Major Schools of Thought#

Deductive Logic#

Over the centuries, a variety of systems of deductive logic have been developed. One of the oldest is Euclid’s Elements, the axiomatization of plane geometry developed around 300 BCE in classical Greece. All of the truths or theorems of plane geometry can be derived from the five fundamental assumptions or axioms of Euclid’s system. Many have attempted to axiomatize other areas of knowledge, including many of the sciences and many areas of mathematics. For example, Giuseppe Peano successfully axiomatized arithmetic in 1889.

The Five Fundamental Axioms of Euclid’s Elements

1. ᾿Ηιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν.
2. Καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾿ εὐθείας ἐκβαλεῖν.
3. Καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράφεσθαι.
4. Καὶ πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις εἶναι.
5. Καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ, ἐκβαλλομένας τὰς δύο εὐθείας ἐπ᾿ ἄπειρον συμπίπτειν, ἐφ᾿ ἃ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες. ²

Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner.²

The original Ancient Greek text of Euclid’s Elements can be found at:

• The University of Texas at Austin offers a PDF of Richard Fitzpatrick’s version
• Euclid’s Elements, Home Page. The original Greek text, Books 13

Studying deductive logic by learning to symbolize natural language sentences (such as English sentences) in a formal language, one becomes more aware and more appreciative of the importance of the structure and complexities of natural languages. The specific words that we use have a direct bearing on whether a piece of reasoning is valid or invalid. For example, it is essential to distinguish between “Roberta will pass if she completes all the homework” and “Roberta will pass only if she completes all the homework” if we want to reason well about Roberta’s prospects for passing. Finally, the concepts that we explore here are abstract concepts. Learning to think about abstract concepts and the relations between them is an important skill that is useful in a wide range of theoretical and applied disciplines.

Now let’s explore the core concepts of deductive logic, starting with basic definitions:

Syntax (of Sentential Logic)#

The syntax of a language specifies the basic expressions of a language and the rules that determine which combinations of those expressions count as sentences of the language. The syntax of a language does not specify how the sentences of the language are to be interpreted; that is a matter for semantics. The syntax of English, and every other natural language, is enormously complex. Fortunately, the syntax of Sentential Logic is simple, straightforward, and easily learned.

When we use a language to talk about a language, we are using that language as a metalanguage, and the language that we are talking about is the object language. For example, in a German class the instructor uses English to talk about German, and in this instance English is the metalanguage and German is the object language. When a grammar instructor uses English to talk about the rules of English grammar English is both the metalanguage and the object language.

Convention

We will use the boldface capital letters "$\boldsymbol{P}$", "$\boldsymbol{Q}$", "$\boldsymbol{R}$", "$\boldsymbol{S}$" with or without subscripts as in

$$\boldsymbol{P} \boldsymbol{P_1} \boldsymbol{Q_3}$$

as metavariables ranging over expressions of object language. These variables are termed “meta” because they are part of the metalanguage.

We now begin introducing formal language of Sentential Logic (SL), starting with the specification of the syntax of SL:

• Sentence Letters: the capital Roman letters ‘A’ through ‘Z’, with or without positive integer subscripts

  CAUTION

  Although both are denoted by capital letters, expressions (bolded such as $\boldsymbol{P}$) and sentences (regular like $P$) are not the same. Expressions can be a sentences but not always.

• Sentential Connectives

  • $\thicksim$ (Negation)
  • $\&$ (Conjunction) - In $(\boldsymbol{P} \& \boldsymbol{Q})$, $\boldsymbol{P}$ and $\boldsymbol{Q}$ are called conjuncts of the conjunction
  • $\vee$ (Disjunction) - In $(\boldsymbol{P} \vee \boldsymbol{Q})$, $\boldsymbol{P}$ and $\boldsymbol{Q}$ are called disjuncts of the conjunction
  • $\supset$ (Material Conditionals) - In $(\boldsymbol{P} \supset \boldsymbol{Q})$,$\boldsymbol{P}$ is called antecedent and $\boldsymbol{P}$ consequent
  • $\equiv$ (Material Biconditionals)

• Sentence of SL, the definition

  1. Every sentence letter of SL is a sentence of SL
  2. If $\boldsymbol{P}$ is a sentence of SL, then $\boldsymbol{\thicksim P}$ is a sentence of SL
  3. If $\boldsymbol{P}$ and $\boldsymbol{Q}$ are sentences of SL, then $(\boldsymbol{P} \& \boldsymbol{Q})$ is a sentence of SL
  4. If $\boldsymbol{P}$ and $\boldsymbol{Q}$ are sentences of SL, then $(\boldsymbol{P} \vee \boldsymbol{Q})$ is a sentence of SL
  5. If $\boldsymbol{P}$ and $\boldsymbol{Q}$ are sentences of SL, then $(\boldsymbol{P} \supset \boldsymbol{Q})$ is a sentence of SL
  6. If $\boldsymbol{P}$ and $\boldsymbol{Q}$ are sentences of SL, then $(\boldsymbol{P} \equiv \boldsymbol{Q})$ is a sentence of SL
  7. Nothing is a sentence of SL unless it can be formed by repeated application of 1 - 6

The followings are not sentences of SL, although they look like very much are

• $B\ \&\ D$ - no parenthesis as definition 3 above requires
• $\vee A$ - no left operand as definition 4 requires
• $(BC \supset D)$ - $BC$ is not a sentence by definition
• $(B \subset (C \vee D))$ - $\subset$ is not a symbol of SL
• $(\boldsymbol{P} \equiv \boldsymbol{Q})$ - neither $\boldsymbol{P}$ nor $\boldsymbol{Q}$ is a sentence of SL

Semantics (of Sentential Logic)#