"ReasonLines" to determine validity in categorical logic

That is actually pretty cool.



But I wonder in what circumstances you would use it as an aid? It seems that the range of application is somewhat limited to few real-life scenarios.
For example, there is this tool https://web.stanford.edu/class/cs103/tools/truth-table-tool/, I use it all the time, because among other things it helps me find equivalencies between different logical statements — Bv¬(A ? B^A) and AvB.
Also it could be better to have the conclusion image translated into statements.

Thanks. As to the conclusion, it (as well as the premises) can be translated into statements by clicking the arrows. I hadn’t thought of showing the translations by default in the conclusion.

The circumstances under which it can be useful is just that of categorical (term) logic generally, rather than propositional logic. Many (most?) introduction to logic textbooks have a chapter on this where students learn to pick out the 15 (or 24 if empty sets are prohibited) valid syllogisms from the 256 that are possible. They do this by appealing to rules or to the Venn or Euler diagrams. Of course, the schematics (of ReasonLines) do this too but, whereas those methods require the syllogisms to be in standard form to be tested (not having both a term and its complement—T and nonT—in the same syllogism) the schematic do not require this. Also, unlike these other methods, the schematics can conveniently handle sorites, i.e.,, multiple premise forms and, in addition, unlike them the schematics can handle numerical quantifiers for arguments such as

At least 10 A’s are B
All but 3 B’s are C
So, At least seven A’s are C.

So, while the applicability of the program is certainly limited, it does widen what the traditional treatment of categorical logical has allowed.

Neat.

I was taught vetting syllogisms with Aristotle's "Six Rules," which I always found very intuitive, but it's also sort of slow and it takes time to think them through. And then if you botch your translation into logical form or identification of the distribution of terms it ends up wrong anyhow. Normally this is easy on class exercises, but fairly difficult when parsing sophisticated natural language arguments. I find the Euler diagram intuitive to look at but not to make, so I'll have to try this out.

Reply to Wallace Murphree

Question: I know that Venn diagrams presuppose the Boolean assumption that all particular propositions have existential import and universal ones lack it. Does this method do the same?

I'll be honest, I understand the distinction but I'm not totally sure how to check it.

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Reply to Lionino In addition to the rules of distribution, negation, and 3-terms that Tim Wood lists, textbooks often give a rule prohibiting drawing a particular conclusion from two universal premises in order to conform to the Boolean assumption. The schematics of ReasonsLines automatically prohibits this too. However, following logician Fred Sommers, the program allows an existential/particular conclusion from universals if an existential assumption is made explicit by adding it as a premise, which thereby changes the syllogism into a sorites. And one can say “Some B’s exist” as a categorical premise by adding “Some B’s are B.” (Note that while “All B’s are B” is a tautology, “Some B’s are B” is not.”) So, while

All B’s are A
All B’s are C
So All A’s are C

is invalid by breaking the additional rule (and is shown to be invalid on both the Venn Diagram and the schematics),

All B’s are A
Some B’s are B
All B’s are C
So Some A’s are B

is shown to be valid on the schematic, and can also be shown to be so on Venn diagrams by breaking it into two syllogisms and having two valid diagrams.

I cover this in the tutorial of the help page.

The problem of translating from natural language into logical form is the same for ReasonLines as it is for the other methods. I also discuss this some in the tutorial.

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Oh no!! So sorry! I think I had this written right and then miscopied (without re-reading) it. It makes two gross mistakes.

Certainly,

All B’s are A
All B’s are C
So All A’s are C

involves illicit process of A. The conclusion I meant instead was “Some A’s are C,” for this is what violates the rule prohibiting a particular conclusion from universal premises.

Then the second error is the wrong conclusion written for

All B’s are A
(Some B’s are B)
All B’s are C
So Some A’s are B.

Sure, the conclusion intended was “Some A’s are C,” as you suggested, and here I claim (following Sommers) that the “Some B’s are B” premise satisfies the existential requirements. I hope you will reconsider this corrected version!

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Being older?? I was born in 1938, retired in 2000, and am trying to bring some attention to ReasonLines before it only happens posthumously, if it happens at all!

Thanks for your comments. Believe it or not, (if I understand correctly) I agree wholeheartedly with them. Certainly, if students just enter the premises and click the AutoSolve button that criticism would be exactly right. However, I actually added the AutoSolve capability as one of the final features of the program. Rather students were to “read” how the premise schematics hang together to determine what, if any, conclusion follows from them. Of course people must be taught how to read them, and I didn’t have time to try to do this in a post; however, instruction is included in the online tutorial. Using the schematics in classes I have had students draw their own arrows, or bring cards arrows on them, for it is in reading them that I feel there is some genuine comprehension.

I think the Venn diagrams are good as they seem to show the logical relationships that hold. I think the schematics, on the other hand, show the process of making valid inference (and how one can err). Also I think they bring to light “behind the scenes” logical connections which don’t show up in the rules. For example, students are told the middle term must be distributed to ensure the extreme terms (the minor and the major) can be connected through it. However, in Baroco

Some A are not B
All C are B
So Some A are not C

B is the distributed middle term while the logical connection is made through the unspoken nonB of the other premise. At least that’s what the schematics show, and the Venn diagram confirms. At least, so it seems to me. Peace.

Quoting tim wood

I have an example of the nature of my complaint. A college level course in statistics that confined itself to instruction in the operation of a certain software package, the instructor, actually a professor, refusing to answer any questions on statistics itself. Needless to say, nothing there learned.

It's useful to reflect on what Aristotelian logic is for and what it is about - a way of testing for nonsense. Presupposed is the student's ability to recognize basic truths and simple nonsense. It seems to me your "device" eliminates the need for such presuppositions, and for such basic knowledge and recognition. Which has been happening for at least fifty years in US education, resulting in a population that cannot tell sense from nonsense and buys the nonsense.

I sympathize with these points. Logical systems should be a tool of reasoning and not something we outsource our thinking to. This is a task that any system or representation must face.

The OP's project looks interesting. I will have a look when I have some more time.

Reply to Wallace Murphree Thanks but I think you meant to tag @Count Timothy von Icarus

Quoting Wallace Murphree

Being older?? I was born in 1938, retired in 2000

Don't pay mind.

@Lionino Thanks!

@tim wood I should have tagged you in a post above.

Reply to Wallace Murphree You can highlight a text and a button should appear over it that you can click to quote the text. You can also press the arrow to reply to a post without quoting anything from it.

@leontiskos Re: ReasonLines.Com

I appreciate that some have looked at and commented on the ReasonLines program; however, I realize I have not provided enough info here for one to get adequate sense of it.. I will attempt to develop it further and hope readers bear with it!.

Although the program could be used to spit out conclusions from premises like a pocket calculator (when the AutoSolve function is ticked), this was never its intended use and I think it is only rarely, if ever, used that way. Rather, when one is familiar enough with the arrows it uses to know which schematics to drag into the premise slots, then how these arrows connect with each other automatically shows the validity/invalidity to the user. As such, I contend it also helps reveal the internal workings of the syllogism, and what makes some inferences valid and others not.

Well, here goes. Individual arrows between terms represent individual statement types. Green arrows represent affirmative and red arrows represent negative statements, and double-ended arrows represent convertible statements (E & I) while single-ended arrows represent nonconvertible ones (A & O). Accordingly,

All A are B is a single-ended green arrow from A to B;
No A are B &No B are A is a double-ended red arrow between A and B,
Some A are B & Some B are A is a double-ended green arrow between A and B,
and Some A are not B is a single-ended red arrow from A to B.

The eight schematics (sets of arrows) on the ReasonLines screen represent the eight propositions (or eight equivalent sets of statements) that can be made using any two terms and their complements, such as A and B and nonA and nonB. Each schematic represents the four equivalent statements of one of the eight proposition.
Using A and B as default terms, clicking a schematic will always show the four statements its ingredient arrows represent. For example, the top left schematic shows All A are B, No A are nonB, No nonB are A, and All nonB are nonA. When this schematic is dragged up to the premise position (where terms A and B are shown in the white circles and nonA and nonB are in the black circles) it can be seen that All A are B is represented by the green single-ended arrow across the bottom, along with the other equivalents. At this point another premise space opens to the right with C and nonC as its terms and that same schematic can be dragged into that second space to form the premises of Barbara, for now another green single-ended arrows extends from B to C, along with arrows for its equivalents. Also now the tips of the green arrows of the first premise connect with the tails of the green arrows of the second at middle terms B and nonB. In this case, these middle terms (B and nonB) can be eliminated to yield the conclusion, for the affirmative connections are complete from A to C across the bottom, and from nonC to nonA across the top.

If AutoSolve is on then the universal affirmatives schematic showing All A are C (and equivalents) will show in the conclusion space; if not, it will be confirmed if it is dragged into that space. But any other schematic put there will be marked incorrect.


These schematics are cleared away by tapping the reset button. Then for Darii the third schematic from the top left is entered in the initial premise space to show Some A are B (and equivalents) and the top left schematic again is then entered in the second space to show All B are C (and equivalents). Now the tip of the double-ended green arrow of the initial premise meets the tail of a single-ended green arrow of the second, and this allows the double-ended green arrow to “stretch” all the way from A to C, eliminating B (and nonB automatically) to get Some A are C (and equivalents) for the conclusion.

Moreover, additional premises can be added to the left or right to turn any syllogism into a sorites of any length (although the screen quickly imposes technological limits). And, as long as there is an unbroken line of tip-to-tail green arrows between the extreme terms, all the middle terms can be eliminated, making the valid conclusion-schematic for the sorites the one that has that green line or lines extending from the term(s) of one extreme to the term(s) of the other. If there is a particular premise, there will be one green, double ended arrow extending in both directions for the conclusion, while if all premises are universal there will be two single-ended arrows extending in the different directions and connecting both sets of extreme terms. Of course, if there is no such line, then the premises do not yield a conclusion. That is, if there is no continuous green line, or if a continuous line contains a tip-to-tip or tail-to-tail connection, then no conclusion follows.

The “Hints” function allows only appropriate unbroken green arrow line(s) to show where the premises are displayed. This button is located beside the conclusion space and can be toggled on and off. On the versions of ReasonLines contained in the iOS and android apps, this function is indicated by a question mark.

Perhaps the greatest obstacle students have to using ReasonLInes is an initial lack of familiarity with the schematics. But I have found as they “play with” the program they soon find themselves comfortable with it. Thanks for looking at it.