Introduction to Symbolic Logic
June 8, 2005
 SL Syntax
 Symbolic Logic is an artificial language.

It is far more precise than a natural language.

In English, there can be several different ways to say the same thing,
but in SL, the symbolization of all of those sentences will be the same.

This allows us to examine the logical properties of arguments and
sentences since we don’t have to worry about natural language
problems.
 Object Language vs. Metalanguage

Remember that SL is the “symbolic language” that we use to give
representations of NL in Sentential Logic

SL is an object language

When we talk about SL, we use a metalanguage, English
 A metalanguage is a language that is used to discuss or describe
an object language

An object language doesn’t have to be artificial.

We could use Spanish to talk about French. In this case, French is
the object language and Spanish is the metalanguage

Use vs. Mention

Usually, we use words to talk about something other than the
words themselves
 Minnesota was the 32nd state admitted to the Union.

When a word or expression is being talked about itself, we mention
that word
 ‘Minnesota’ is an Indian word.
 Notice that the word being mentioned is in single quotes
 Without the quotes to signify that a word is being mentioned
and not used, a sentence could be made false
‘Saratoga’ contains 4 syllables.
Saratoga contains 4 syllables.

A metalanguage statement that discusses SL could look like this:
‘~P’ is a negation.
 In this case, ‘~P’ is the name of an SL expression. ‘~P’ is not
an SL expression, but ~P is.

‘rtxf’ is not a possible word in English.

Here, the expression mentioned is not a part of the object
language English, but ‘rtxf’ is an expression in English
since we can use it in the metalanguage
 Metavariables

Remember that in the definition of the sentential connectives, bold
capital letters are used to represent any SL sentence.


P can be an atomic sentence or a molecular sentence
We call these bold letters metavariables because they range over the
expressions in our symbolic language.

A metavariable allows us to talk about SL expressions more
generally instead of having to name each one (which would be
quite difficult!)
 If ‘~(HI)’ is an expression of SL consisting of a tilde followed
by a sentence of SL, then ‘~(HI)’ is a negation
 We don’t want to have to say this for every sentence of this
form, so we can replace the expression that is mentioned
with a metavariable.
 If P is an expression of SL consisting of a tilde followed by a
sentence of SL, then P is a negation.
 The Language SL

Now we are ready to formally define the sentences of SL

First, specify the vocabulary
 What are the basic expressions of SL? These are the building
blocks of all SL sentences in the same way that words and
punctuation are the building blocks of English sentences.
 Sentence Letters – Capitalized Roman letters with or without
a positive integer subscript
 Truth-functional Connectives – ~, &, , , 
 Punctuation Marks – ( )

Second, specify the grammar
 The grammar tells us how we are allowed to string basic
expressions together. If the rules of the grammar are not
followed, the resulting expression won’t make sense, even if it is
composed of legal vocabulary building blocks
 ‘Some and vanity men will left’ is not an English sentence
just as ‘~(GP~(‘ is not an SL sentence
 The Rules (Recursive Definition)
 1. Every sentence letter is a sentence.
 2. If P is a sentence, then ~P is a sentence.
 3. If P and Q are sentences, then (P&Q) is a sentence.
 4. If P and Q are sentences, then (PQ) is a sentence.
 5. If P and Q are sentences, then (PQ) is a sentence.
 6. If P and Q are sentences, then (PQ) is a sentence.
 7. Nothing is a sentence unless it can be formed by repeated
application of 1-6.

Well-formed Formula (wff)

An expression that follows the rules is a sentence of SL. We say
that it is a well-formed formula, or a wff.
 Since we have a full specification of what a sentence of SL is, we
have an effective method for deciding if an expression is a
sentence.
 Show that (~B&(~BA)) is a sentence
 Begin with the sentence letters and work your way out.

‘A’ and ‘B’ are sentences by 1.

‘~B’ is a sentence by 2.

‘(~BA)’ is a sentence by 4.

‘(~B&(~BA))’ is a sentence by 3.
 Some SL expressions that are not sentences
 (BCD)
 ~&A
 (BCD)
 (pq)
 ~~(P&Q) (wff)
 By convention, we can drop the outermost parenthesis of a
sentence whenever that sentence occurs by itself. It cannot be
part of another sentence.
 By convention, brackets [] may be used in place of parentheses.
This is often useful when there are a lot of parentheses floating
around in a very complex sentence.
 What’s special about syntax?

It is important to remember that at no time during the discussion of the
syntax of SL did we ever talk about what the sentences mean or what
the truth tables of the connectives are.

We have an idea of what we intend the interpretation of SL to be like

For example, we intend ‘&’ to symbolize the truth-functional
connective ‘and’

But the syntax of SL (or any language) is completely independent of
any possible interpretation of the expressions

When we look at these interpretations, this is called Semantics.
 The truth tables for each of the SL connectives are part of the
semantics of SL
 Final Syntactic Concepts

Main Connective, immediate sentential components, sentential
components, atomic components

If P is an atomic sentence, P contains no connectives and hence
does not have a main connective. P has no immediate sentential
components.

If P is of the form ~Q, where Q is a sentence, then the main
connective of P is the tilde that occurs before Q, and Q is the
immediate sentential component of P.

If P is of the form Q&R, QR, QR, or QR, where Q and R are
sentences, then the main connective of P is the connective that
occurs between Q and R, and Q and R are the immediate sentential
components of P.

The sentential components of a sentence are the sentence itself, its
immediate sentential components, and the sentential components
of its immediate sentential components.

The atomic components of a sentence are all the sentential
components that are atomic sentences.
 Complex Symbolizations
 Paraphrase Stage

Each paraphrase will be either a simple sentence, a truth-functionally
compound sentence, or a non-truth-functionally compound sentence

Simple sentences and non-truth-functionally compound sentences
are symbolized as atomic sentences in SL

Truth-functionally compound sentences are symbolized as
molecular sentences in SL

Paraphrasing Suggestions

Try to eliminate ambiguities in your paraphrase by using
parentheses

Try to use the minimum number of sentential letters to symbolize
the paraphrase
 Either Jim will not pass the test or Jim spent last night studying
logic. Jim’s night was not spent poring over his logic text.
Hence, Jim will fail the test.
 Either it is not the case that Jim will pass the test or Jim spent
last night studying logic.
~JS
It is not the case that Jim spent last night studying logic.
(It is not the case that Jim’s night was spent poring over his
logic text.)
~S
 It is not the case that Jim will pass the test.
(Jim will fail the test.)
~J

Dealing with Tense
 SL is not able to deal with temporal aspects of NL, but
sometimes the tense used in NL doesn’t really give any
temporal information
 If this is the case, try to make all of the verbs the same tense in
your paraphrase
 The British will win the race if neither of the other two major
competitors wins.
Model: There are just 3 major competitors.
M: The Americans win.
R: The British win.
N: The Canadians win.
A: The Americans have good luck.
B: The British have good luck.
C: The Canadians have good luck.
E: Everyone is surprised.
T: A major tradition is broken.
 If both it is not the case that the Americans win and it is not
the case that the Canadians win, then the British win the
race.
 (~M&~N)R

There can be more than one correct paraphrase for an NL sentence
and this could lead to symbolizations that look different.
 Even though the symbolizations look different, they will be
equivalent, if the paraphrases were correct.
 Example: ‘neither…nor…’
 Both it is not the case that P and it is not the case that Q
~P&~Q
 It is not the case that either P or Q
~(PQ)
 More Practice

The Americans will win unless the British have good luck, in which
case the British will win.

‘in which case’ behaves like ‘and’ here (and, if the British have good
luck, then they will win)

‘unless’ can be represented as PQ, ~PQ, or ~QP

Both (either the Americans win or the British have good luck) and,
(if the British have good luck, then the British win).


(MB)&(BR)
A major tradition will be broken if but only if no major competitor
wins.

A major tradition is broken if and only if it is not the case that
[either (either the Americans win or the British win) or the
Canadians win].

T~[(MR)N]
 Quantity Terms

SL doesn’t do well with quantifier words like ‘some’ and ‘all’. These
words can be part of an SL sentence, but any truth-functional
information that they impart gets lost.

If we know the exact number of things we are talking about, we can
represent some quantity terms such as ‘at least’ in SL

Examples

At least one of the major competitors will have good luck.
 Either the Americans will have good luck or (either the British
will have good luck or the Canadians will have good luck).
 A(BC)

Exactly one of the major competitors will have good luck.
 Either [both the Americans have good luck and it is not the case
that (either the British have good luck or the Canadians have
good luck)] or …
 [A&~(BC)] …
 Symbolizing an argument

If the Australians raise their spinnaker then if the wind doesn’t
increase they will win the race, but if they raise their spinnaker and the
wind does increase they will lose the race and look foolish. The wind
will increase and the Australians will reef their main and strike their
jib, and will not raise their spinnaker. So if they don’t capsize the
Australians will win the race.

Put in Standard Form
If the Australians raise their spinnaker then if the wind doesn’t
increase, they will win the race.
If the Australians raise their spinnaker and the wind does increase,
they will lose the race and look foolish.
The wind will increase.
The Australians will reef their main and strike their jib, and will not
raise their spinnaker.
 If the Australians don’t capsize, they will win the race.

What sentences do we need?
R: The Australians raise their spinnaker.
I: The wind increases.
A: The Australians win the race.
L: The Australians look foolish.
M: The Australians reef their main.
J: The Australians strike their jib.
C: The Australians capsize.

Paraphrase each sentence in the standard argument form and give
its symbolization
 If R then (if it is not the case that I then A).
R(~IA)
 If (both R and I) then (both it is not the case that A and L).
(R&I)(~A&L)
 The wind increases.
I
 Both (both the Australians reef their main and the Australians
strike their jib) and it is not the case that the Australians raise
their spinnaker.
(M&J)&~R
 If it is not the case that the Australians capsize then the
Australians win the race.
~CA