Logic & Critical Reasoning
Translation into Propositional Logic
Steps for Translating into PL
Step 1: Read the passage in English and determine all significant
units within the passage.
Step (2): Set a scheme of abbreviation by assigning propositional
letters of PL, such as P, Q, R,… to sentences of the passage you are
translating, such as ‘John is tall.’ or ‘Mary is smart.’
Step (3): Using the rules of translation, translate the sentences of
English into formulas of PL.
Step 1
When reading a passage or a sentence of English one needs to pay
attention to the significant units that can be translated into PL.
Since PL is primarily about simple sentences of English, and how they are
combined by propositional connectives, such as ‘and’ and ‘or’, it is
important to pay attention to those when reading. Be sure to circle these.
However, one cannot forget the central point of argument identification.
In translating an argument, one still needs to pay attention to and circle
significant units that mark parts of an argument, such as premise and
conclusion indicators.
Step 1.1: mark all premise and conclusion indicators to distinguish premises
from the conclusion.
Step 1.2: mark all logical operators of compound sentences.
Step 2
A map is useless without a legend that tells you what the symbols on the
map mean. If you don’t know whether dotted lines mean trails you
should go on or not go on, seeing them on the map is useless.
A scheme of abbreviation is a legend for a translation of an English
sentence into PL. A scheme of abbreviation basically involves assigning a
propositional letter of PL to a simple non-compound sentence of English.
For example:
If the sentence is ‘John is tall’, one can set ‘P’ of PL to the sentence as
follows. P: John is tall.
If the sentence is ‘Mary is smart’, one can set ‘Q’ of PL to the sentence as
follows. Q: Mary is smart.
Step 2: Rules for Schemes of Abbreviation and the
Maxim of Logical Revelation
Schemes of abbreviation are governed by two main rules:
DO NOT SET A SINGLE PROPOSITIONAL LETTER TO TWO OR
MORE DIFFERENT SENTENCES IN THE SAME SCHEME.
For example, do not set ‘P’ as follows. P: John is tall. P: Mary is smart.
The Principle of Maximal Logical Revelation: Always translate so as to reveal
as much logical structure as the language into which you are translating
allows for.
If the sentence is ‘John is not tall’, set P to ‘John is tall’ and translate as P.
Do not set P to ‘John is not tall’, since ‘not’ can be translated out with ‘’.
If the sentence is ‘John is tall and Mary is smart’, set P to ‘John is tall’ and Q
to ‘Mary is smart’ and translate as (P  Q). Do not set P to ‘John is tall and
Mary is smart’, since ‘and’ can be translated out with ‘’.
Step 3: ‘Not’
The English word ‘not’ and its variants, such as ‘it is not the case that’,
should be translated with the broken arrow symbol ‘’.
For example, where the scheme is P: John is tall, both
It is not the case that John is tall.
and
John is not tall.
should be translated as P.
Step 3: ‘And’
The English word ‘and’ and its variants, such as ‘but’, should be
translated with the carrot symbol ‘’.
For example, where the scheme is P: John is tall, and Q: Mary is smart,
both
John is tall and Mary is smart.
and
John is tall but Mary is smart.
should be translated as (P  Q).
Step 3: ‘Or’
The English word ‘or’ and its variants, such as ‘either or’, should be
translated with the wedge symbol ‘’.
For example, where the scheme is P: John is tall, and Q: Mary is smart,
both
John is tall or Mary is smart.
and
Either John is tall or Mary is smart.
should be translated as (P  Q).
Step 3: ‘If…, Then’
The English phrase ‘If…, then…’ and its variants, such as ‘…only if….’,
should be translated with the arrow symbol ‘’.
For example, where the scheme is P: John is tall, and Q: Mary is smart,
both
If John is tall, then Mary is smart.
and
John is tall only if Mary is smart.
should be translated as (P  Q).
Step 3: ‘If and Only If’
The English phrase ‘if and only if’ and its variants, such as ‘just in case’,
should be translated with the triple bar symbol ‘’.
For example, where the scheme is P: John is tall, and Q: Mary is smart,
both
John is tall if and only if Mary is smart.
and
John is tall just in case Mary is smart
should be translated as (P  Q).
Step 3: ‘Neither Nor’
The English phrase ‘neither…nor’ is often confusing when learning
translation. However, it is easy to translate.
For example, where the scheme is P: Ben is tall, and Q: Mark is tall, both
Neither Ben is tall nor Mark is tall.
and
Neither Ben nor Mark is tall.
can be translated as either (P  Q) or (P  Q).
Step 3: ‘It is not the case that both’
The English phrase ‘it is not the case that both’ is often confusing in the
beginning. However, it is easy to translate it.
For example, where the scheme is P: Ben is tall, and Q: is Mark is tall,
both
It is not the case that both Ben is tall and Mark is tall.
and
It is not the case that both Ben and Mark are tall.
can be translated as either (P  Q) or (P  Q).
Step 3: Conditionals
In English there are a variety of ways in which one can express something
that can be said through the use of the conditional ‘if…, then….’. In order
to understand this one needs to learn the parts of a conditional.
The ‘if……’ part of a conditional is called the antecedent and it expresses
that something is a sufficient condition.
The ‘then……’ part of the conditional is called the consequent and it
expresses that something is a necessary condition.
P is a necessary for Q
(Q  P)
P is sufficient for Q
(P  Q)
P only if Q
(P  Q)
P if Q
(Q  P)
Step 3: Master Table For Translation
Not P, It is not the case that P
P
P and Q, P but Q, Although P, Q
(P  Q)
Either P or Q, P or Q
(P  Q)
If P, then Q
(P  Q)
P if Q
(Q  P)
P only if Q
(P  Q)
P if and only if Q, P just in case Q
(P  Q)
Neither P nor Q
(P  Q) or (P  Q)
It is not the case that both P and Q
(P  Q) or (P  Q)
Both not P and not Q
(P  Q)
Only if P, Q
(Q  P)
P is both necessary and sufficient for Q
(P  Q)
P unless Q
(P  Q)
Step 3: Keep in mind
When translating from English to PL you are most concerned with
capturing the logical meaning of the sentence, and not with the pragmatic
or conveyed meaning. Keeping these separate in your mind is essential for
being a good logical translator.
When translating pay attention to punctuation, such as commas.
Punctuation helps determine what should be grouped together.
That sometimes a sentence may be ambiguous between two or more
different groupings. Your goal is to learn to become sensitive to grouping
and logical ambiguity.
That most often a sentence says more than what it logically means.