Is there a standard method for solving syllogisms?

[mack_10]

syllogism solving

Is there a standard method for solving syllogisms?

ie

premise 1: socrates is a man
premise 2: all men are mortal
conclusion: therefore socrates is mortal

While it is easy to see this is true, is there a formal approach to proving it?

the set theory approach with Venn/Euler diagrams shown in

http://en.wikipedia.org/wiki/Syllogisms

is good but I would like to be able to automate it on a computer, also worth noting that closer inspection of the the wikipedia entry failed to identify what were essential the same syllogism forms duplicated in the "classic" set.

Or is there already a program that will do this for me?

 

[honestrosewater]

You can check http://www.philosophypages.com/lg/e08b.htm" (the paper is old, just new to me). (I was reading the paper because the concept of distribution strikes me as ad-hoc/a posteriori/inelegant/not well-motivated, so this would not be my method of choice.)

Why not just use the diagrams? You can represent them by setting some variables and evaluating a conditional. You follow rules for shading in the diagrams, right? What does your input look like? Do you get to decide? Parsing the input if it isn't in a standard form is the only difficulty I can see.

On a broader note, how could a logical deduction not be computable?

 

[mack_10]

thankyou for your reply
Couldn't get the Notre Dame paper to download for some reason, try again later.

The shaded diagrams are fine for me, very difficult for the computer.

It is mostly a case of, I can see why the syllogism is true I just cannot automate the process.

Parse the syllogism for keywords, extract the noun and verb phrase so I can put it in the forms
A>B, A.B etc
but cannot see where to go next, don't really want to write a comprehensive set of rules for every specific form but its looking as if that is actually the way to go.

I am a little surprised that after 60 years of AI research that no-one has done this before?

 

[honestrosewater]

You don't have to write rules for each argument form. The necessary conditions listed in the first link can be turned into what should be a straightforward algorithm. (If there are exactly three terms and the middle term is distributed in at least one premise and... then it's valid else it's not.)

But you don't need to use those rules if you, like me, dislike them. You can turn the Venn diagram method into an algorithm. Your propositions define circles that you make intersections out of. An intersection can have three states: shaded, marked, unknown/unset/nil.

"All men are mortal" => shade(intersection(men))
"No men are mortal" =>shade(intersection(men,mortal))

Make some predicates: shaded? marked? unknown?

Use these to test for a valid conclusion. You need four rules to turn each type of proposition into three objects (the sections of the two intersecting circles) and either set their states if the proposition is a premise or check their states if the proposition is a conclusion.

Where is your input coming from? Are you extracting syllogistic arguments from random internet text? Are you inputting it yourself? Is it in English? Are the propositions already in standard form?

 

[g_edgar]

Lewis Carroll's small book, A GAME OF LOGIC.
Here we see how to do syllogisms by placing markers in a diagram.

 

[JeremyHorne]

For standardizing a syllogism - putting it into standard form - simply type in "standard form for syllogisms" and boodles of tutorials will pop up. It is best to know this method, as well as that for Venn diagrams; both can cross check each other, and most logic professors will require you to know both. Keep in mind also whether you are looking for Boolean form or forms where each set has at least one member. The first has 15 valid instances, the second 24. Also, keep in mind the 64 moods and the "butterfly" figure into which the major, middle, and minor terms fit.

 

[JThompson]

A quick search yielded two programs that may be what you're looking for..

http://webapp5.rrz.uni-hamburg.de/syllogism/aristotelianlogic/"

http://www.efn.org/~bsharvy/rsharvy/syll.html"

 

[JeremyHorne]

These, and there are numerous interactive ones, both with Venns and using the standard form method. I learned all this stuff PRE-internet, but this spring I was tutoring someone in logic, and it is amazing how many resources are available; we never had it this good! Kids today are spoiled, but in many ways in good ways.

 

[mack_10]

JThompson thankyou that is exactely what I was looking for.

Just so I know what search terms did you use to find these?

 

[JThompson]

No problem. I believe that those two sites popped up searching for 'syllogism solver program' with Google.

 

[onyx1491]

Try this interactive Syllogistic Machine
http://www.theotherscience.com/syllogism-machine/"

 

[praeclarum]

premise 1: socrates is a man
premise 2: all men are mortal
conclusion: therefore socrates is mortal

There are a few types of syllogisms. You can translate some of them into propositional calculus; others require predicate logic. This one requires predicate logic and some elementary set theory.

Socrates is a man becomes
S \in Men
or Socrates is a member of the set that contains all men.

All men are mortal.
\forall x \in Men Mortal(x)
using restricted quantification, or using regular quantification,
\forall x ( x\inMen \rightarrow Mortal(x) )
which is read, "For all x that are Men, x is mortal" or "For all values of x, if x is a man, x is mortal"

Therefore, Socrates is mortal.
Mortal (S).

You can actually prove this (quite simply) through substitution. The \forallx means you can substitute any value of a variable (i.e. Socrates) for instances of that variable, (x) in the formula. What I mean to say is, say you have \forallx \varphi (x) you can infer \varphi (y) where \varphi (x) stands for some function of x, and \varphi (y) is some function of y. Suppose phi is
x = x
\forallx x = x means that no matter what you substitute for x, x = x is still true. So you can infer
y = y

Take our original equation:
\forall x ( x \in Men \rightarrow Mortal(x) )
The ( x \in Men \rightarrow Mortal(x) ) part is a function of x, so since it is true for all x, we can infer that it is true when we substitute Socrates for x.
( S \in Men \rightarrow Mortal(S) )
And remember, we already know that S\inMen, so, Modus Ponens, we know Mortal(S) [Socrates is Mortal]

 

[evagelos]

syllogism solving

Is there a standard method for solving syllogisms?

ie

premise 1: socrates is a man
premise 2: all men are mortal
conclusion: therefore socrates is mortal

While it is easy to see this is true, is there a formal approach to proving it?

the set theory approach with Venn/Euler diagrams shown in

http://en.wikipedia.org/wiki/Syllogisms

is good but I would like to be able to automate it on a computer, also worth noting that closer inspection of the the wikipedia entry failed to identify what were essential the same syllogism forms duplicated in the "classic" set.

Or is there already a program that will do this for me?

Here is a formal proof using set theory:

Let A = set of men , B = set of mortals , let S stand for Socrates.

Then we have :

Premise 1 : Socrates is a man : SεΑ

Premise 2 : All men are mortal : A\subset B

3) A\subset B \Longleftrightarrow\forall x[x\in A\Longrightarrow x\in B].........Definition of subsets

4) A\subset B \Longrightarrow\forall x[x\in A\Longrightarrow x\in B]......... from (3) and using biconditional elimination

5)\forall x[x\in A\Longrightarrow x\in B].........from (2) and (4) and using M.Ponens

6) S\in A\Longrightarrow S\in B ...........from (5) and using Universal Elimination where we put x=S

7)S\in B................from (6) and (1) and using M.Ponens

 

[evagelos]

And here is a formal proof using predicates:

Let :

Mx = x is a man

MRx = x is a mortal

Ms = Socrates is a man

Premise 1 : Socrates is a man : Ms

Premise 2: All men are mortal : \forall x [Mx\Longrightarrow MRx]


3) Ms\Longrightarrow MRs............From (2) and using Universal Elimination where we put x =s


4) MRs.................From (3) and (1) and using M.Ponens

 

[onyx1491]

An trully interactive online Syllogism Solver can be found at http://thefirstscience.org/syllogistic-machine/
It demonstrates all of the fallacies, modes, and figures and so should do the trick.