Equivalence and Syllogisms Term Logic Terms and Connectives Equivalence and Syllogisms Relational Terms Compound Terms Proof

Introduction The classical logic of categorical syllogisms was originated by the ancient Greek philosopher Aristotle (384-322 BC). Modern deductive logic (symbolic logic) was developed by Frege (1848-1925 AD). There seems to be a tradeoff between the greater naturalness of the classical syllogisms and the greater logical power of modern logic. Fred Sommers developed the classical term logic so that it has a comparable degree of logical power.

1. Terms and Connectives The logic of a deductive argument is governed by the logical form of the statements involved. So our first task is to abstract the logical form of a categorical statement from its content. As usual, we symbolize the subject and predicate terms with the letters S and P. In term logic, the quantity (universal and particular) and the quality (affirmative and negative) are represented by plus (+) and minus (-).

Translations First, we take plus and minus as unary connectives and apply them to terms: S is P = (+S) is (+P) S is non-P = (+S) is (-P) Non-S is P = (-S) is (+P) Non-S is non-P = (-S) is (-P) As in arithmetic, the minus sign changes the term into its opposite, and the law of double negation holds: (the plus sign often be omitted) (--P) = (+P) = P

Secondly we use the plus and minus signs to represent the quality and quantity: Some S is P = +S+P Some fabrics are washable = +F+W A baby is crying = +B+C The two plus signs represents a binary connective that operates on a pair of terms to produce a statement, just as the plus sign of addition is a binary connective for numbers: The first plus sign stands for “some” The second plus sign stands for “is”

Two Ways to Represent O O: Some S is not P. We can represent it either as “some S is non-P” or as “some S is not P”: i) +S+(-P) Ii) +S-P

The Minus for Universal? As indicated above, we use the plus sign to represent “some,” but this suggests that we can use a minus sign for universal quantity (“all” or “every”). Recall the square of opposition: An A proposition contradicts an O proposition; An I proposition contradicts an E proposition.

A: All S are P E: No S is P contraries sub-alternate sub-alternate I: Some S are P O: Some S are not P sub-contraries contradictories

If we use a plus sign for “some,” it follows that we can use a minus sign for “all”: O: Some S is not P = +(+S-P) I: Some S is P = +(+S+P) A: -(+S-P) E: -(+S+P) Since -(+S-P) contradicts +(+S-P) and –(+S+P) contradicts +(+S+P), it fits the square of opposition.

Note that the plus and minus signs outside the parentheses represent unary connectives, applying to the statement inside the parentheses as a single unit. Therefore, an A proposition “-(+S-P)” can be changed into “-S+P,” saying that “Every (all) S is P”—this is exactly what an A proposition expresses.

Exercise Every machine is an artifact. Some countries are not at peace. Some trees are deciduous. Alan has brown eyes. No perishable item is taxable. The cat next door is a killer. Not a returnable thing is a sale item.

2. Equivalence and Syllogisms In Categorical Syllogisms, we have learned four rules for telling whether a categorical syllogism is valid. Now we have learned how to translate categorical propositions in Term Logic. In the following, we can replace the four rules with two simple rules: Principle of Equivalence (PEQ) Rule of Syllogistic Validity (SV)

Equivalence We use a minus sign for a universal statement because it is a denial of a particular statement. A particular statement asserts the existence of something, while a universal statement asserts nonexistence of something—recall the Aristotelian view (Gensler, Syllogistic Logic). This feature of a statement is called its valence: Particular statements have positive valence; Universal statements have negative valence.

Valence Function Valence is a function of the first two signs: If they are the same (both positive or both negative), then the statement has positive valence; If they are different, the valence is negative. However, we normally won’t use the fully explicit form, and if this is the case, valence is the same as quantity and can be determined simply by looking at the quantity sign.

Principle of Equivalence (PEQ) With the concept of valence in hand, the rule is: Two categorical statements are equivalent if and only if (i) they have the same valence, and (ii) they are algebraically equal. The first part is easy to apply; we have just seen how to determine the valence of statements. The second part exploits the fact that we have used plus and minus signs to represent different aspects of logical form.

For example All graduates are competent people. All incompetent people are non-graduates. We should translate them into: “-G+C,” “-(-C)+(-G).” Firstly, (1) and (2) are both universal, so they have the same valence. Secondly, they are algebraically equal.

Exercise Some non-adults are thumb-suckers. All adults are thumb-suckers. Every non-sale item is returnable. Not a returnable thing is a sale item.

Validity The classical syllogism consists of three statements, which contain altogether three terms (S, P, M), each term occurring twice in different statements—the middle term occurs in both premises but not the conclusion. As in the principle of equivalence, equality is not enough, and we also need a condition pertaining to quantity.

Rule of Syllogistic Validity (SV) A syllogistic argument in affirmative standard form is valid if and only if: It contains only universal statements, or else just one particular premise and a particular conclusion; The premises are equal to the conclusion.

For example Some bananas are ripe. Some apples are not ripe. Some apples are bananas.

For example +B+R. +A-R. +A+B. +B+R+A-R = +B+A = +A+B Though the sum of the premises equals the conclusion, the syllogism is obviously invalid. The problem is that it violates (i) of SV.

Every felony is a crime. Every embezzlement is a felony. Every embezzlement is a crime.

-F+C. -E+F. -E+C. Since –F+C+(-E+F) = +C-E = -E+C, and it meets (i) of SV, it is valid. On the other hand, it also meets PEQ.

Advantage In addition to its simplicity, this rule has one advantage: The rule is not restricted to classical syllogisms consisting of three statements and three terms.

For example Only evidence that is admissible in court is reliable. No hearsay evidence is admissible in court. All gossip is hearsay evidence. All gossip is unreliable.

For example -(-A)-R. -H-A. -G+H. -G+(-R).

Exercise Ideas are not physical things. But some ideas are patented, and anything patented privately owned. So it is not true that only physical things are privately owned.

3. Relational Terms In term logic, every statement has a dyadic structure: it consists of a subject and a predicate term, linked by a binary connective. But a term may also be complex, having its own internal dyadic structure.

Consider the statement: (1) A hailstorm ruined some of the crops. This tells us about the relationship between a hailstorm and the crops. The basic structure is “+H+X,” where “X” stands for the complex term “ruined some of the crops”—within the basic structure of (1), this is a predicate term.

“Ruined Some of the Crops” We understand “ruined some of the crops” as “some crops are ruined,” which we should translate into “+C+R.” Therefore, (1) should be translated as “+H+(+C+R),” where “X” is replaced with “+C+R.”

Passive or Active Nothing in the notation tells us to read R in a passive way, as meaning “were ruined by,” rather than in an active way, as meaning “ruined.” We avoid this problem by using subscripts on the term letters: R gets two subscripts—R12—to indicate that it stands for a relationship in which something (1) ruins something (2).

(1) A hailstorm ruined some of the crops. On the other hand, we also put the subscript on H and C to indicate the passive or active relationship between them. Therefore, (1) should be translated as: +H1+(R12+C2)

4. Compound Terms Categorical statement may contain compound terms, so one may desire those involving compound terms can be expressed in term logic. Consider this statement: (2) Some people are both knaves and fools. How should we translate it in term logic?

First Attempt (2) Some people are both knaves and fools. (2) can be translated as “+P+B,” where “B” stands for “both knaves and fools.” However, this is not neat, for we may have a syllogism like this: Some people are both knaves and fools. Every knave is dishonest. Therefore, some people are both dishonest and fools.

Some people are both knaves and fools. Every knave is dishonest. Therefore, some people are both dishonest and fools +P+B -K+D +P+B?

Second Attempt The structure “both…and…” is a binary connective that indicates a conjunction of terms, so it is natural to represent it as “+…+…”. In other words, we can use “some…is…” to express “both…and…”. Therefore, (2) should be translated as: +P+<+K+F> The symbols insider the corner, “<“ and “>,” represents “both…and…”.

Some people are both knaves and fools. Every knave is dishonest. Therefore, some people are both dishonest and fools +P+<+K+F> -K+D +P+<+D+F>

Exercise Conrad is a writer who is underrated. A mighty oak has fallen. Mary had a little lamb. A woman I know was elected. Some books are worth reading, and some are not.

If…then…/either…or… Not both P and not Q = if P then Q: -(+P-Q) = -P+Q If not-p then not not-q = either p or q: --P--Q

5. Proof As showed before, the Principle of Syllogistic Validity gives us a decisive test for the validity of any argument that fits the syllogistic pattern. However, it won’t work for inferences involving relational terms, or compound terms (“both knaves and fools”). If we want to test a syllogism involving relational terms for its validity, we need to learn how to construct proofs in term logic.

Dictum de Omni The basic principle of inference is called Dictum de Omni, a Latin phrase that means “rule concerning the all.” It says that whatever is true of all X is true of whatever is X. The validity of this rule is obvious in the basic syllogism: Every M is P. -M+P Some/Every S is M S+M Therefore, some/every S is P. S+P

The first premise says that P is true of every M The first premise says that P is true of every M. Since some or all S are M, we may conclude that some or all S are P. Every valid syllogism can be put into this form, and thus proven valid by Dictum de Omni (DDO), if we use immediate inferences to transform the premises and/or the conclusion.

In order to use DDO with more complex inferences, Sommers has formulated the rule in a more general way: The rule requires a pair of premises with a common term M. In one of the premise, M must have a positive occurrence; in the other it must occur with universal quantity.

The premise with the positive occurrence of M is the host premise; the premise in which M occurs with universal quantity is the donor premise (because it contributes part of its content to the host premise).

For example -M+P this is the Donor premise +S+M this is the Host premise The +P in the donor premise is what it’s going to contribute. And we are going to replace M in the host premise with it. In other words, DDO is a matter of replacement: Replace M (in the host premise) with +P (in the donor premise).

Exercise -M-P -S+M  -M+P +M+S  -M-P +M-S 

Exercise with Relational Terms Every Monet painting is a valuable thing. I am the owner of a Monet painting. I am the owner of a valuable thing.

Exercise with Relational Terms -M+V. I1+(O12+M2) I1+(O12+V2)

Review In term logic, to derive a conclusion from a set of premises, as we have seen, there are two basic kinds of step we can take: Transforming one statement into another that is equivalent; Combining two premises by means of DDO. Now we will introduce more inference rules.

Simplification (Simp) +P+Q/P +P+Q/Q In term logic, Simp also allows the following inferences: +<+S+P>Q/+SQ +<+S+P>Q/+PQ +<+S+P>Q/+S+P

Hypothetical Syllogism -P+Q -R+P -R+Q

Modus Ponens -P+Q +P +Q

Disjunctive Syllogism -(-P)-(-Q) +(-P) +-(-Q)

Equivalence Rules Double Negation (DN): Commutation (Com): --X = X Commutation (Com): +X+Y = +Y+X Association (Assoc): +X+(+Y+Z) = +(+X+Y)+Z External Negation (EN): -(XY) = + X + Y

Internal Negation (IN): Tautology (Taut): X-(Y) = X+Y Tautology (Taut): X = +X+X X = -(-X)-(-X) Contra-position (Contra): -X+Y = -(-Y)+(-X) Predicate Distribution (PD): -X+<+Y+Z> = +[-X+Y]+[-X+Z] +X+<-(-Y)-(-Z)> = --[+X+Y]--[+X+Z]

Exercise (To Prove their Equivalence) Some Greeks are shipping magnates. Some shipping magnates are Greeks. It’s not the case that every tree is deciduous. Some trees are not deciduous. No man is an island. No island is a man.

Exercise (To Prove their Equivalence) +S+<-Q+R> +<+S+S>+<-Q+R> +P+(-(-Q)-(-R)) --(+P+Q)—(+P+R) -[+S+P]+[+Q+R] -[-Q-R]+[-S-P]

Exercise (Construct a Proof) -<-(-S)-(-P)>+Q +R+S /+R+Q

Exercise (Construct a Proof) -<-(-S)-(-P)>+Q +R+S /+R+Q -(-Q)+(-<-(-S)-(-P)>) 1 Contra

Exercise (Construct a Proof) -<-(-S)-(-P)>+Q +R+S /+R+Q -(-Q)+(-<-(-S)-(-P)>) 1 Contra -(-Q)+<+(-S)+(-P)> 3 EN

Exercise (Construct a Proof) -<-(-S)-(-P)>+Q +R+S /+R+Q -(-Q)+(-<-(-S)-(-P)>) 1 Contra -(-Q)+<+(-S)+(-P)> 3 EN +[-(-Q)+(-S)]+[-(-Q)+(-P)] 4 PD

Exercise (Construct a Proof) -<-(-S)-(-P)>+Q +R+S /+R+Q -(-Q)+(-<-(-S)-(-P)>) 1 Contra -(-Q)+<+(-S)+(-P)> 3 EN +[-(-Q)+(-S)]+[-(-Q)+(-P)] 4 PD -(-Q)+(-S) 5 Simp

Exercise (Construct a Proof) -<-(-S)-(-P)>+Q +R+S /+R+Q -(-Q)+(-<-(-S)-(-P)>) 1 Contra -(-Q)+<+(-S)+(-P)> 3 EN +[-(-Q)+(-S)]+[-(-Q)+(-P)] 4 PD -(-Q)+(-S) 5 Simp -(-(-S))+(-(-Q)) 6 Contra

Exercise (Construct a Proof) -<-(-S)-(-P)>+Q +R+S /+R+Q -(-Q)+(-<-(-S)-(-P)>) 1 Contra -(-Q)+<+(-S)+(-P)> 3 EN +[-(-Q)+(-S)]+[-(-Q)+(-P)] 4 PD -(-Q)+(-S) 5 Simp -(-(-S))+(-(-Q)) 6 Contra -S+Q 7 DN

Exercise (Construct a Proof) -<-(-S)-(-P)>+Q +R+S /+R+Q -(-Q)+(-<-(-S)-(-P)>) 1 Contra -(-Q)+<+(-S)+(-P)> 3 EN +[-(-Q)+(-S)]+[-(-Q)+(-P)] 4 PD -(-Q)+(-S) 5 Simp -(-(-S))+(-(-Q)) 6 Contra -S+Q 7 DN +R+Q 2, 8 DDO

Exercise –S+<+R+T> -R+P -T+Q /-S+<+P+Q>

Exercise 1. +S1-(R12-Q2) 2. -Q+T /+S1-(R12-T2)