Philosophy - Philosophy Course Notes

Sentential Logic - Critical Thinking

An Introduction to Formal Mathematical Logic

Lecture 10

May 28, 2007

 

The goal of the next three lectures is to look at modern systems of logic, including sentential logic and predicate logic.

1. Will discuss main concepts of logic.

2. Will look at why we need to develop a system of logic.

3. Will introduce a language of sentential logic, and learn how to translate English statements into this language.

 

 

I. Key Concepts of Logic

 

Logic has been pretty much static since Aristotle discovered categorical logic.  There have been some improvements, but not many major ones until the nineteenth century.

 

II. Systems of Logic

 

Systems of logic are useful because sometimes it is hard to look at an argument in English and analyze its validity.

There is no single system of logic that contains all valid arguments. (thanks a lot Godel!)

There is a trade-off between ease of use and the number of arguments a system can handle.  (simplicity vs. strength)

 

III. Propositional or Sentential Logic (SL)

 

1. Definitions

Sentential logic is a formal deduction system whose atomic formulas are propositional variables. (Capital letters)

Predicate logic is a formal deduction system whose atomic formulas are propositional functions. (see lecture 12.)

In sentential logic you take three connectives (three simple, binary connectives for declarative sentences are one sentential (unary) connective) and create a new sentence.  You can repeat those steps as often as is needed.

Sentential = of or relating to a sentence. ie. the sentential subject.  It refers to a sentence or clause that functions as a topic for a larger sentence.

SL is the simplest system that allows for the study or a large subset of arguments.

 

Sentential Logic has two basic components:

a. Capital letters which represent simple, declarative sentences. (P,Q)

b. Sentential connectives.  These allow us to connect the sentences from (1).

 

2. Binary Connectives

& (ampersand) and,but,although

v (vell) or, unless

=> (arrow) conditional statement (if A, then B)

 

3. Unary Connective

¬  (negation sign) not A

 

 

IV. Translating English Arguments into SL

 

The grammatical role of sentential connectives in SL is to take declarative sentences and link them together using binary connectives to form another grammatical sentence.

You can take lots of different English language connectives and translate them all by using the small number of binary connectives.  You can gain a lot of efficiency by doing this.

 

1. Method

a. Take the sentences and assign a unique capital letter to them.

b. Assign a binary connective between each group of sentence.

 

2. Note on Textbook Symbols

The textbook uses different (incorrect :P ) symbols.  Author of textbook is not a logician.

a. Instead of => , uses ∩ (sideways u instead of arrow)

b. Instead of &, uses .  (dot instead of ampersand)

c. Instead of ¬, uses - (hyphen instead of sideways L)

d. Uses v correctly.

 

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