessarily the case that, if the premises are true, then the conclusion is true. In other words, it is logically impossible for the premises to be true and the conclusion false” (11). Agler  gives this example, “If John is a crooked lawyer, then he will hide evidence. John is a crooked lawyer. Therefore, John will hide evidence” (11).

We should note also that a conclusion follows from its premises in a way that is different from how events follow from one another chronologically (11-12).

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

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