An introduction to formal logic
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whitefork
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More scholastic material
The modern way of validating a syllogism is with the use of Venn Diagrams, and there are some good web sites that show how to use them, but in the spirit of the middle ages, I'll present the traditional method.
The traditional syllogism has three statements and three terms (Subject, Middle, and Predicate).
The conclusion is of the form Subject / operator / Predicate
Premise 1 relates the middle and predicate terms,
Premise 2 relates the middle and subject,
the conclusion relates subject and predicate.
There are four Figures:
1. Middle / Predicate
Subject / Middle
Subject / Predicate
2. Predicate / Middle
Subject / Middle
Subject / Predicate
3. Middle / Predicate
Middle / Subject
Subject / Predicate
4. Predicate / Middle
Middle / Subject
Subject / Predicate
We have a concept of "distribution". The members of a class are distributed according to the type of proposition, as follows:
"All" distributes the subject of a proposition.
"None" distributes the subject and predicate.
"Some" distributes neither subject nor predicate.
"Some - not" distributes the predicate (well, not in the sense of the other three, but you just have to accept this)
Distribution, with the exception of type O, thus says something about all members of a class. There are problems with this bald characterization, but you just have to trust me.
Now, four rules for validity (assuming non-empty classes):
Middle term distributed at least once
If a term is distributed in the conclusion, it must be distributed in a premise
At least one non-negative (A or I) premise
If the conclusion is negative (E or O) there must be a negative premise
Example 1 - :
All A are B
All C are A
All C are B
Known as Barbara. A is the Middle term, B the predicate and C the subject. All the valid figures have traditional names, the vowels of which correspond to the operators in the statements.
I never showed why this was valid did I?
A, the middle term, is distributed in premise 1. C is distributed in the conclusion, and in premise 2. Affirmative premises and conclusion. Valid.
This is of course an instance of the well-known
All men are mortal
All Greeks are men
All Greeks are mortal (or is that "all greeks are Socrates" - I forget)
The modern way of validating a syllogism is with the use of Venn Diagrams, and there are some good web sites that show how to use them, but in the spirit of the middle ages, I'll present the traditional method.
The traditional syllogism has three statements and three terms (Subject, Middle, and Predicate).
The conclusion is of the form Subject / operator / Predicate
Premise 1 relates the middle and predicate terms,
Premise 2 relates the middle and subject,
the conclusion relates subject and predicate.
There are four Figures:
1. Middle / Predicate
Subject / Middle
Subject / Predicate
2. Predicate / Middle
Subject / Middle
Subject / Predicate
3. Middle / Predicate
Middle / Subject
Subject / Predicate
4. Predicate / Middle
Middle / Subject
Subject / Predicate
We have a concept of "distribution". The members of a class are distributed according to the type of proposition, as follows:
"All" distributes the subject of a proposition.
"None" distributes the subject and predicate.
"Some" distributes neither subject nor predicate.
"Some - not" distributes the predicate (well, not in the sense of the other three, but you just have to accept this)
Distribution, with the exception of type O, thus says something about all members of a class. There are problems with this bald characterization, but you just have to trust me.
Now, four rules for validity (assuming non-empty classes):
Middle term distributed at least once
If a term is distributed in the conclusion, it must be distributed in a premise
At least one non-negative (A or I) premise
If the conclusion is negative (E or O) there must be a negative premise
Example 1 - :
All A are B
All C are A
All C are B
Known as Barbara. A is the Middle term, B the predicate and C the subject. All the valid figures have traditional names, the vowels of which correspond to the operators in the statements.
I never showed why this was valid did I?
A, the middle term, is distributed in premise 1. C is distributed in the conclusion, and in premise 2. Affirmative premises and conclusion. Valid.
This is of course an instance of the well-known
All men are mortal
All Greeks are men
All Greeks are mortal (or is that "all greeks are Socrates" - I forget)
a_unique_person
Director of Hatcheries and Conditioning
Soubrette said:
Like there is something I can do to stop him? I had part of the question answere, which I was happy about, and it is finally getting back to the original topic, so I am starting to read it again. I don't think it was a slight detour, or a detour into an area of equal interest, which is something that often happens in a thread, but which I don't mind.
a_unique_person
Director of Hatcheries and Conditioning
a_unique_person
Director of Hatcheries and Conditioning
BillHoyt said:
Gee, I gave up on this thread ages ago, and have only just got back into it. Maybe we should start up not just a Franko thread, but a Franko forum. Then it should be pretty easy to keep out of his way.
a_unique_person
Director of Hatcheries and Conditioning
whitefork said:
CICS, did you pronounce it C.I.C.S, like the traning videos, or KICKS, like we all did. That product caused us more trouble than anything else I have come across. Letting cobol programmers loose on unprotected memory is like giving babies razor blades to play with.
CICS joke.
Name two films stars and a dog.
Benji, Lassie and CICS.
a_unique_person
Director of Hatcheries and Conditioning
Tricky said:
Ditto, this answers the second part of my request at the start of this thread.
a_unique_person
Director of Hatcheries and Conditioning
whitefork said:
Talk about hijacking threads, that guy hijacked a whole TV series. They orinally wanted him around for a couple of shows at the start to get it moving, then made himself indispensible. It would never have been half the show it was without him.
Apparently, he was also improvising his part, and after the initial shock, they realised he was on to something.
whitefork
None of the above
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Hey Phineas!
Nice to see you back. I'm having fun with this, but all the research has brought back the dreaded "final exam" dreams.
CICS - the Canadians and Brits say Kicks, I used to say see-eye-see-ess, but they renamed it Transaction Server a few years back. It's a whole lot more stable than it was back in the 80's, but since my business is performance and stress testing, we find all sorts of opportunities for mayhem. Nobody should be working in that environment unless they have a few years of rigorous programming experience. That said, it's pretty much the only game in town for dealing with huge masses of data and very large transaction volume. (there's room for other opinions, but I'm an IBM mainframe bigot).
Thanks for the attachment tip.
Nice to see you back. I'm having fun with this, but all the research has brought back the dreaded "final exam" dreams.
CICS - the Canadians and Brits say Kicks, I used to say see-eye-see-ess, but they renamed it Transaction Server a few years back. It's a whole lot more stable than it was back in the 80's, but since my business is performance and stress testing, we find all sorts of opportunities for mayhem. Nobody should be working in that environment unless they have a few years of rigorous programming experience. That said, it's pretty much the only game in town for dealing with huge masses of data and very large transaction volume. (there's room for other opinions, but I'm an IBM mainframe bigot).
Thanks for the attachment tip.
whitefork
None of the above
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Back to the syllogism
The first line of the poem: "Barbara celarent darii ferio que prioris"
Next in figure 1 is Celarent
No A are B
All C are A
No C are B
Middle (A) is distributed, C and B are distributed in the conclusion and the premisses, negative conclusion with negative premise.
Darii (see how this is going?)
All A are B
Some C is A
Some C is B
Ferio
No A is B
Some C is A
Some C is not B
Middle (A) distributed, predicate (B) distributed in conclusion and premise 1.
Completing the valid syllogisms in figure 1. With four figures, and four operators taken three at a time, we have I believe 256 different possible syllogisms, only a small number of which are valid.
Figure 2: Cesare camestres festino baroco secundae
Now it's a little more interesting, because the names contain mnemonics for transformation rules.
Cesare
No A are B
All C are B
No C are A
The "s" in Cesare is an instruction to convert the first premise (switch the subject and predicate). Conversion is a valid rule of inference for type E and I propositions, but not, of course for A and O (think about it: if no A are B, then no B are A, and if some A is B then some B is A. But if all A are B, then it's not necessary that all B are A)
After conversion, we get
No B are A
All C are B
No C are A
which is the same structure as Celarent.
Camestres
All A are B
No C are B
No C are A
The "m" says "change the order of the premisses", and the "s" says "convert" (since there are two, convert a premise and the conclusion)
After transposing:
No C are B
All A are B
No C are A
After conversion
No B are C
All A are B
No A are C = Celarent
Festino
No A are B
Some C is B
Some C is not A
again, conversion gives
No B are A
Some C is B
Some C is not A = Ferio
(Imagine a room full of wise guys like Hamlet being force-fed this material)
Baroco
All A are B
Some C is not B
Some C is not A
the "c" means reduction per impossibilis: That is, if we deny the conclusion, giving "All C are A", then by premise 1, and the validity of Barbara, we must conclude that "All C are B", contradicting premise 2. The validity of Barbara proves the validity of Baroco.
Notice that the first letter of the name indicates the syllogism of figure 1 to which the other figures reduce.
Figure 3: Tertia darapti disamis datisi felapton bocardo freison habet (Tertia is not a form, but means "third" - "habet" means "has")
Darapti
All A are B
All A are C
Some C are B
"P" means conversion per accidens, switching the subject and predicate, and changing the universal operator to the particular (A to I or E to O). Thus, All A is B becomes Some B is A, and No A is B becomes Some B is not A.
Thus
All A are B
Some C is A
Some C is B = Darii
Disamis
Some A is B
All A are C
Some C is B
Converting premise 1 and the conclusion (s twice) and transposing the premisses (m)
All A are C
Some B is A
Some B is C = Darii
(you wonder why they don't teach this anymore?)
Datisi
All A are B
Some A is C
Some C is B
Convert premise 2 (the s)
All A are B
Some C is A
Some C is B = Darii
Felapton
No A are B
All A are C
Some C is not B
Convert premise 2 per accidens (the p)
No A are B
Some C is A
Some C is not B = Ferio
Bocardo
Some A is not B
All A are C
Some C is not B
Per impossibilis (C) - Deny the conclusion giving All C are B. Then premise 2 and Barbara give All A is B, contradicting premise 1.
Ferison
No A are B
Some A are C
Some C is not B
Convert premise 2
No A are B
Some C are A
Some C is not B = Ferio
Figure 4: Quarta insuper addit Bramantip camenes dimaris fesapo fresison.
Bramantip
All A are B
All B are C
Some C are A
Tranpose premises (m) and (here's a bit of cheating) convert the conclusion of Barbara per accidens to get Some C are A
All B are C
All A are B
Some C are A = Barbara with converted conclusion
Camenes
All A are B
No B are C
No C are A
Transpose premises, convert conclusion
No B are C
All A are B
No A are C = Celarent
(just a few more, I'm on a roll here)
Dimaris
Some A is B
All B are C
Some C is A
Transpose premises, convert conclusion
All B are C
Some A is B
Some A is C = Darii
Fesapon
No A are B
All B are C
Some C are not A
convert premise 1, convert premise 2 per accidens
No B are A
Some C is B
Some C is not A = Ferio
(last one)
Fresison - I like this because it sounds dirty
No A are B
Some B are C
Some C are not A
convert both premises
No B are A
Some C are B
Some C are not A = Ferio
And now you know everything there is to know about the classical syllogism. (right).
If an argument does not fit one of these forms, then it requires "other methods" to demonstrate its validity (wink-wink)
Here endeth the lesson for the day.
I hope to continue by showing how some of these argument forms translate into Predicate logic, which is more in the Victor/LW line.
The first line of the poem: "Barbara celarent darii ferio que prioris"
Next in figure 1 is Celarent
No A are B
All C are A
No C are B
Middle (A) is distributed, C and B are distributed in the conclusion and the premisses, negative conclusion with negative premise.
Darii (see how this is going?)
All A are B
Some C is A
Some C is B
Ferio
No A is B
Some C is A
Some C is not B
Middle (A) distributed, predicate (B) distributed in conclusion and premise 1.
Completing the valid syllogisms in figure 1. With four figures, and four operators taken three at a time, we have I believe 256 different possible syllogisms, only a small number of which are valid.
Figure 2: Cesare camestres festino baroco secundae
Now it's a little more interesting, because the names contain mnemonics for transformation rules.
Cesare
No A are B
All C are B
No C are A
The "s" in Cesare is an instruction to convert the first premise (switch the subject and predicate). Conversion is a valid rule of inference for type E and I propositions, but not, of course for A and O (think about it: if no A are B, then no B are A, and if some A is B then some B is A. But if all A are B, then it's not necessary that all B are A)
After conversion, we get
No B are A
All C are B
No C are A
which is the same structure as Celarent.
Camestres
All A are B
No C are B
No C are A
The "m" says "change the order of the premisses", and the "s" says "convert" (since there are two, convert a premise and the conclusion)
After transposing:
No C are B
All A are B
No C are A
After conversion
No B are C
All A are B
No A are C = Celarent
Festino
No A are B
Some C is B
Some C is not A
again, conversion gives
No B are A
Some C is B
Some C is not A = Ferio
(Imagine a room full of wise guys like Hamlet being force-fed this material)
Baroco
All A are B
Some C is not B
Some C is not A
the "c" means reduction per impossibilis: That is, if we deny the conclusion, giving "All C are A", then by premise 1, and the validity of Barbara, we must conclude that "All C are B", contradicting premise 2. The validity of Barbara proves the validity of Baroco.
Notice that the first letter of the name indicates the syllogism of figure 1 to which the other figures reduce.
Figure 3: Tertia darapti disamis datisi felapton bocardo freison habet (Tertia is not a form, but means "third" - "habet" means "has")
Darapti
All A are B
All A are C
Some C are B
"P" means conversion per accidens, switching the subject and predicate, and changing the universal operator to the particular (A to I or E to O). Thus, All A is B becomes Some B is A, and No A is B becomes Some B is not A.
Thus
All A are B
Some C is A
Some C is B = Darii
Disamis
Some A is B
All A are C
Some C is B
Converting premise 1 and the conclusion (s twice) and transposing the premisses (m)
All A are C
Some B is A
Some B is C = Darii
(you wonder why they don't teach this anymore?)
Datisi
All A are B
Some A is C
Some C is B
Convert premise 2 (the s)
All A are B
Some C is A
Some C is B = Darii
Felapton
No A are B
All A are C
Some C is not B
Convert premise 2 per accidens (the p)
No A are B
Some C is A
Some C is not B = Ferio
Bocardo
Some A is not B
All A are C
Some C is not B
Per impossibilis (C) - Deny the conclusion giving All C are B. Then premise 2 and Barbara give All A is B, contradicting premise 1.
Ferison
No A are B
Some A are C
Some C is not B
Convert premise 2
No A are B
Some C are A
Some C is not B = Ferio
Figure 4: Quarta insuper addit Bramantip camenes dimaris fesapo fresison.
Bramantip
All A are B
All B are C
Some C are A
Tranpose premises (m) and (here's a bit of cheating) convert the conclusion of Barbara per accidens to get Some C are A
All B are C
All A are B
Some C are A = Barbara with converted conclusion
Camenes
All A are B
No B are C
No C are A
Transpose premises, convert conclusion
No B are C
All A are B
No A are C = Celarent
(just a few more, I'm on a roll here)
Dimaris
Some A is B
All B are C
Some C is A
Transpose premises, convert conclusion
All B are C
Some A is B
Some A is C = Darii
Fesapon
No A are B
All B are C
Some C are not A
convert premise 1, convert premise 2 per accidens
No B are A
Some C is B
Some C is not A = Ferio
(last one)
Fresison - I like this because it sounds dirty
No A are B
Some B are C
Some C are not A
convert both premises
No B are A
Some C are B
Some C are not A = Ferio
And now you know everything there is to know about the classical syllogism. (right).
If an argument does not fit one of these forms, then it requires "other methods" to demonstrate its validity (wink-wink)
Here endeth the lesson for the day.
I hope to continue by showing how some of these argument forms translate into Predicate logic, which is more in the Victor/LW line.
a_unique_person
Director of Hatcheries and Conditioning
Re: Back to the syllogism
Are these all just stating the permutations and combinations of that venn diagram you were referring to?
whitefork said:
Are these all just stating the permutations and combinations of that venn diagram you were referring to?
whitefork
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Re: Re: Back to the syllogism
What we have are four basic syllogisms of the first figure (Barbara, etc). That's the first line of the poem. (I will see about putting up the corresponding Venn diagrams). The other three lines are the mnemonic rules for transforming the syllogisms of the other three figures into those of figure one.
They're rules of inference, if you will.
a_unique_person said:
What we have are four basic syllogisms of the first figure (Barbara, etc). That's the first line of the poem. (I will see about putting up the corresponding Venn diagrams). The other three lines are the mnemonic rules for transforming the syllogisms of the other three figures into those of figure one.
They're rules of inference, if you will.
whitefork
None of the above
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Baroco:
All A are B
Some C is not B
Some C is not A
The X is in a position within C that is not within B. Since all A is void except where it overlaps B, that X must outside of A's domain.
From these two example you should be able to guess at the others. The reduction of syllogisms to figure 1 does not imply that their Venn diagram representations are the same.
All A are B
Some C is not B
Some C is not A
The X is in a position within C that is not within B. Since all A is void except where it overlaps B, that X must outside of A's domain.
From these two example you should be able to guess at the others. The reduction of syllogisms to figure 1 does not imply that their Venn diagram representations are the same.
a_unique_person
Director of Hatcheries and Conditioning
Re: Venn Diagrams
Thanks, so what is the mathematical representation of this?
whitefork said:
Thanks, so what is the mathematical representation of this?
whitefork
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I'd suggest a rigorous treatment in terms of set theory, I guess. (which I am not in a good position to provide - it's been a while). Given a universe consisting of three sets A, B and C, you have the intersection of not-A and B null, not-B and C null, so the only members of A must remain in the intersection of A, B and C, so all the members of A are also members of C.
whitefork
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transition to predicate logic
the first time I heard the word logic was from my father who is a highly contentious man and likes to argue like nobody's business. Of course I have very little in common with him. He said once apropos of nothing "All squares are rectangles, but not all rectangles are squares". I don't remember how old I was at the time.
this sort of wisdom stays with you for a long time. Let us try to apply it.
In sentential logic we have propositions, which are either true or false. We can determine the truth value of a complex proposition by analysis. Predicate logic allows us to represent complex categorical propositions symbolically (lot of syllables there).
For instance, All A are B.
the universe consists of a lot of "x" and other funny letters. They are in lower case. Properties are represented by upper case letters like A, B, etc.
(x)(IF Ax Then Bx) - would be read as "If anything is A, then it is B" - the symbol - (x) - meaning "for every x". This does not mean that an x satisfying the condition "Ax" actually exists.
"All the Samoan Cy Young winners are bald" is a true statement.
But take this: "Some Samoan Cy Young winner is bald." This statement is symbolized as
(Ex)(Sx AND Bx) -- the "E" in "Ex" is supposed to be reversed -- .
that is, "there is at least one entity that is Samoan and a Cy Young winner AND that person is bald." A false statement.
So you have two operators, (x) and (Ex), the universal and existential quantifiers.
Barbara
All A are B
All C are A
All C are B
becomes
(x)(IF Ax Then Bx)
(x)(If Cx Then Ax)
(x)(If Cx Then Bx) the If - then being generally represented by a horseshoe symbol with open end to the left.
Celarent
No A are B
All C are A
No C are B
(x)(IF Ax then NOT-Bx) - if x is A then x is not B.
(x)(IF Cx then Ax)
(x)(IF Cx then NOT-Bx)
Darii
All A are B
Some C are A
Some C are B
(x)(IF Ax then Bx)
(Ex)(Cx AND Ax)
(Ex)(Cx AND Bx)
Ferio
No A are B
Some C are A
Some C are not B
(x)(IF Ax then NOT-Bx)
(Ex)(Cx AND Ax)
(Ex)(Cx AND NOT-Bx)
There are rules of inference. If you have a universal quantifier like (x) you can infer that any variable such as "a" has the same attributes.
So from
(IF Ax Then Bx)
we can infer IF Ay then By - if y has property A then y has property B.
from (IF Cx then Ax)
we can infer IF Cy then Ay
Since Ay, By, and Cy are propositions, we can construct a truth table for
(IF A then B) AND (IF C THEN A) implies (IF C THEN B)
and infer
IF Cy THEN By
Or, you can build a conditional proof by
1. Cy - assumption
2. IF Cy then Ay - assumption
3. Ay - 1,2,Modus Ponens
4. IF Ay then By - assumption
5. By - 3,4,Modus Ponens
6. IF Cy then By - 1,5,Conditional Proof
and by a rule known as Universal Instantiation we have
(x)(IF Cx THEN Bx)
More coming
the first time I heard the word logic was from my father who is a highly contentious man and likes to argue like nobody's business. Of course I have very little in common with him. He said once apropos of nothing "All squares are rectangles, but not all rectangles are squares". I don't remember how old I was at the time.
this sort of wisdom stays with you for a long time. Let us try to apply it.
In sentential logic we have propositions, which are either true or false. We can determine the truth value of a complex proposition by analysis. Predicate logic allows us to represent complex categorical propositions symbolically (lot of syllables there).
For instance, All A are B.
the universe consists of a lot of "x" and other funny letters. They are in lower case. Properties are represented by upper case letters like A, B, etc.
(x)(IF Ax Then Bx) - would be read as "If anything is A, then it is B" - the symbol - (x) - meaning "for every x". This does not mean that an x satisfying the condition "Ax" actually exists.
"All the Samoan Cy Young winners are bald" is a true statement.
But take this: "Some Samoan Cy Young winner is bald." This statement is symbolized as
(Ex)(Sx AND Bx) -- the "E" in "Ex" is supposed to be reversed -- .
that is, "there is at least one entity that is Samoan and a Cy Young winner AND that person is bald." A false statement.
So you have two operators, (x) and (Ex), the universal and existential quantifiers.
Barbara
All A are B
All C are A
All C are B
becomes
(x)(IF Ax Then Bx)
(x)(If Cx Then Ax)
(x)(If Cx Then Bx) the If - then being generally represented by a horseshoe symbol with open end to the left.
Celarent
No A are B
All C are A
No C are B
(x)(IF Ax then NOT-Bx) - if x is A then x is not B.
(x)(IF Cx then Ax)
(x)(IF Cx then NOT-Bx)
Darii
All A are B
Some C are A
Some C are B
(x)(IF Ax then Bx)
(Ex)(Cx AND Ax)
(Ex)(Cx AND Bx)
Ferio
No A are B
Some C are A
Some C are not B
(x)(IF Ax then NOT-Bx)
(Ex)(Cx AND Ax)
(Ex)(Cx AND NOT-Bx)
There are rules of inference. If you have a universal quantifier like (x) you can infer that any variable such as "a" has the same attributes.
So from
(IF Ax Then Bx)
we can infer IF Ay then By - if y has property A then y has property B.
from (IF Cx then Ax)
we can infer IF Cy then Ay
Since Ay, By, and Cy are propositions, we can construct a truth table for
(IF A then B) AND (IF C THEN A) implies (IF C THEN B)
and infer
IF Cy THEN By
Or, you can build a conditional proof by
1. Cy - assumption
2. IF Cy then Ay - assumption
3. Ay - 1,2,Modus Ponens
4. IF Ay then By - assumption
5. By - 3,4,Modus Ponens
6. IF Cy then By - 1,5,Conditional Proof
and by a rule known as Universal Instantiation we have
(x)(IF Cx THEN Bx)
More coming
Peter Soderqvist
Critical Thinker
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Can Aristotelian logic deal with asymmetrical relations?
MASS = ENERGY
Mass is made up of energy, photons are made up of energy, photons are made up of mass.
Photons can only travel at light speed, because they have no mass, so I am wondering about, if Aristotelian logic can deal with asymmetrical relations?
MASS = ENERGY
Mass is made up of energy, photons are made up of energy, photons are made up of mass.
Photons can only travel at light speed, because they have no mass, so I am wondering about, if Aristotelian logic can deal with asymmetrical relations?
whitefork
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If I understand your question, the answer is "no". An assymetrical relation (something like X is the father of Y) cannot be represented within the traditional syllogism.
X is the father of Y
Y is the father of Z
X is the grandfather of Z
requires other methods for evaluation.
Even symmetrical relations ("X is married to Y" implies "Y is married to X") don't work out.
Same for non-symmetrical relations (X loves Y)
(the above paraphrasing Kahane, page 160-161)
For such relations, predicate logic is required.
(I don't want to talk about mass/energy. The scientists here are much better qualified)
X is the father of Y
Y is the father of Z
X is the grandfather of Z
requires other methods for evaluation.
Even symmetrical relations ("X is married to Y" implies "Y is married to X") don't work out.
Same for non-symmetrical relations (X loves Y)
(the above paraphrasing Kahane, page 160-161)
For such relations, predicate logic is required.
(I don't want to talk about mass/energy. The scientists here are much better qualified)