The
Logic Calculator
The Logic Calculator is a free app on the iOS (iPhones and
iPads), Android (phones, tablets, etc.) and Windows (desktops, laptops, tablets, xbox ones)
platforms. I coded it to allow users of
propositional logic to perform operations with the same ease as that
offered by a mathematical calculator. Its design is such that it
hopefully
facilitates learners of logic by providing feedback on what makes input
well-formed. For more details on this and, more generally, on how to
operate the calculator, please consult the information section below.
As of this version, the calculator is restricted to semantic operations.
To report bugs or to request the addition of new functions in
future versions of this app please send your e-mail to:
logic(/a-t\)votsis.org
Copyright: (c) 2019 Ioannis Votsis
**Privacy Policy: **In a nutshell,
the app is
free and collects no information. For more details read the
Privacy Policy PDF.
**Information:**
(A) INPUT ALLOWED:
Just like a mathematical calculator, the logic calculator
allows its users to compute results from various inputs. The inputs in
this case are the logical formulae of propositional logic. For
simplicity, we can call these formulae 'sentences'. The calculator
offers a wide range of input. One can use up to six different letter
variables ('P', 'Q', 'R', 'S', 'T', 'U') to denote atomic sentences.
Moreover, the calculator allows the user to give as input compound
(also known as 'complex' or 'molecular') sentences. These are sentences
that employ one or more of the five logical connectives ('¬',
'&', '∨', '→', '↔'), e.g. sentences '¬P', 'P&Q' and
'(P∨Q)↔R'. Additionally, the calculator allows input in the form of
sets of sentences. The sets are denoted by curly brackets, e.g. '{P,
P→Q}', '{P, ¬P}', '{Q, R→¬Q, R∨¬Q}'. Finally, the calculator allows as
input arguments where the symbol '╞' signifies the entailment relation.
If there are two or more sentences that make up the premises of an
argument, they must be listed as a set followed by the entailment
relation symbol and, finally, the sentence purporting to be the
conclusion. Here are three examples: '{P}╞P', '{P, P→Q}╞Q', '{P, P→Q,
Q→R, R→S}╞S'.
(B) OUTPUT PRODUCED:
Three types of output are produced:
(1) The calculator determines whether the input is
well-formed. In other words, it determines whether the input is
syntactically correct and lets the user know. When the input is not
well-formed, errors are indicated. More on this below.
(2) In case the input is indeed well-formed, the calculator
proceeds to determine its properties. For individual sentences (atomic
or compound), it determines whether they are tautologies,
contradictions or contingent. For sets, it determines whether they are
consistent or inconsistent. For arguments, it determines whether they
are valid or invalid.
(3) Provided the input is well-formed, the calculator is also
able to produce truth tables
which the user can copy and paste into other apps and documents.
(C) INSTRUCTIONS:
Only two buttons produce output: 'ENTER' and 'TABLE'. Here's
how to use them. First, we need
some input. Suppose that input is 'P&Q'. To insert this input
is as easy as it sounds. Press button 'P', then press button
'&' and finally press button 'Q'. To determine whether the
input is well-formed, you can then press either the 'ENTER' button or
the 'TABLE' button. If you press the 'ENTER' button, the calculator
will remain on the same page and will just indicate the output just
below the input and above the calculator buttons. As input
'P&Q' is indeed well-formed, the calculator will produce the
output 'contingent sentence' as the input sentence is neither a
tautology nor a contradiction. If you press the 'TABLE' button instead,
you will also be shown that sentence's truth table on another page in
addition to being told that it is a contingent sentence. The
truth-table also indicates, through the use of the downward arrow '↓',
where the main connective is located in each sentence. In the case at
hand, the main connective is the ampersand symbol '&'. In cases
of atomic sentences, where there is no main connective, the downward
arrow appears over each atomic sentence. Whenever a truth table is
produced, the user has the choice of returning to the calculator or
copying the truth table. Take another input that is not a well-formed
sentence, e.g. 'P&&Q'. The app will indicate this on
the main calculator page just below the input with the phrase 'not
well-formed'. This is regardless of whether the user pressed 'ENTER' or
'TABLE' as no truth table is produced when the input is not
well-formed. Also indicated in cases where the input is not well-formed
is/are the location(s) of error(s) through the use of red underlining.
In the case at hand, the second '&' is underlined in red. More
information about errors can be found below. To determine whether a set
is consistent or inconsistent, the user must provide as input a set.
Suppose the set is {P, ¬P}. To open a set press the '{...}' button.
Then insert the two sentences and separate them by a comma using the
comma ',' button. To close a set, press the '{...}' button a second
time. Once more, the user can either press the 'ENTER' or 'TABLE'
buttons. The first button yields the output that, in this case, the set
is inconsistent. The second outputs the corresponding truth table in
addition to the judgement that, in this case, the set is inconsistent.
To determine whether an argument is valid or invalid, one needs to
provide an argument as input. Suppose that argument is {P∨Q, ¬Q}╞P. As
before, the user can either press 'ENTER' or 'TABLE' to produce output.
The first button yields the output that the argument in this case is
valid. The second outputs the truth table in addition to the judgement
that the argument in this case is valid. Finally, the user can employ
the delete button '⌫' to delete the last input, the clear button 'C' to
clear the whole screen, the memory store button 'MS' to put a copy of
the input into memory, the memory recall button 'MR' to retrieve any
stored input from memory and the memory clear button 'MC' to empty the
memory store.
(D) UNDERSTANDING ERRORS:
If the input is not well-formed, the calculator indicates
that this is
so and, where possible, provides red underlining to pin-point the
location of the error(s). In most cases, specific descriptions of the
nature of non-well-formedness are also given. To be precise, the
following specific errors are displayed: 'missing brackets', 'missing
conclusion', 'missing connectives', 'missing parentheses', 'missing
variables', 'needless parentheses', 'no { } after ╞', 'no ╞ within {
}', 'no comma after ╞', 'no content outside { }', 'unequal parentheses'
and 'scope ambiguity'. If, for example, the input is '(P)', then the
output will be: 'not well-formed: needless parentheses'. Whenever a
plus sign '+' precedes such descriptions, this means that more than one
type of error has been found. To keep the output neat and simple, only
one type of error is explicitly stated through a description. For
example, input '((P&&Q))' contains two errors, namely
'needless parentheses' and 'missing variables', but only one is shown:
'not well-formed: + needless parentheses'.
Here is an explanation of the types of specific errors in
detail:
The 'missing brackets' error indicates that one or both of
the curly
brackets '{ }' that denote the beginning and end of a set is/are
missing. That usually happens because the user inserted a comma in the
input to identify different sentences, e.g. 'P,Q', but forgot to
enclose this input within a pair of brackets, i.e. '{P, Q}'. If there
is a solitary sentence on the left side of a double turnstile '╞', the
brackets can be dropped. That is, either 'P╞P' or '{P}╞P' is
well-formed, but 'P, Q╞P' is not.
The 'missing conclusion' error comes up when the
user inserts a double turnstile '╞' in the input without a sentence
following it. Such a sentence is what we call a 'conclusion' in an
argument. For example, the input '{P, Q}╞' is missing a conclusion. To
correct this error, just insert a sentence after the previously
mentioned input. For example, '{P, Q}╞P'.
The 'missing connectives' error is indicated
when the input contains consecutive letter variables, e.g. 'PP', but no
binary connective, i.e. '&', '∨', '→', '↔', to bind them
together. To correct the error, place a binary connective between the
letter variables, e.g. 'P→P'.
The 'missing parentheses' error points out that the absence
of one or more pairs of parentheses, e.g.
'P&Q&R'. Any instance of a binary connective (i.e.
'&', '∨', '→', '↔') together with the letter variables it binds
must be enclosed within parentheses. The sole exception to this rule
concerns cases where the outermost parentheses may be dropped if the
user so wishes to. Thus, to correct the earlier input we can just add
one or more pairs of parentheses. For example, either
'(P&Q)&R' or '((P&Q)&R)'. Note also
that the parentheses could have been placed around different variables.
For example, either 'P&(Q&R)' or
'(P&(Q&R))'.
The 'missing variables' error signifies that the
input doesn't have enough letter variables. Here's one example: 'P↔∨Q'.
To correct it, the user can put whatever letter variable they desire in
between any two neighbouring binary connectives, making sure to also
add the appropriate parentheses in the process, e.g. 'P↔(R∨Q)'. Another
way to correct this error is to remove one of the logical connectives,
e.g. 'P↔Q'.
The 'needless parentheses' error appears when the user has
given too many
pairs of parentheses to the input. For example: '((P&Q))'. This
input can be corrected by removing one or both pairs of parentheses.
That is, either '(P&Q)' or 'P&Q' is correct. The second
of these illustrates the exception mentioned above, namely that the
outermost pair of parentheses of a sentence may be dropped. Note also
that input that consists solely of a single letter variable should
never be enclosed in parentheses. That is, '(P)' is not well-formed but
'P' is well-formed.
The 'no { } after ╞' error signifies that the user has
inserted one or more
curly brackets '{ }' after the double turnstile '╞', e.g.
'P&Q╞{P,Q}'. Since, in this app, we only consider whether a set
of one or more premises entails only one conclusion at a time, such
input is prohibited. To correct it, remove the curly brackets plus any
commas and insert just one sentence after the double turnstile, e.g.
'P&Q╞P'.
The 'no ╞ within { }' error is reported whenever the user
places a double
turnstile '╞' within curly brackets '{ }'. For example, '{Q╞Q∨P}'. The
double turnstile must appear only after such brackets. To correct this
error shift all input beginning with '╞' immediately to the right of
bracket '}'. In the example just given this results in: '{Q}╞Q∨P'.
The 'no comma after ╞' error shows up when a comma is
employed after the double turnstile '╞'.
For example, '{P, P→Q}╞P,Q'. As explained above, this app evaluates
only one conclusion at a time. To correct this error, remove any commas
appearing after the double turnstile. For example, '{P, P→Q}╞Q'.
The 'no content outside { }' error is expressed whenever
letter variables or connectives appear
either to the left of the curly brackets, e.g. 'P&{Q}' or to
the right of curly brackets but not after the double turnstile '╞',
e.g.'{Q}&P' or '{Q}&P╞Q'. To correct this error, remove
such content or place it within the curly brackets, e.g. '{Q}' or
'{Q&P}', '{Q}╞Q' or '{Q&P}╞Q'.
The 'unequal parentheses' error is produced whenever the
input contains parentheses that do not form pairs. For
example, 'R∨S)' or '((P∨Q)&R'. To correct this error ensure
that parentheses come in pairs. In the examples just given: '(R∨S)' and
'((P∨Q)&R)'.
Finally, the 'scope ambiguity' error is generated whenever
the input
contains binary connectives whose range conflicts. For example, the
conditional connectives '→' in '(P→(P∨Q)→Q)' are conflicting. Two
binary connectives in a sentence cannot share the same content as one
of the two sentences each connects. In the example just given, (P∨Q) is
both the consequent for P→(P∨Q) and the antecedent for (P∨Q)→Q. To
resolve this problem, the user can use parentheses to force a unique
interpretation, e.g. either P→((P∨Q)→Q) or (P→(P∨Q))→Q. Note that these
two sentences have different truth-tables. Though this is not always
the case, it is worth bearing in mind.
Philosophy Faculty, New College of the Humanities, 19 Bedford
Square, London WC1B 3HH, ioannis.votsis(/a-t\)nchlondon.ac.uk