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PHL245H1 Chapter Notes -Nicomachean Ethics, Presupposition, The Conscious Mind


Department
Philosophy
Course Code
PHL245H1
Professor
Niko Scharer

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Logic Unit 2: Symbolization
©2011 Niko Scharer
1
UNIT 2
SENTENTIAL LOGIC: SYMBOLIZATION
2.1 WHAT IS SENTENTIAL LOGIC?
Sentential Logic (SL): A branch of logic in which sentences or propositions are used as the basic
units. It is also called Propositional Logic or Propositional Calculus.
We will use a symbolic language that will let us move from English sentences to symbolic sentences
and back again. Each truth-valuable English sentence (statements that can be true or false, rather than
questions, orders, exclamations, etc.) will be assigned a symbol and then we can use symbols for
logical operators to combine those sentences together.
Sentential logic allows us to focus on the logical relations between sentences. By symbolizing truth-
functional sentences of natural languages (English, French, Mandarin…), we can focus on the logical
structures without being distracted by what the sentences mean. Of course, the disadvantage of this is
that it ignores the logical structures within a sentence. Some of those logical relations within sentences
will be addressed in the second part of the course (Predicate Logic).
The logical connectives that join the simple sentences are ‗truth-functional‘ they operate on the truth-
values of sentences rather than their meanings. For that to work, the atomic sentences need to be
simple and (for classical logic) bivalent. The logical operators work on the simple sentences in a
systematic way allowing us to calculate the truth-value of complex sentences from the truth-values of
simple sentences. Then, we can use the techniques of sentential logic to determine whether sets of
complex sentences are consistent and whether arguments are valid or invalid, etc.
The truth-functional nature of the logical operators of sentential logic makes it relatively easy to
interpret the logical operations electronically or mechanically. Logic gates (AND, OR, NOT…)
control the flow of information (truth-values) and are used in logical circuits, calculators and
computers. They work by taking the truth-values of one or two sentences as input and outputting truth-
values according to the logical function of the ‗gate‘. By assembling and arranging such logic gates,
so that the output of one gate is the input for another, one can build
more and more complex computing devices. Indeed, you can build
simple computing machines out of wooden levers and balls,
dominoes, Popsicle sticks, or out of Lego® or Meccano®.
Logic gates made out of Lego®. Babbage difference engine made of Meccano®
Constructed by Tim Robinson

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Logic Unit 2: Symbolization
©2011 Niko Scharer
2
2.2 THE SYMBOLS FOR SL
In sentential logic we need three types of symbols
1. Symbols for sentences or propositions: capital letters.
2. Symbols for the logical relationships between those sentences: ~
3. Symbols to keep things clear and organized: brackets and parentheses.
It is just a matter of convention which symbols get used for sentences and logical operators and
there are many different conventions. But, the different conventions share many features, and once
you get used to one set of conventions, it is easy to understand symbolic sentences or arguments
symbolized according to different conventions.
Sentence Letters: Capital letters P through Z (with or without numerical subscripts)
P, Q, R … Z
or we can use them with numerical subscripts so that we have an infinite supply:
P1, Q1, R1…, P2, Q2, R2, …
Each letter symbolizes a complete sentence or proposition. These are ‗atomic sentences‘, the
basic building blocks of the symbolic language we are using.
‗P‘ could symbolize ‗Plato is a Greek philosopher.‘, ‗Frank is the fastest runner in town. or
‗Toronto is in Ontario.‘ or any other sentence.
Likewise, ‗Q‘ could symbolize ‗Anteaters eat termites.‘ or ‗Suzy loves Adam.‘ or ‗Plato is a
Greek philosopher.‘ or any other sentence.
Most symbolic languages use letters some use small letters, others use capitals. We will use
the capital letters in the latter part of the alphabet for atomic sentences.
Sentential Connectives or Logical Operators:
Conditional sign (if ... then)
Negation sign (not)
Conjunction sign (and)
Disjunction sign (or)
Biconditional (if and only if)
These operate syntactically on sentences creating compound symbolic sentences (well-formed
formulas). Most of them are binary operators, used to combine two sentences together. the
negation sign is unary operator, operating on a single sentence.

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Logic Unit 2: Symbolization
©2011 Niko Scharer
3
Parentheses or Brackets
(round brackets/parentheses) [square brackets]
These keep things clear in complex symbolic sentences by indicating the order of operation for
the logical operators. Whatever is inside a set of brackets or parentheses must be done first.
This is just like in math: 3 (2 + 4) is equal to 3 6.
Connecting Sentences in Ordinary Language
We connect sentences or clauses together informally with words such as ‘and’,
‘or’ and ‘not’. For instance, consider the following two sentences:
1.1 It’s raining.
1.2 It’s windy.
We use logical operators to connect them, to form compound sentences:
2.1 It’s raining and it’s windy.
2.2 It’s raining or it’s windy
2.3 It’s not windy.
The meaning of these new sentences depends on the meaning of the two
simple sentences 1.1 and 1.2 and on the logical operation performed by the
word connecting them.
The difference between 2.1 and 2.2 is due just to the difference in the logical
operators. The logical connective (and, or) determines the inferential relations
between sentences. For instance, from 2.1 we can infer either 1.1 or 1.2. (If
we are told that it is raining and it’s windy, we can infer that it is raining.) From
2.2 we cannot! (If we are told that it is raining or it is windy, we cannot
conclude that it is raining and we cannot conclude that it is windy.) However,
from 2.2 and 2.3 together, we can infer 1.1
In other textbooks or in philosophical articles you might
see other symbols. These are other symbols that might be
used for the logical operators we are using.
Negation
~
~ ¬
Conditional
Disjunction
Conjunction
&
Biconditional
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