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- Introduction..................................................................................................... SENTENTIAL LOGIC
- The Grammar of Sentential Logic; A Review ................................................
- Conjunctions ...................................................................................................
- Disguised Conjunctions ..................................................................................
- The Relational Use of ‘And’...........................................................................
- Connective-Uses of ‘And’ Different from Ampersand...................................
- Negations, Standard and Idiomatic ...............................................................
- Negations of Conjunctions............................................................................
- Disjunctions ..................................................................................................
- ‘Neither...Nor’...............................................................................................
- Conditionals ..................................................................................................
- ‘Even If’ ........................................................................................................
- ‘Only If’ ........................................................................................................
- A Problem with the Truth-Functional If-Then..............................................
- ‘If And Only If’.............................................................................................
- ‘Unless’ .........................................................................................................
- The Strong Sense of ‘Unless’........................................................................
- Necessary Conditions....................................................................................
- Sufficient Conditions ....................................................................................
- Negations of Necessity and Sufficiency .......................................................
- Yet Another Problem with the Truth-Functional If-Then.............................
- Combinations of Necessity and Sufficiency .................................................
- ‘Otherwise’....................................................................................................
- Paraphrasing Complex Statements................................................................
- Guidelines for Translating Complex Statements ..........................................
- Exercises for Chapter 4 .................................................................................
- Answers to Exercises for Chapter 4 ..............................................................
92 Hardegree, Symbolic Logic
1. INTRODUCTION
In the present chapter, we discuss how to translate a variety of English state- ments into the language of sentential logic.
From the viewpoint of sentential logic, there are five standard connectives – ‘and’, ‘or’, ‘if...then’, ‘if and only if’, and ‘not’. In addition to these standard con- nectives, there are in English numerous non-standard connectives, including ‘unless’, ‘only if’, ‘neither...nor’, among others. There is nothing linguistically special about the five "standard" connectives; rather, they are the connectives that logicians have found most useful in doing symbolic logic.
The translation process is primarily a process of paraphrase – saying the same thing using different words, or expressing the same proposition using different sentences. Paraphrase is translation from English into English, which is presumably easier than translating English into, say, Japanese.
In the present chapter, we are interested chiefly in two aspects of paraphrase. The first aspect is paraphrasing statements involving various non-standard connec- tives into equivalent statements involving only standard connectives.
The second aspect is paraphrasing simple statements into straightforwardly equivalent compound statements. For example, the statement ‘it is not raining’ is straightforwardly equivalent to the more verbose ‘ it is not true that it is raining’. Similarly, ‘Jay and Kay are Sophomores’ is straightforwardly equivalent to the more verbose ‘Jay is a Sophomore, and Kay is a Sophomore’.
An English statement is said to be in standard form , or to be standard , if all its connectives are standard and it contains no simple statement that is straightfor- wardly equivalent to a compound statement; otherwise, it is said to be non- standard.
Once a statement is paraphrased into standard form, the only remaining task is to symbolize it, which consists of symbolizing the simple (atomic) statements and symbolizing the connectives. Simple statements are symbolized by upper case Roman letters, and the standard connectives are symbolized by the already familiar symbols – ampersand, wedge, tilde, arrow, and double-arrow.
In translating simple statements, the particular letter one chooses is not terribly important, although it is usually helpful to choose a letter that is suggestive of the English statement. For example, ‘R’ can symbolize either ‘it is raining’ or ‘I am running’; however, if both of these statements appear together, then they must be symbolized by different letters. In general, in any particular context, different letters must be used to symbolize non-equivalent statements, and the same letter must be used to symbolize equivalent statements.
94 Hardegree, Symbolic Logic
3. CONJUNCTIONS
The standard English expression for conjunction is ‘and’, but there are numer- ous other conjunction-like expressions, including the following.
(c1) but (c2) yet (c3) although (c4) though (c5) even though (c6) moreover (c7) furthermore (c8) however (c9) whereas
Although these expressions have different connotations, they are all truth- functionally equivalent to one another. For example, consider the following state- ments.
(s1) it is raining, but I am happy (s2) although it is raining, I am happy (s3) it is raining, yet I am happy (s4) it is raining and I am happy
For example, under what conditions is (s1) true? Answer: (s1) is true pre- cisely when ‘it is raining’ and ‘I am happy’ are both true, which is to say precisely when (s4) is true. In other words, (s1) and (s4) are true under precisely the same circumstances, which is to say that they are truth-functionally equivalent.
When we utter (s1)-(s3), we intend to emphasize a contrast that is not empha- sized in the standard conjunction (s4), or we intend to convey (a certain degree of) surprise. The difference, however, pertains to appropriate usage rather than seman- tic content.
Although they connote differently, (s1)-(s4) have the same truth conditions, and are accordingly symbolized the same:
R & H
Chapter 4: Translations in Sentential Logic 95
4. DISGUISED CONJUNCTIONS
As noted earlier, certain simple statements are straightforwardly equivalent to compound statements. For example,
(e1) Jay and Kay are Sophomores
is equivalent to
(p1) Jay is a Sophomore, and Kay is a Sophomore
which is symbolized:
(s1) J & K
Other examples of disguised conjunctions involve relative pronouns (‘who’, ‘which’, ‘that’). For example,
(e2) Jones is a former player who coaches basketball
is equivalent to
(p2) Jones is a former (basketball) player, and Jones coaches basketball,
which may be symbolized:
(s2) F & C
Further examples do not use relative pronouns, but are easily paraphrased using relative pronouns. For example,
(e3) Pele is a Brazilian soccer player
may be paraphrased as
(p3) Pele is a Brazilian who is a soccer player
which is equivalent to
(p3') Pele is a Brazilian, and Pele is a soccer player,
which may be symbolized:
(s3) B & S
Notice, of course, that
(e4) Jones is a former basketball player
is not a conjunction, such as the following absurdity.
(??) Jones is a former, and Jones is a basketball player
Sentence (e4) is rather symbolized as a simple (atomic) formula.
Chapter 4: Translations in Sentential Logic 97
By contrast, each of (r1)-(r5) states that a particular relationship holds be- tween Jay and Kay. The relational quality of (r1)-(r5) may be emphasized by restating them in either of the following ways.
(r1') Jay is a cousin of Kay (r2') Jay is a sibling of Kay (r3') Jay is a neighbor of Kay (r4') Jay is a roommate of Kay (r5') Jay is a lover of Kay
(r1) Jay and Kay are cousins of each other (r2) Jay and Kay are siblings of each other (r3) Jay and Kay are neighbors of each other (r4) Jay and Kay are roommates of each other (r5) Jay and Kay are lovers of each other
On the other hand, notice that one cannot paraphrase (c1) as
(??) Jay is a Sophomore of Kay (??) Jay and Kay are Sophomores of each other
Relational statements like (r1)-(r5) are not correctly paraphrased as conjunc- tions. In fact, they are not correctly paraphrased by any compound statement. From the viewpoint of sentential logic, these statements are simple ; they have no internal structure, and are accordingly symbolized by atomic formulas.
[NOTE: Later, in predicate logic, we will see how to uncover the internal structure of relational statements such as (r1)-(r5), internal structure that is inaccessible to sentential logic.]
We have seen so far that ‘and’ is used both conjunctively , as in
Jay and Kay are Sophomores,
and relationally , as in
Jay and Kay are cousins (of each other).
In other cases, it is not obvious whether ‘and’ is used conjunctively or relationally. Consider the following.
(s2) Jay and Kay are married
There are two plausible interpretations of this statement. On the one hand, we can interpret it as
(i1) Jay and Kay are married to each other ,
in which case it expresses a relation, and is symbolized as an atomic formula, say: M. On the other hand, we can interpret it as
(i2) Jay is married, and Kay is married, (perhaps, but not necessarily , to each other),
98 Hardegree, Symbolic Logic
in which case it is symbolized by a conjunction, say: J&K. The latter simply reports the marital status of Jay, independently of Kay, and the marital status of Kay, independently of Jay.
We can also say things like the following.
(s3) Jay and Kay are married, but not to each other.
This is equivalent to
(p3) Jay is married, and Kay is married, but Jay and Kay are not married to each other,
which is symbolized:
(J & K) & ~M
[Note: This latter formula does not uncover all the logical structure of the English sentence; it only uncovers its connective structure, but that is all sentential logic is concerned with.]
6. CONNECTIVE-USES OF ‘AND’ DIFFERENT FROM
AMPERSAND
As seen in the previous section, ‘and’ is used both as a connective and as a separator in relation-statements.
In the present section, we consider how ‘and’ is occasionally used as a connective different in meaning from the ampersand connective (&). There are two cases of this use.
First, sentences that have the form ‘P and Q’ sometimes mean ‘P and then Q’. For example, consider the following statements.
(s1) I went home and went to bed (s2) I went to bed and went home
As they are colloquially understood at least, these two statements do not express the same proposition, since ‘and’ here means ‘and then’.
Note, in particular, that the above use of ‘and’ to mean ‘and then’ is not truth- functional. Merely knowing that P is true, and merely knowing that Q is true, one does not automatically know the order of the two events, and hence one does not know the truth-value of the compound ‘P and then Q’.
Sometimes ‘and’ does not have exactly the same meaning as the ampersand connective. Other times, ‘and’ has a quite different meaning from ampersand.
(e1) keep trying, and you will succeed
(e2) keep it up buster, and I will clobber you
(e3) give him an inch, and he will take a mile
100 Hardegree, Symbolic Logic
7. NEGATIONS, STANDARD AND IDIOMATIC
The standard form of the negation connective is
it is not true that _____
The following expressions are standard variants.
it is not the case that _____
it is false that _____
Given any statement, we can form its standard negation by placing ‘it is not the case that’ (or a variant) in front of it.
As noted earlier, standard negations seldom appear in colloquial-idiomatic English. Rather, the usual colloquial-idiomatic way to negate a statement is to place the modifier ‘not’ in a strategic place within the statement, usually immediately after the verb. The following is a simple example.
statement: it is raining idiomatic negation: it is not raining standard negation: it is not true that it is raining
Idiomatic negations are symbolized in sentential logic exactly like standard negations, according to the following simple principle.
If sentence S is symbolized by the formula A, then the negation of S (standard or idiomatic) is symbolized by the formula ~A.
Note carefully that this principle applies whether S is simple or compound. As an example of a compound statement, consider the following statement.
(e1) Jay is a Freshman basketball player.
As noted in Section 2, this may be paraphrased as a conjunction:
(p1) Jay is a Freshman, and Jay is a basketball player.
Now, there is no simple idiomatic negation of the latter, although there is a standard negation, namely
(n1) it is not true that (Jay is a Freshman and Jay is a basketball player)
The parentheses indicate the scope of the negation modifier.
However, there is a simple idiomatic negation of the former, namely,
(n1′) Jay is not a Freshman basketball player.
We consider (n1) and (n1′) further in the next section.
Chapter 4: Translations in Sentential Logic 101
8. NEGATIONS OF CONJUNCTIONS
As noted earlier, the sentence
(s1) Jay is a Freshman basketball player,
may be paraphrased as a conjunction,
(p1) Jay is a Freshman, and Jay is a basketball player,
which is symbolized:
(f1) F & B
Also, as noted earlier, the idiomatic negation of (p1) is
(n1) Jay is not a Freshman basketball player.
Although there is no simple idiomatic negation of (p1), its standard negation is:
(n2) it is not true that (Jay is a Freshman, and Jay is a Basketball player),
which is symbolized:
~(F & B)
Notice carefully that, when the conjunction stands by itself, the outer parentheses may be dropped, as in (f2), but when the formula is negated, the outer parentheses must be restored before prefixing the negation sign. Otherwise, we obtain:
~F & B,
which is reads:
Jay is not a Freshman, and Jay is a Basketball player,
which is not equivalent to ~(F&B), as may be shown using truth tables.
How do we read the negation
~(F & B)?
Many students suggest the following erroneous paraphrase,
Jay is not a Freshman, and Jay is not a basketball player, WRONG!!!
which is symbolized:
~J & ~B.
But this is clearly not equivalent to (n1). To say that Jay isn't a Freshman basketball player is to say that one of the following states of affairs obtains.
Chapter 4: Translations in Sentential Logic 103
9. DISJUNCTIONS
The standard English expression for disjunction is ‘or’, a variant of which is ‘either...or’. As noted in a previous chapter, ‘or’ has two senses – an inclusive sense and an exclusive sense.
The legal profession has invented an expression to circumvent this ambiguity
- ‘and/or’. Similarly, Latin uses two different words: one, ‘vel’, expresses the inclusive sense of ‘or’; the other, ‘aut’, expresses the exclusive sense.
The standard connective of sentential logic for disjunction is the wedge ‘∨’, which is suggestive of the first letter of ‘vel’. In particular, the wedge connective of sentential logic corresponds to the inclusive sense of ‘or’, which is the sense of ‘and/or’ and ‘vel’.
Consider the following statements, where the inclusive sense is distinguished (parenthetically) from the exclusive sense.
(is) Jones will win or Smith will win (possibly both)
(es) Jones will win or Smith will win (but not both)
We can imagine a scenario for each. In the first scenario, Jones and Smith, and a third person, Adams, are the only people running in an election in which two people are elected. So Jones or Smith will win, maybe both. In the second scenario, Jones and Smith are the two finalists in an election in which only one person is elected. In this case, one will win, the other will lose.
These two statements may be symbolized as follows.
(f1) J ∨ S
(f2) (J ∨ S) & ~(J & S)
We can read (f1) as saying that Jones will win and/or Smith will win, and we can read (f2) as saying that Jones will win or Smith will win but they won't both win (recall previous section on negations of conjunctions).
As with conjunctions, certain simple statements are straightforwardly equiva- lent to disjunctions, and are accordingly symbolized as such. The following are examples.
(s1) it is raining or sleeting (d1) it raining, or it is sleeting R ∨ S
(s2) Jones is a fool or a liar (d2) Jones is a fool, or Jones is a liar F ∨ L
10. ‘NEITHER...NOR’
Having considered disjunctions, we next look at negations of disjunctions. For example, consider the following statement.
104 Hardegree, Symbolic Logic
(e1) Kay isn't either a Freshman or a Sophomore
This may be paraphrased in the following, non-idiomatic, way.
(p1) it is not true that (Kay is either a Freshman or a Sophomore)
This is a negation of a disjunction, and is accordingly symbolized as follows.
(s1) ~(F ∨ S)
Now, an alternative, idiomatic, paraphrase of (e1) uses the expression ‘neither...nor’, as follows.
(p1') Kay is neither a Freshman nor a Sophomore
Comparing (p1') with the original statement (e1), we can discern the following principle.
‘neither...nor’ is the negation of ‘either...or’
This suggests introducing a non-standard connective, neither-nor with the fol- lowing defining property.
neither A nor B is logically equivalent to ~(A ∨ B)
Note carefully that neither-nor in its connective guise is highly non-idiomatic. In particular, in order to obtain a grammatically general reading of it, we must read it as follows.
neither A nor B is officially read: neither is it true that A nor is it true that B
This is completely analogous to the standard (grammatically general) reading of ‘not P’ as ‘it is not the case that P’.
For example, if R stands for ‘it is raining’ and S stands for ‘it is sleeting’, then ‘neither R nor S’ is read
neither is it true that it is raining nor is it true that it is sleeting
This awkward reading of neither-nor is required in order to insure that ‘neither P nor Q’ is grammatical irrespective of the actual sentences P and Q. Of course, as with simple negation, one can usually transform the sentence into a more colloquial form. For example, the above sentence is more naturally read
106 Hardegree, Symbolic Logic
Whereas A is called the antecedent of the conditional, C is called the consequent of the conditional. Note that, unlike conjunction and disjunction, the constituents of a conditional do not play symmetric roles.
There are a number of idiomatic variants of ‘if...then’. In particular, all of the following statement forms are equivalent (A and C being any statements whatso- ever).
(c1) if A, then C
(c2) if A, C (c2') C if A
(c3) provided (that) A, C (c3') C provided (that) A
(c4) in case A, C (c4') C in case A
(c5) on the condition that A, C (c5') C on the condition that A
In particular, all of the above statement forms are symbolized in the same manner:
A→C
As the reader will observe, the order of antecedent and consequent is not fixed: in idiomatic English usage, sometimes the antecedent goes first, sometimes the consequent goes first. The following principles, however, should enable one systematically to identify the antecedent and consequent.
‘if’ always introduces the antecedent
‘then’ always introduces the consequent
‘provided (that)’, ‘in case’, and ‘on the condition that’ are variants of ‘if’
12. ‘EVEN IF’
The word ‘if’ frequently appears in combination with other words, the most common being ‘even’ and ‘only’, which give rise to the expressions ‘even if’, ‘only if’.
In the present section, we deal very briefly with ‘even if’, leaving ‘only if’ to the next section.
Chapter 4: Translations in Sentential Logic 107
The expression ‘even if’ is actually quite tricky. Consider the following ex- amples.
(e1) the Allies would have won even if the U.S. had not entered the war (in reference to WW2)
(i1) the Allies would have won if the U.S. had not entered the war
These two statements suggest quite different things. Whereas (e1) suggests that the Allies did win, (i1) suggests that the Allies didn't win. A more apt use of ‘if’ would be:
(i2) the Axis powers would have won if the U.S. had not entered the war.
Notwithstanding the pragmatic matters of appropriate, sincere usage, it seems that the pure semantic content of ‘even if’ is the same as the pure semantic content of ‘if’. The difference is not one of meaning but of presupposition , on the part of the speaker. In such examples, we tend to use ‘even if’ when we presuppose that the consequent is true, and we tend to use ‘if’ when we presuppose that the consequent is false. This is summarized as follows.
it would have been the case that B
if
it had been the case that A
pragmatically presupposes
~B
it would have been the case that B
even if
it had been the case that A
pragmatically presupposes
B
To say that one statement A pragmatically presupposes another statement B is to say that when one (sincerely) asserts A, one takes for granted the truth of B.
Given the subtleties of content versus presupposition, we will not consider ‘even if’ any further in this text.
Chapter 4: Translations in Sentential Logic 109
A only if B
is paraphrased
not A if not B
In other words, the ‘if’ stays put, and in particular continues to introduce the antecedent, but the ‘only’ becomes two negations, one in front of the antecedent (introduced by ‘if’), the other in front of the consequent.
With this in mind, let us go back to original examples, and paraphrase them in accordance with this principle. In each case, we use a colloquial form of negation.
(p1) I will not get an A in logic if I do not take all the exams (p2) I will not get into law school if I do not take the LSAT
Now, (p1) and (p2) are not in standard form, the problem being the relative position of antecedent and consequent. Recalling that ‘A if B’ is an idiomatic variant of ‘if B, then A’, we further paraphrase (p1) and (p2) as follows.
(p1') if I do not take all the exams, then I will not get an A in logic (p2') if I do not take the LSAT, then I will not get into law school
These are symbolized, respectively, as follows.
(s1) ~T → ~A (s2) ~T → ~L
Combining the paraphrases of ‘only if’ and ‘if’, we obtain the following prin- ciple.
A only if B
is paraphrased
not A if not B which is further paraphrased
if not B, then not A
which is symbolized ~B → ~A
110 Hardegree, Symbolic Logic
14. A PROBLEM WITH THE TRUTH-FUNCTIONAL IF-THEN
The reader will recall that the truth-functional version of ‘if...then’ is characterized by the truth-function that makes ‘A→B’ false precisely when A is true and B is false. As noted already, this is not a wholly satisfactory analysis of English ‘if...then’; rather, it is simply the best we can do by way of a truth- functional version of ‘if...then’. Whereas the truth-functional analysis of ‘if...then’ is well suited to the timeless, causeless, eventless realm of mathematics, it is not so well suited to the realm of ordinary objects and events.
In the present section, we examine one of the problems resulting from the truth-functional analysis of ‘if...then’, a problem specifically having to do with the expression ‘only if’.
We have paraphrased ‘A only if B’ as ‘not A if not B’, which is para- phrased ‘if not B, then not A’, which is symbolized ‘~B→~A’. The reader may recall that, using truth tables, one can show the following.
~B → ~A
is equivalent to A → B
Now, ~B→~A is the translation of ‘A only if B’, whereas A→B is the translation of ‘if A, then B’. Therefore, since ~B→~A is truth-functionally equivalent to A→B, we are led to conclude that ‘A only if B’ is truth- functionally equivalent to ‘if A, then B’.
This means, in particular that our original examples,
(o1) I will get an A in logic only if I take the exams (o2) I will get into law school only if I take the LSAT
are truth-functionally equivalent to the following, respectively:
(e1) if I get an A in logic, then I will take the exams (e2) if I get into law school, then I will take the LSAT
Compared with the original statements, these sound odd indeed. Consider the last one. My response is that, if you get into law school, why bother taking the LSAT!
The oddity we have just discovered further underscores the shortcomings of the truth-functional if-then connective. The particular difficulty is summarized as follows.
