Talk:Truth table/Archive 1

Language and meta-language ; meaning of the connectives

Including a logical operator cheat sheet might be helpful. See the full table and explanation at http://jerrywickey.com/test/logicoperators.html

False FFFF NOR TFFF AND FFFT XNOR TFFT ELSE FFTF Other NOT TFTF NULL FFTT Other NELSE TFTT Other ELSE FTFF NOT TTFF Other NULL FTFT NELSE TTFT XOR FTTF NAND TTTF OR FTTT True TTTT

-- jerry@jerrywickey.com

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The comments about "finite mathematics" are silly. "Finite mathematics" is not a field within mathematics, but rather a collection of diverse topics in elementary mathematics that the curriculum brings togetther in a single undergraduate course for business students. Truth tables are not different in "finite mathematics" than in other disciplines. -- Mike Hardy

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The "arrow" connective, it is to be understood as a truth-functional operator, should not be described as "implication." Call it "conditional" or "if-then." Doing otherwise involves confusing the use-mention distinction that Quine first noticed and spent his whole career trying to enforce (Perhaps hopelessly: quantified modal logic is deeply infected with use-mention confusions.) See his Mathematical Logic, Section 5.

In any case, "if...then" is not the same as "implies." "Implies" is a relation between sentences: a two-place predicate that takes sentences as the values of its variables and produces a sentence from them: it is a function from names of sentences--terms--to a sentence.

By constrast "if then" is a not a predicate but a connective; it is a funtion from sentences to a sentence. It does not take anything as values because it does not contain variables.

"Implies" talks about--mentions--two sentences, and can only be used in a meta-language. "If...then" uses two sentences; it mentions whatever the sentences mention, and is itself a term within the object language. Shortly:

If A then B. If the light goes out then the monsters will come.

but

"A" implies "B". "The light goes out" implies "The monsters will come".

Sorry for the rant. If anyone sees this mistake elsewhere, please correct it.

(Added) Same goes for equivalence. Sentences are equivalent to one another; but the things they say are related as "...if and only if ..."

Also, variables don't generally have truth-values. That's too confused to explain at all. P, Q, and R here aren't being used as variables, (though if they were, they'd have sentences, not truth-values, as their values). They're schemata; their standing in for sentences, but there's no assumption that you can quantify over them. The best way to explain this stuff is using Quine-corners and his Greek-letter metavariables. But, alas, no one cares about being rigorous anymore. Sigh.

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I do not think that the authors who (following Russell) call "if-then" an implication are so confused. What Quine calls a conditional, they call an implication ; and what Quine calls an implication, they call an inference. But they are equally rigorous. The distinction between language and meta-language is preserved.- Michel42 21:05, 27 June 2007 (UTC)

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If there is confusion about what "implication" means, this post adds to it. OP only talks about his/her opinions, not what is generally agreed upon among logicians. OP's opinions are as good or bad as anyone else's. In my opinion, it is extremely confusing to talk about "connectives" and "relations" in this way. If Quine futilely spent his career to enforce his view, this only proves that his usages of the terms were not the most appropriate. When most mathematicians and logicians today talk about implication, they mean a connective, not a relation, whatever that would mean. Of course, one can use the word "conditional", if that feels better, but in my opinion, this only adds to the confusion, since we get several words for the same notion.

It is often claimed that in everyday language, it is not meaningful to talk about an implication P -> Q if P is false, or if there is no logical or causal connection between P and Q. It is true that mostly, people don't say "If P, then Q" or "P implies Q" in these cases, but there are exceptions, even in everyday language. Consider the following dialogue:

A: I have received incontrovertible evidence that Lyndon B. Johnson killed John F. Kennedy.

B: Ha, if LBJ killed JFK, then I am Mickey Mouse!

Dialogues of this kind occur in everyday language, yet the sentence "LBJ killed JFK" is false (at least according to B, who pronounced the implication), and there is no logical or causal connection between this sentence and the (obviuosly false) sentence "B is Mickey Mouse". Still, when B pronounces the implication, (s)he intends it to be true. The point of pronouncing it is to ridicule A's original statement. Or take the sentence in the beginning of my post: "If there is confusion about what "implication" means, this post adds to it". Here, "There is confusion about what "implication" means" is true, and "This post adds to it" is true (in my opinion). There is no logical or causal connection between these sentences, yet I intend the implication to be true. As in the previous case, it is used as a rhetorical device.

When I teach logic, I give an example of the first kind, and after some initial questioning, most students accept implication as a connective. Many words are used vaguely in everday language. It is the task of mathematicians and logicians to make their usage precise. To import the vague usage in everyday language to logic and mathematics is no good idea. — Preceding unsigned comment added by Erland Gadde (talk • contribs) 11:33, 3 November 2012 (UTC)

Erland Gadde — Preceding unsigned comment added by Erland Gadde (talk • contribs) 11:36, 3 November 2012 (UTC)

Peirce

Can anyone provide a reference for Charles Peirce’s development of truth tables? I would like to check the form that they took. It appears from a quick bit of research that the tables he developed were substantially different in form to those presented here (see http://plato.stanford.edu/entries/peirce-logic/ ) Those sources that support the claim appear to derive from the Wikipedia. Was it Wittgenstein who developed the form that is now used? Banno 22:59, May 15, 2004 (UTC)

Discussed in depth in this thread: http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr00/0117.html

An excellent summary of the issue. Thanks for pointing it out to me. The conclusion appears to be that Frege, Peirce, and Schröder all played a part in the development of the truth table, and so the attribution of them to Peirce alone should be altered. Wittgenstein perhaps had the role of popularising their use. Banno 00:04, May 16, 2004 (UTC)

I propose restoring the following paragraph from earlier versions of the article: "The pattern of reasoning that the truth table tabulates was Frege's, Peirce's, and Schröder's by 1880. The tables have been prominent in literature since 1920 (Lukasiewicz, Post, Wittgenstein)" (Quine, 39). Lewis Carroll had formulated truth tables as early as 1894 to solve certain problems, but his manuscripts containing his work on the subject were not discovered until 1977 [1]. Wittgenstein's Tractatus Logico-Philosophicus uses them to place truth functions in a series. The wide influence of this work led to the spread of the use of truth tables." This paragraph is much more informative than just crediting Wittgenstein for the use of truth tables in Logic as is implied in the current version of the article. See also: Talk:Tractatus_Logico-Philosophicus_(5.101)#Truth_Tables. --Omnipaedista (talk) 13:10, 28 June 2011 (UTC)

It may also be worthwhile to point out that Peirce's Ph.D. student Christine Ladd-Franklin found the truth tables in TLP 5.101 40 years earlier than Wittgenstein. Christine Ladd (1881), "On the Algebra of Logic", p.62, Studies in Logic, C. S. Peirce ed., 1883 --Ancheta Wis (talk) 21:06, 28 June 2011 (UTC)

This quotation from the Peirce paper appears to be the cited "indirect truth table" for the conditional (which Pierce symbolizes -<): "the proposition a-<b is true if a is false or if b is true, but is false if a is true while b is false". It is a complete description of how the truth values combine, but is not a systematic presentation thereof. Only with hindsight is it recognizable as describing a truth table.

If Peirce counts, then Boole probably does, too. He writes: "If we associate the Propositions X and Y, the total number of conceivable cases will be found as exhibited in the following scheme. [Consider @ to be a tab character.]

Cases @ Elective expressions

1st X true, Y true, @ xy,

2nd X true, Y false, @ x(1-y),

3rd X false, Y true, @ (1-x)y,

4th X false, Y false, @ (1-x)(1-y).

...

Consider what are those distinct and mutually exclusive cases of which it is implied in the statement of the given Proposition, that some one of them is true, and equate the sum of their elective expressions to unity. This will give the equation of the given Proposition." Mdmi (talk) 02:02, 10 October 2023 (UTC)

Truth table for most commonly used logical operators

Including a logical operator cheat sheet might be helpful. See the full table and explanation at http://jerrywickey.com/test/logicoperators.html

False FFFF NOR TFFF AND FFFT XNOR TFFT ELSE FFTF Other NOT TFTF NULL FFTT Other NELSE TFTT Other ELSE FTFF NOT TTFF Other NULL FTFT NELSE TTFT XOR FTTF NAND TTTF OR FTTT True TTTT

-- jerry@jerrywickey.com

The given truth table gives definitions of the 6 (NOT 7) of the 16 possible truth functions of 2 binary variables.

∨ (XNOR or exclusive nor) and ↔ (biconditional or "if-and-only-if") have identical values 202.173.204.250 08:36, 7 November 2005 (UTC)

For consistency (just a suggestion) since the other sections do so, the section that mentions the "equal" table is the same as the XNOR table, however, it doesn't mention that as it does for the "AND", "OR" tables, etc. Just_Me@75.81.38.135 09:26, 30 August 2007 (UTC)

AND / OR symbols

The symbols arent working for the most part they show up as squares in my browser.

JA: The formats used are fairly standard for WP. I tried substituting TeX formats. Let me know if it's any better. Jon Awbrey 14:46, 25 April 2006 (UTC)

Use TeX! And a small concern

Firstly, keep consistent with usage the <math> tags; if it's simple enough (like in most examples on this page), WP will render it as plain text if the user so wishes; otherwise, it is renderred into a PNG via TeX. The symbols for boolean algebra typically are \lor ($\lor$), \land ($\land$), and \neg ($\neg$). You can use the \underline{} and \overline{} functions as you wish.

Secondly, I'm concerned that there doesn't appear to be a uniform method of notating XOR, XNOR, and NAND. Some use overline, some use underline, and other articles use a plus sign. I'm no boolean algebra expert, so I don't know the common notation used, nor can I find a predefined symbol for such in any TeX packages I can find. -Matt 06:43, 20 May 2006 (UTC)

JA: JVz, all of the pages on boolean logic, propositional logic, and so on are in the process of being cleaned up and formatted in a more or less standard way, but nobody got around to this one yet until now. Thanks for the improvements. As a rule though, local practice tries to avoid in-line use of TeX eXcepT when it can't be helped. Thanks again, Jon Awbrey 16:24, 20 May 2006 (UTC)

Consistency

There needs to be consistency with the use of the operators on this page. Each one should have it's own section and used the same as they are represented in their own section. Right now it is very confusing, because some things are mentioned, others are not, and some are introduced into the article at the end with different symbolic representations. 70.111.238.17 14:52, 1 October 2006 (UTC)

Consistency in Truth Tables

Consistency needed for T and F order in tables, whether F first or T first.

F first might be preferred, since it corresponds with the natural order of its binary value (00, 01, 10, 11 - 0, 1, 2, 3).

p	q	binary	value
F	F	00	0
F	T	01	1
T	F	10	2
T	T	11	3

"T first" has reversed order.

p	q	binary	value
T	T	11	3
T	F	10	2
F	T	01	1
F	F	00	0

118.136.65.187 (talk) 22:36, 20 March 2009 (UTC)

I agree with this comment. I'm even willing to implement the changes if that's ok. (I'm still a noob editor, so I want to verify that moving forward with this change is acceptable before I put in the effort.) Ryanker (talk) 16:00, 19 October 2022 (UTC) Ryanker

It is possible to reorder Truth table#Truth table for all binary logical operators by flipping the top 4 rows, viz.

F F

F T

T F

T T

Ancheta Wis   (talk | contribs) 05:44, 21 October 2022 (UTC)

Agreed. "How" is the easy part. The question is whether or not we should. Ryanker (talk) 19:16, 8 November 2022 (UTC)

Truth table and logic gates

Hello friends I have a query, that how a logic gate relates to a truth table. If any one have an answer, kindly send answer at xact_solutions@hotmail.com

Plea for better explanation

Hi, this article seems quite well made, but there is a problem (IMHO). I am 28, have a science degree, have done a bit of basic programming and have competent though relativly basic maths and I simply can't work out what truth tables are from this article (certainly not in less than 5 minutes or even 10). I am guessing that for some poor kid to find out what they are from this article is going to be even harder. It is so often the case that maths cannot be taught by maths geniuses as they simply don't understand how to explain things to us mere mortals. Is there any chance that someone could put in a simple to understand explanation of a truth table here? Many thanks --82.69.113.120 02:15, 22 May 2007 (UTC)

Ugly green tables

Those tables are ugly. There is no reason to use green/teal, and they are unnecessarily wide. —Preceding unsigned comment added by 74.202.89.125 (talk) 22:15, 19 December 2007 (UTC)

External Links

I suggest a tidy up of the external links.

• Kmap minimizer (Flash applet), published 2009 (remove - WP:CONFLICT - WP:EL#ADV - not relevant)
• Truth Table Constructor (Java applet) (remove - WP:CONFLICT - WP:ELNO: personal site)
• Web-based truth table generator (remove - WP:CONFLICT - WP:ELNO: personal site)
• Powerful logic engine (remove - WP:CONFLICT - WP:ELNO: personal site)
• Boolean expression evaluator, generates truth table (Java applet) (remove - WP:CONFLICT - WP:ELNO: personal site)
• Free logic minimization program Minilog (remove - broken link)

Many of these sites require JAVA to work, which also makes it difficult for many users to use the resource.—Preceding unsigned comment added by 60.234.246.33 (talk • contribs)

WP:CONFLICT is not a reason to remove, except maybe when the link was added by the site maintainer himself. the external links guideline gives hints on which pages are appropriate. I'll remove the last. --Dirk Beetstra ^{T C} 11:32, 13 April 2010 (UTC)

Actually - it clearly states in WP:ELNO that personal sites should be avoided.

External Links - Interactive Truth Table Solver

The following link may be a useful addition to this page:

• Samuel Williams' Truth Table Solver

Don't care X

No mention of things like "Don't care" inputs represented as X? —Preceding unsigned comment added by 71.167.71.69 (talk) 20:23, 13 July 2010 (UTC)

Are you talking about a case where one bit is irrelevant? In which case, the article implies that, and using extra terminology is just likely to confuse 88.108.100.104 (talk) 23:49, 29 September 2010 (UTC)

More likely we need to take a look at "don't care" outputs. For instance, let's say you have a four-bit decimal counter. i.e. It counts from FFFF (representing 0) to TFFT (representing 9) and then resets back to FFFF (0) on the next count. You want to work out the logic for each segment of a single-digit seven-segment LED. The truth table for each segment would have specific values in the truth table rows for inputs FFFF - TFFT, but X's representing "don't care" entries for the TFTF - TTTT (10 - 15) rows. That's because the state of a segment for those rows doesn't matter because those combinations of inputs will never happen under normal operation.

The truth table for the top horizontal LED segment might look like this:

b3	b2	b1	b0	counter value	segment a of LED
F	F	F	F	0	T
F	F	F	T	1	F
F	F	T	F	2	T
F	F	T	T	3	T
F	T	F	F	4	F
F	T	F	T	5	T
F	T	T	F	6	T
F	T	T	T	7	T
T	F	F	F	8	T
T	F	F	T	9	T
T	F	T	F	10	X
T	F	T	T	11	X
T	T	F	F	12	X
T	T	F	T	13	X
T	T	T	F	14	X
T	T	T	T	15	X

Ryanker (talk) 02:26, 21 October 2022 (UTC) Ryanker

Please take care to preserve Truth table#TLP, a 16-row table of 16 logical operators, each with a known Boolean function, with well-known notation stemming from (Tractatus Logico-Philosophicus 5.101, but known from the 19th century) Ancheta Wis   (talk | contribs) 05:22, 21 October 2022 (UTC)

The article called Don't-care term addresses the X value. -- Ancheta Wis   (talk | contribs) 05:55, 21 October 2022 (UTC)

Good catch. Where is this article would be the right place to reference the Don't-Care Term article? Ryanker (talk) 19:19, 8 November 2022 (UTC)

Truth table#See also Ancheta Wis   (talk | contribs) 21:47, 8 November 2022 (UTC)

Contextualisation

It might be nice to contextualise the discussion by describing for *which* kinds of logics truth tables are appropriate (and to show a TT for a logic *other than* Classical Logic to demonstrate that they are more-widely applicable as a tool ... e.g., Belnap's 4-valued relevant logic). One cannot, e.g., use a TT in constructive logics that have no notion of truth (i.e., that have provability as the closest analogous aspect) or in logics that have a transfinite set of truth values (e.g., fuzzy logics).

Truth Table

I don't understand this junk in the main truth table for the 16 Boolean functions. Since when is a tautology designated by XAND, or contradiction with XNAND? According to what definition. There's also stuff like XP, XNP, etc. A brief search of Google only provides various discussions which state that there is no such function as XAND or XNAND. I wish to have it clarified what formal definition the (X) prefix is supposed to be taking in this article. --24.212.154.2 (talk) 00:15, 28 January 2013 (UTC)

Removing XAND and XNAND, as no reference has been given and no counterargument has been made. 72.234.110.47 (talk) 07:37, 2 October 2013 (UTC)

I propose to replace the remaining 4 fields of non-mnemonic notation of the third column of the Key in the 16-row truth table Xp, etc with the symbols ←, →, etc. OK? --Ancheta Wis   (talk | contribs) 16:08, 9 August 2014 (UTC)

Truth table for all binary logical operators is missing iff or ↔︎ double arrow — Preceding unsigned comment added by 24.148.132.57 (talk) 13:30, 19 July 2019 (UTC)

Please see Key, line 9 of Truth_table#Truth_table_for_all_binary_logical_operators. It already has if and only if, i.e. iff --Ancheta Wis   (talk | contribs) 16:28, 19 July 2019 (UTC)

Many-valued logics

I'm missing truth tables for many-valued logics, see also three-valued logic. Paradoctor (talk) 12:15, 17 January 2014 (UTC)

Why?

Why are the values for FT and FF what they are? The article just gives them by fiat. What is the reason? (Even a clear example would help.) 211.225.33.104 (talk) 07:42, 6 May 2014 (UTC)

What do you mean by "FT" and "FF"? Paradoctor (talk) 14:46, 6 May 2014 (UTC)

A better order for the Unary Operators

It seems better to introduce "logical true" as the first rather than the fourth unary operator. That would allow it to be introduced before it is used in the identity and negation tables. Thanks! --Lbeaumont (talk) 15:22, 12 April 2016 (UTC)

 Done --Ancheta Wis   (talk | contribs) 13:46, 13 April 2016 (UTC)