cohen book

Copyright 2004, S. Marc Cohen Revised 9/29/04 1-1

Chapter 1: Atomic Sentences

We begin by describing a simple artificial language. It is technically what is known as a first-order language; well call it FOL.

Although FOL can contain a much larger vocabulary than well use, well mostly be restricting ourselves to a portion of FOL that can be used to describe the worlds we can build using the software program Tarskis World.

At the very simplest level, FOL contains sentences that contain two kinds of ingredients: individual constants (names) and predicate symbols (property and relation words). Examples will make this clearer.

1.1 Individual constants

a, b, c, d, e, f, n1, n2, n3, etc. We will use these as the names of the various blocks that inhabit the Tarski worlds we will be examining. If we use one of these constants in describing a Tarski world, it must name some actually existing block a block that exists in the world that we are evaluating. Note these requirements:

x Every world must contain at least one block. x Any name that we use must name some block. x In a given Tarski world, no name refers to more than one block. x A block may have more than one name. x Some blocks may not have names.

1.2 Predicate symbols

They are listed on p. 21. Notice that they come in three arities.

Arity 1:

Cube, Tet, Large, etc. Correspond to property words is a cube, is a tetrahedron, is large, etc.

Arity 2:

Smaller, Larger, LeftOf, SameSize, etc. Correspond to relational words is smaller than, is larger than, is to the left of, is the same size as, etc.

Arity 3:

Between

Correspond to relational words is between and .

Notice that the arity of a predicate is the same as the number of individual constants (names) it takes to combine with the predicate to form a complete sentence.


1.3 Atomic sentences

We write atomic sentences in the blocks language by combining a predicate (which always begins with a capital letter), followed by (in parentheses) one or more individual constants (which always begin with a lower case letter). The number of individual constants matches the arity of the predicate. For examples, look at the chart on p. 22.

Note that in writing atomic sentences, we use prefix notation:

Cube(a) Larger(b, c)

the predicate comes first, followed by (in parentheses) one or more names. The one exception is in the case of the identity symbol, =. In this case, we use infix notation: a = b, rather than =(a, b).

Now it is time to start learning about atomic sentences, and when they are true and when they are false. It is also time to start learning about the program Tarskis World.

Introduction to Tarskis World

1. Open the program. Click Start, Programs, LPL Software, Tarskis World 5.6. (Alternatively, click on Tarski.exe. Its in the Tarskis World Folder, inside the LPL Software Folder.) You will find an empty world and an empty sentence file. In the world, add two blocks, of different shapes and sizes. E.g., a large cube and a small tetrahedron.

2. Name them a and b. (To name the cube a, select the cube, then put a check-mark in the box labeled a in the Inspector and click on OK.)

3. In a new sentence file, write four sentences, describing their size and shape. You may type them in, or click on the keyboard on the screen (faster, easier, and more accurate than typing), or even copy (ctrl-C)-and-paste (ctrl-V) from another program.

Cube(a) Tet(b)

Large(a) Small(b)

4. Now, verify the four sentences (ctrl-F). The letter T that appears next to each sentence tells you that the sentence makes a true statement about the world you have constructed.

5. Return to the world. Alter the shapes and sizes of the blocks, as to make all of the sentences false.

6. Back to the sentences. Change them so that they still describe the shapes and sizes of the blocks, but make them all true.

7. Notice the shape and size of a. Add another block of that size and shape, and name it c.

8. Now write the sentence a = c.

9. Predict its truth-value: do you expect it to be true? If so, why? Do you expect it to be false? If so, why?

10. Verify the sentences again. Now you know that a = c is false. What can you do to the world to make it true?

11. Now you see that for a = c to be true, a and c have to name the same block. The equal sign (=) means one and the same object, not two objects that are exactly alike.


Chapter 2: The Logic of Atomic Sentences

2.1 Valid and sound arguments Conclusion

An argument is a piece of reasoning (a sequence of statements) attempting to establish a conclusion. The conclusion is what the arguer is trying to establish. It is indicated by words like therefore, so, hence, thus, consequently. What immediately follows these words is usually the conclusion.

Premises The premises are the reasons the arguer gives in support of the conclusion. They may be given before the conclusion, or after it. Premises are typically preceded by words like because, since, after all.

Validity In a valid argument, the conclusion follows from or is a logical consequence of the premises. Here is our definition of validity:

An argument is valid if it is impossible for its premises to be true and its conclusion false.

The word impossible is important here. The fact that an arguments conclusion is actually true does not make the argument valid validity requires that there be no possible circumstance in which the premises would be true and the conclusion false. Similarly, the fact that an argument contains a false premise means nothing about the arguments validity or invalidity. Some arguments with false premises are valid, and others are invalid. What matters is whether there is any possible circumstance in which the premises would be true and the conclusion false.

Soundness A sound argument is a valid argument with true premises. So, every sound argument is valid, but not every valid argument is sound.

Fitch bar notation In many books, arguments are written up using the 3-dot symbol: ? So, for example, you might see:

Socrates is a man. All men are mortal. ?Socrates is mortal.

In LPL, well use the Fitch bar notation. The premises are written above the horizontal line (the Fitch bar), and the conclusion below:

Socrates is a man. All men are mortal. Socrates is mortal.


Examples Here are two examples of arguments: one valid, one invalid.

Example 1a: a valid argument Cube(a)

Large(a)

SameShape(a, b)

Cube(b)

Example 1b: an invalid argument Cube(a)

Large(a)

SameShape(a, b)

Large(b)

Well return to these arguments later. Well see how to prove the first one valid, and how to show that the second one is invalid. Our method of proof of validity is very different from our method of showing invalidity.

2.2 Methods of proof A step-by-step demonstration showing that the conclusion follows from the premises. In a proof, a series of intermediate conclusions are reached, leading in a chain from the premises to the (ultimate) conclusion. The intermediate conclusions are also written below the Fitch bar. At each step, there must be absolute certainty. That is, there must be no chance that any conclusion (intermediate or otherwise) does not follow from the sentences it is inferred from. Our steps must be such that there is never a possibility that we might be inferring a false sentence from true ones.

Proofs involving the identity symbol Our language so far contains only atomic sentences, which limits our ability to come up with rules for deriving conclusions from premises. But we can take advantage of some features of the identity relation to put our first two rules (concerning the symbol =) into play.

x First, note that the identity relation (the relation that holds between a and b in virtue of which a = b is true) is reflexive. That is, each thing is identical to itself. In other words, sentences like b = b are always true.

x Second, the identity relation is symmetrical. That is, if a = b, then b = a. x Third, the identity relation is transitive. That is, if a = b and b = c, then a = c. x Finally, if a = b, then whatever holds of a also holds of b. This is called the

indiscernibility of identicals. We will enshrine these features of identity in our system of proof by introducing rules that take advantage of them.


2.3 Formal proofs We will be developing a deductive system for writing up formal proofs. We call the system L, and we will be employing a computer program called Fitch that is a somewhat more user-friendly version of L. In a formal proof in L, we use the Fitch bar notation. The premises are written above the (horizontal) Fitch bar; the subsequent steps (intermediate conclusions and the ultimate conclusion) are written below the Fitch bar. Each step in a formal proof must be entered in accordance with some precisely stated rule of the formal system of rules. By applying a rule to some previous line or lines in a proof, we provide a justification for entering a new step in a proof. A justification, then, cites a rule and the lines to which the rule is being applied in order to generate the line being introduced. Our first rules can now be stated:

Identity Introduction (= Intro) n = n The triangle points to the sentence that the rule entitles you to enter. This rule says, in effect, that you may enter a sentence of the form n = n at any point you wish. Obviously, this rule emodies the principle of reflexivity of identity.

Identity Elimination (= Elim) P(n)

n = m

P(m) This rule tells you that you may substitute m for n wherever you like, provided that you have the sentence n = m. This rule embodies the principle of indiscernibility of identicals. Notice that although the rule is called an elimination rule, nothing is really being eliminated. The idea is that we have used (eliminated?) an identity sentence in the process of arriving at a conclusion. That is, we are arguing from an identity sentence, and in that sense we are eliminating it. In L, each logical symbol has a pair of rules associated with it: an introduction rule, which tells you how to get a sentence containing that logical symbol into a proof, and an elimination rule, which tells you how to deduce something from a sentence containing that logical symbol. (For this reason the rules in a system like L are sometimes called int-elim rules.) Thus, = Intro tells us how to enter an identity sentence (we can enter a = a), and = Elim tells us how to use an identity sentence (n = m) as a premise. Dont worry that our two rules seem to have ignored the symmetry and transitivity of identity. In fact, symmetry and transitivity follow from reflexivity and indiscernibility. That is, using only = Intro and = Elim, you can prove that b = a follows from a = b, and that a = c follows from a = b and b = c. (You will be proving transitivity in exercise 2.16 in H3.)


For an illustration of how = Elim works, open Ch2Ex3.prf. Point the focus slider at line 3 and click on both premises; they will be highlighted, meaning that they are your support sentences. Then choose rule = Elim, and click on Check Step. Notice which sentence Fitch inserts, by default. Is this the sentence you expected? (Perhaps you were surprised to see both occurrences of a replaced by b). Try entering a new line that makes only one replacement in line 1, and ask Fitch to check it out. (Be sure to highlight your two support lines by clicking on them.) Then enter yet another line that makes a different single replacement in line 1 and have Fitch check it out. You will notice that = Elim licenses all three of these inferences. Please be aware that (unlike Fitch) L is a very strict system. Its rule = Elim permits us to substitute the name that occurs to the right of the equals sign for the one that occurs to the left, but does not permit us to substitute the name on the left for the one on the right. That is, the rule does not strictly apply to the pair of sentences Cube(b) and a = b. It only applies to the sentences Cube(a) and a = b. Since the Fitch program is more liberal about this fine detail than L is, we will be able to ignore it when were using Fitch.

2.4 Constructing proofs in Fitch You try it

Work the problem on p. 58, using the file Identity 1 (its in the Fitch Exercise Files folder). To see what your proof should look like, open the file Proof Identity 1.prf. (Either click on the link or find the file on the Supplementary Exercises page of the course web site.)

Ana Con This is a mechanism that is built into Fitch. It basically checks to see whether a conclusion does indeed follow from its premises. Ana Con has some limitations: it does not understand the predicates Adjoins and Between, and some complicated arguments may stump it. As we will see, Ana Con uses a broader notion of logical consequence than is strictly allowed in FOL. For example, in FOL we cannot deduce Larger(a, b) from Smaller(b, a). This is because this inference depends on the meaning of the predicates, and FOL is ignorant of the meanings of the predicates in the arguments it examines. But given the meanings of Larger and Smaller, we may note that it is not possible for the first sentence to be true and the second false. So there is a clear sense in which the inference in question is valid. Ana Con takes the meanings of the predicates into account. So well say that Larger(a, b) is an analytic consequence of Smaller(b, a), even though it is not a first-order consequence of it. Try to show this in Fitch by opening Ch2Ex2.prf and using Ana Con to complete the proof.

2.5 Demonstrating nonconsequence We dont use proofs

We do not give proofs of nonconsequence; we do this by means of counterexample. This is because of the following fact:

When we establish that an argument is valid, we establish something quite general. That is, that it is impossible for the premises to be true and the conclusion false. To put it another way,

we establish that in every possible situation in which the premises are true, so is the conclusion.


Conversely, to establish that an argument is invalid, we must show that it is not valid. That is, that it is possible for the premises to be true and the conclusion false. To put it another way, we must establish that there is some possible situation in which the premises are true and the conclusion is false. So when we show an argument to be invalid, we need not prove anything general. It is sufficient to describe a possible situation in which the premises are true and the conclusion is false. Since demonstrating nonconsequence does not involve proofs, we will not be using the program Fitch to show that an argument is invalid. For example, suppose we are using Fitch, and we examine some purported proof of a given argument, and we see that the proof contains a mistake or a misapplication of the rules. That doesnt show the argument to be invalid. Perhaps it is just a faulty proof of a valid argument! In that case, some other proof could be found. But we can use Fitchs Con mechanisms to tell us that an argument is invalid. Lets apply the Ana Con mechanism to our previous examples, Example 1a and Example 1b. To see how, open Ch2Ex1a.prf and Ch2Ex1b.prf.]

We construct counterexamples To demonstrate nonconsequence, we use the program Tarskis World. This program lets us create counterexamples: possible situations, or worlds, in which the premises of an argument are true and its conclusion is false. Well continue with Examples 1a and 1b. Open Ch2Ex1a.sen, Ch2Ex1b.sen, and Ch2Ex1.wld. (1a) is valid, (1b) is invalid. If we change the world slightly, we can make the conclusion of (1b) false while leaving the premises true. (Just make b into a small cube.) But there is no way of making the conclusion of (1a) false while leaving the premises true, for that is a valid argument, and any world in which its premises are true will also make its conclusion true. Now do the You try it on p. 64 to construct your own demonstration of nonconsequence. You will be constructing a counterexample to Bills Argument. If you solved this problem, you should have ended up with a world that looks something like this: Bills Argument.wld.

Deductive vs. Inductive Reasoning In this course, we will be studying deductive reasoning, where we try to determine whether a given conclusion does or does not follow from a certain set of premises. But a good deal of reasoning is not deductive; often, one is interested in something weaker than absolute certainty. One may be interested in whether a set of premises makes the truth of a conclusion more probable, rather than in whether it guarantees the truth of the conclusion. For a good illustration of the difference between these two modes of reasoning, look at the world Deductive vs Inductive.wld and the accompanying sentence file Deductive vs Inductive.sen. We cant see whether there is a block behind the medium cube in the column on the right, but we know from the fact that the sentences in the file are true that there must be one there. For we know that there is a block named b that is in the same column as a and the same row as c.


Can we tell anything about the size and shape of b? Using inductive reasoning, we might conclude that b is a cube. After all, in the left-hand column, all the blocks are of the same shape, and in the middle column, all the blocks are of the same shape. So one might reason, inductively, that the third column will be like the other two in this respect. If it is, the hidden block will be a cube, for the two that are visible are cubes. But there is no certainty here, for we can imagine a world in which b is not a cube. Can we tell what size b is? Here, we can do better. For b must be small. If b were medium or large, it would be at least partially visible. But we cannot see anything of b. Therefore, it must be small. Here, we have used deductive reasoning to establish that a certain conclusion follows from the information that we already have (and not just that it is more likely to be true, given that information). So we have given an inductive argument that b is likely to be a cube, and a deductive argument that b must be small. We can now rotate the world 90 degrees (or switch to a 2-D view) to find out the size and shape of b. As to the size, there can be no surprisestheres no possibility that b can be anything but small, consistent with the information we already have. But as to the shape, we may well be in for a surprise. Note that this is a feature of all inductive arguments: no matter how good the argument is, there is always a possibility, however remote, that the conclusion may be false even though all the premises are true.


12. Write these additional sentences:

Adjoins(a, b) FrontOf(a, b)

SameSize(a, b) Between(a, b, d)

13. Now play with the blocks (move them around) and verify the sentences (ctrl-F) each time you move them. That way, youll see under what conditions these sentences are true, and learn first-hand the meanings of the predicates. Write some more sentences, using the other predicates in the blocks language, and continue to experiment.

14. You will notice, for example:

x Adjoins(a, b) requires that a and b be on squares that share a side; they cannot be diagonally adjacent.

x FrontOf(a, b) requires no more than that a be closer to the front than b; it does not have to be anywhere near b, or even in the same row or column.

x A sentence containing a name that does not name any block in a given world does not have any truth value in that world. To make Between(a, b, d) have a truth value, we had to assign the name d to one of the blocks in the world.

x Between(a, b, d) requires that a, b, and d be in a straight line: either in the same row, column, or diagonal. Note that it is the first named block (a in the sentence above) that is the one in the middle.

x If you try to move blocks in such a way that a large block adjoins another block, you cannot do it! In Tarskis World, no large block can adjoin any other block. (That is because the large blocks are so large they overlap their borders and infringe on the adjacent block.)

x SameSize(a, a), SameShape(a, a), SameRow(a, a), SameCol(a, a), and a = a are always true.

x Larger(a, a), Smaller(a, a), Adjoins(a, a), FrontOf(a, a), BackOf(a, a), RightOf(a, a), LeftOf(a, a), and a z a are always false.

x Between(a, a, a), Between(a, a, b), Between(a, b, a), and Between(b, a, a), etc. are always false. A Between sentence cannot be true unless it contains three different names. (Although even then it may still be false.)

15. These facts all express features of the meanings of the predicates in the blocks language, which closely (although not exactly) match the meanings of their English counterparts. For example, it is part of the meaning of larger than that a thing cannot be larger than itself; it is part of the meaning of is in the same row as that a thing cannot fail to be in the same row as itself.

16. The predicates of the blocks language are determinate, not vague. There is no gradation of sizes between small and medium, and any two objects that are both are small are considered to be of the same size. Hence, every sentence of the blocks language is either true or false. Nor are there degrees of truth and falsitya sentence is either (entirely) true or (entirely) false, and no true sentence is truer than another.

Be sure to do the You try it on p. 24.


Chapter 3: The Boolean Connectives

These are truth-functional connectives: the truth value (truth or falsity) of a compound sentence formed with such a connective is a function of (i.e., is completely determined by) the truth value of its components.

3.1 Negation symbol:

The negation of a true sentence is false; the negation of a false sentence is true. This information is recorded in the truth-table on p. 69. (Here and in other such tables we will abbreviate true by T and false by F.)

P P

T F

F T

This table tells us that the negation of a sentence has the opposite truth value.

Some terminology: If P is atomic, then both P and P are called literals. Thus, Cube(a) and Cube(a) are literals, but Cube(a) is not a literal.

3.2 Conjunction symbol:

Writing conjunctions in FOL and in English

In English, conjunction is expressed by and, moreover, and but.

George is wealthy and John is not wealthy

George is wealthy but John is not wealthy

are both translated in FOL as Wealthy(george) Wealthy(john)

Note that we read this FOL sentence as: Wealthy George and not wealthy John. This way of reading FOL sentences will make it much easier later when we come to write them using Tarskis World.

vs. and

In English, and often conveys a temporal meaning: and then or and next. Thus, these arent equivalent:

Max went home and Claire went to sleep

Claire went to sleep and Max went home

The first suggests that Claire retired after Max left; the second suggests that Max didnt leave until after Claire retired.

But in FOL, the following sentences are equivalent:

WentHome(max) WentToSleep(claire)

WentToSleep(claire) WentHome(max)

That is, requires nothing more than joint truth, not temporal order.


The semantics of

See the truth table for on p. 72.

P Q P Q

T T F F

T F T F

T F F F

This table shows that a conjunction P Q is true in just one case: the case in which P is true and Q is true.

To see how this works, try playing the game in Tarskis World. Do the You try it on p. 72.

in FOL where theres no corresponding English connective

How do we translate d is a large cube into FOL? Although the English sentence has no connective, we treat it as if it had an and in it: d is a cube and d is large. The advantage of this is that it makes translation easyour FOL translation looks like this:

Cube(d) Large(d).

We can do this because in Tarskis World, we treat the size of an object as being entirely independent of its shape. Whether an object is a cube or a tetrahedron has no effect on whether it is counted as large, medium, or small.

This approach to translation into FOL keeps things simple, but it does not always give satisfactory results. Suppose we try putting Dumbo is a small elephant into FOL as:

Elephant(dumbo) Small(dumbo)

But small elephants are still large objects, so one might plausibly assert: Although Dumbo is a small elephant, Dumbo is large. If we put this into FOL using the scheme above, we get:

Elephant(dumbo) Small(dumbo) Large(dumbo)

This translation, however, is problematic. For one thing, this FOL sentence never comes out true, since nothing can be simultaneously, and without qualification, both small and large.

To confirm this, try the following experiment: open Fitch and start a new proof with no premises. Add a new line, and enter the sentence (Elephant(dumbo) Small(dumbo) Large(dumbo)). Now justify the line using Ana Con and click on Check Step. You will see that it checks out, which means that it is always true. Hence the sentence it negates, Elephant(dumbo) Small(dumbo) Large(dumbo), is always false.

For another thing, it is unclear what this FOL sentence is supposed to mean. Since the order of the conjuncts in an FOL conjunction has no effect on its meaning, we could translate it equally well in either of the following ways:

Although Dumbo is a small elephant, Dumbo is large.

Although Dumbo is a large elephant, Dumbo is small.


These English sentences are certainly not equivalent, so they cannot both correspond to the same FOL sentence. In English, Dumbo is a small elephant really means that Dumbo is small for an elephant. But there is no way to express Dumbo is small for an elephant in FOL using only the predicates Small and Elephant and the truth-functional connectives.

3.3 Disjunction symbol:

Writing disjunctions in FOL and in English

In English, disjunction is expressed by or.

George is wealthy or John is wealthy

Either George or John is wealthy

are both translated in FOL as Wealthy(george) Wealthy(john)

We read this FOL sentence as: Wealthy George or wealthy John.

vs. or

In English, or is sometimes used in an exclusive sense, meaning one or the other but not both. But it will be our practice to use it in the (more common) inclusive sense, in which it means one or the other or both. (This is sometimes called and/or.)

Thus, in our example above, the sentence comes out true in the event that both George and John are wealthy. If we need to say that exactly one of the two is wealthy (either George or John but not both), we can always write in FOL:

(Wealthy(george) Wealthy(john)) (Wealthy(george) Wealthy(john))

The semantics of

See the truth table for on p. 75.

P Q P Q

T T F F

T F T F

T T T F

This table shows that a disjunction P Q is true in three cases: P true and Q true, P true and Q false, and P false and Q true. That is, it is false in just one case: the case in which P is false and Q is false.

To see how this works, try playing the game in Tarskis World. Do the You try it on p. 76.

Some connectives that are not truth-functional

Lots of English connective words are not truth-functional. That is, if you use one of these words as the main connective in a compound sentence, the truth-value of the resulting sentence does not depend in all cases solely on the truth-values of the component sentences. An easy way to see that a connective is not truth-functional is to try to construct a truth-table for a compound in which it is the main connective. You will notice that you cannot complete all the rows.


Claire fed Scruffy while Max slept.

Fed(claire, scruffy) Slept(max)

Fed(claire, scruffy) while Slept(max)

T T F F

T F T F

? F F F

In this case, we know that if either component is false, the whole compound must be false. For example, if Claire did not feed Scruffy, it is false that she fed Scruffy while Max slept. The problem occurs when both components are true. It may be true that Claire fed Scruffy and true that Max slept, and nothing follows about whether the feeding and sleeping took place at the same time or not. The truth of both component sentences is compatible with either the truth or the falsity of the entire compound sentence.

Claire went home because she found Max boring

WentHome(claire) Bored(max, claire) WentHome(claire)

because Bored(max, claire)

T T F F

T F T F

? F F F

Once again, we know that if either component is false, the whole compound must be false. For example, if Claire did not find Max boring, it is false that she went home for that reason. Again, the problem occurs when both components are true. It may be true that Claire went home and true that Max bored her, and nothing follows about whether or not his boring her was the reason she went home. The truth of both component sentences is compatible with either the truth or the falsity of the entire compound sentence.

3.4 Remarks about the game

The game rules for , , and are summarized on p. 78. There is no need to memorize them, though, as Tarskis World will always tell you what your commitments are (after you choose your initial commitment), and will tell you when it is your turn to move.

To play the game and be sure of winning, you will need to know not only that a sentence has the truth value you say it has (your commitment), but also why it does. This means, for example, that if you know that a disjunction is true, you will need to know which disjunct is true in order to be sure of winning. Similarly, if you know that a conjunction is false, you will need to know which conjunct is false in order to be sure of winning.


Sometimes, however, you may know the truth value of an entire compound sentence without knowing the truth values of its components. Suppose you have the sentence Cube(d) Cube(d). You know that this is true even though you dont know which disjunct is the true one. If d is a cube, the left disjunct is true; otherwise, its the right disjunct thats true. But you may not be able to see d; perhaps it is small, and hidden behind a larger object. Try exercise 3.11 to see how this works.

3.5 Ambiguity and parentheses

In FOL, we need to be able to avoid ambiguities that can arise in English. The form

P and Q or R

is ambiguous. Does it mean P, and either Q or R? Or does it mean either both P and Q, or R? Notice how the auxiliary words either and both, working with or and and, respectively, remove the ambiguity. (You will see these at work in exercise 3.21, problems 1, 8, and 10.)

FOL does not have such auxiliary words. We use parentheses to remove ambiguity:

P (Q R) (P Q) R

The effect is the same. The parentheses remove the ambiguity by showing which is the main connective, and which the subsidiary. (As we will say, they show which connective has the larger scope.)

Scope is especially important with negation. Compare these sentences:

Cube(a) Cube(b) (Cube(a) Cube(b))

The first says that a is not a cube, but b is a cube. The second does not give us such definite information about a and b. All it tells us is that that arent both cubes. That is, either a is not a cube, or b is not a cube, or perhaps neither is a cube. The first is a much more informative claim.

Practice

Lets check out some sentences in a sample world. Download the files Sentences TF1 and World TF1 from the course web sitetheyre on the Supplementary Exercises page. Then predict the truth values of these sentences in this world, and play the game with Tarskis World.

3.6 Equivalent ways of saying things

There are many different ways of saying the same thing in FOL. That is, for any given FOL sentence, we can come up with a different but equivalent FOL sentence. (Equivalent here means comes out true or false in exactly the same cases, or has the same truth table. Here are some of the more common equivalent pairs. ( represents equivalence).

P P Double negation

(P Q) (P Q) DeMorgans law

(P Q) (P Q) DeMorgans law

Note that these can be combined to yield more equivalences:

(P Q) (P Q) defined in terms of

(P Q) (P Q) defined in terms of


3.7 Translation

Under what conditions do we count an FOL sentence to be a correct translation of an English sentence? The only rule is that the two sentences must agree in truth value in all possible circumstances.

Notice that this requires more than that the two sentences both be true, or both be false. Agreement in (actual) truth value may be due to accidental circumstances that happen to obtain. The two sentences must agree even if you change the facts.

This means that any two equivalent FOL sentences will be equally correct translations of any English sentence that either of them correctly translates. That is, if an FOL sentence S is a good translation of an English sentence S, and S is equivalent to some other FOL sentence S, then S also counts as a correct translation of S.

A result of this policy is that some rather unnatural sounding translations will count as correct. Consider the English sentence b is a cube and c is a tetrahedron. The most natural translation of that into FOL is:

Cube(b) Tet(c)

But given the DeMorgan and Double Negation equivalences noted above, we can see that:

(Cube(b) Tet(c)) (Cube(b) Tet(c))

Hence, our sentence is equally accurately translated as:

(Cube(b) Tet(c))

But even though this is (technically) correct, it is not the best or most natural translation, for it introduces three nots and an or, none of which were present in the English original.

Still, both Tarskis World and I will follow the policy of counting any translation that is equivalent to the right one as correct.

[Note that later in the term, when the sentences get more complicated, the Grade Grinder may not always be able to tell whether an answer you give is equivalent to the correct answer. If that happens, it will tell you that it timed outi.e., couldnt figure out whether your answer was correct. Bring any such cases to your instructor for evaluation.]


Chapter 4: The Logic of Boolean Connectives

4.1 Tautologies and logical truth

Logical truth

We already have the notion of logical consequence. A sentence is a logical consequence of a set of sentences if it is impossible for that sentence to be false when all the sentences in the set are true. We will define logical truth in terms of logical consequence.

Suppose a given sentence is a logical consequence of every set of sentences. That means that it is impossible for that sentence to be false it comes out true in every possible circumstance. Hence:

A sentence is a logical truth if it is a logical consequence of every set of sentences.

Tautology

A tautology is a logical truth that owes its truth entirely to the meanings of the truth-functional connectives it contains, and not at all to the meanings of the atomic sentences it contains.

For example, Cube(a) Cube(a). No matter what shape a is, this sentence comes out true. And it owes its truth entirely to the meanings of or and not. You could replace Cube with any other predicate and a with any other name, and the resulting sentence would still be true. Indeed, you could replace Cube(a) with any other sentence and the resulting sentence would still be true.

Tautologies and truth tables

To show that an FOL sentence is a tautology, we construct a truth table. Look at the example of the table for Cube(a) Cube(a) on p. 96.

Features of truth tables

The number of rows in the table for a given sentence is a function of the number of atomic sentences it contains. If there are n atomic sentences, there are 2n rows.

Each row represents a possible assignment of truth values to the component atomic sentences.

On each row, the values of the atomic sentences determine the values of the compounds of which they are components. The values of the compounds of atomic sentences in turn determine the values of the larger compounds of which they are components. In the end, a unique value for the entire sentence is determined on each row.

A tautology is a sentence that comes out true on every row of its truth table.

Do the You try it on p. 100: Open the program Boole and build the truth table. You will confirm that (A (A (B C))) B is a tautology.


Tautologies, logical truths, and Tarskis World necessities

When we looked at the sentence Cube(a) Cube(a), we noted that it owes its truth entirely to the meanings of or and not. You could replace Cube (both occurrences, of course) with any other predicate, and the resulting sentence would still be true. Indeed, you could replace the two occurrences of Cube(a) with any other sentence and the resulting sentence would still be true.

Contrast this with Cube(a) Tet(a) Dodec(a). Although this sentence always comes out true in Tarskis World, we can imagine a circumstance in which it is not true: suppose that a is a sphere. So this sentence is not even logically true. We can say that it is a Tarskis World necessity, because it comes out true in every world in Tarskis World. (It is a special feature of Tarskis World that there are no objects other than cubes, tetrahedra, and dodecahedra.)

So Tarskis World necessities form a large set of sentences that includes the tautologies as a (smaller) part: every tautology is a Tarskis World necessity, but not conversely.

Note that there are some necessary truths that are not tautologies, but dont depend for their truth on any special features of Tarskis World. For example:

(Larger(a, b) Larger(b, a))

This is not a tautology, for it depends on the meaning of the predicate larger than. But its necessity is not limited to Tarskis World, for it can never be true that both a is larger than b and b is larger than a.

Why Boole cant identify all logical truths

Boole is sensitive to the meaning of the truth-functional connectives, but not to the meanings of the predicates contained in atomic sentences. (In particular, Boole does not recognize the meaning of the identity symbol =, nor does Boole recognize the meanings of the quantifier symbols and that well be studying in chapter 9.)

So when Boole sees a sentence like (Larger(a, b) Larger(b, a)), it does not see the predicate larger than. Instead, all Boole sees is the negation of a conjunction of two different atomic sentences. In effect, all Boole sees is sentence of the form (P Q). And when Boole sees this sentence, it thinks, I know how to make this sentence falseI just assign T to P and T to Q. That makes P Q true, and so it makes (P Q) false. Since Boole cant see inside the atomic sentences and doesnt understand the predicates they contain, he doesnt know that its impossible for both Larger(a, b) and Larger(b, a) to be true.

Now when it comes to tautologies, Boole rules! So any logical truth that Boole doesnt recognize as coming out true in every circumstance is a non-tautology. A row on a truth-table that contains a T under the main connective, then, may not represent a genuine logical possibility.

We have discovered that there is a set of logical truths that falls between the tautologies and the Tarskis World necessities. It is best to picture the situation in terms of a nested group of concentric circles (Euler circles) collecting together a subset of all the true sentences:

The outer, largest, circle: the Tarskis World necessities (sentences that are TW-necessary). It also contains the contents of all the inner circles.

The next largest circle: the logical truths or logical necessities.

If we say truth is a function between a statement and a set of sentences, then:

1. Tautological truth ranges over the broadest class of sentences.2. Logical truth ranges over only sentence sets certain definitions of relation symbols.3. World truths only range over certain worlds.

World necessity: must be true within a given world. But may not be in other worlds?

Logical necessity: must be true given the meaning of a set of relation symbols. But could be true across other universes as long as the meaning of the relation symbols remain the same.

Tautology necessity: must be true given the meaning of logical operators. But could be true across all universes and different interpretations of relations

In a way, tautology is the broadest sense of necessary truth. It is simply truth based on the primitive logical notions of the logical operators.

Truth at the logical necessity entails additional notions of what relation symbols mean.

Truth at the world necessity level entails yet an additional layer of notions about the nature of the universe.

Necessary truths based on:

1. The nature of universe set Tarski world.2. The nature of relation symbols describing the relative "positions" of objects inside universe.3. The nature of logical connective symbols.


The innermost circle: the tautologies (TT-necessary). The relationship is depicted graphically on p. 102 in figure 4.1. You should be able to give examples of each kind of necessary truth.

Note that every tautology is also a logical truth, and every logical truth is also a TW-necessity. But the converse is not true: some logical truths are not tautologies, and some TW-necessities are not logical truths.

Three kinds of possibility

Notice that if we are considering possibility, rather than necessity, we have a similar nest of Euler circles. The difference is that the TT-possible sentencesthe ones that come out true on at least one row of their truth tableare included in the largest circle, and the TW-possible sentences are in the smallest circle. That is, a sentence may be TT-possible without being logically possible or TW-possible, although all TW-possibilities are also logically possible and TT-possible.

Look at exercise 4.10. We are asked to locate these three classes of sentences in an Euler diagram. To see what the circles look like, open Possibility.pdf (on the Supplementary Exercises page of the course web site).

The outer, largest, circle: the TT-possible sentences. It also contains the contents of all the inner circles.

The next largest circle: the logically possible sentences. The innermost circle: the Tarskis World possibilities (TW-possible sentences).

Again, you should be able to give examples of each kind of possibility. You can test your understanding of these different kinds of possibility by completing exercise 4-9 (not assigned for homework).

Id suggest downloading and printing a copy of Possibility.pdf for your notes.

4.2 Logical and tautological equivalence

Logically equivalent sentences

Sentences that have the same truth value in every possible circumstance are logically equivalent.

Tautologically equivalent sentences

Logically equivalent sentences whose equivalence is due to the meanings of the truth functional connectives they contain are tautologically equivalent.

Tautological equivalence and truth tables

To see whether a pair of FOL sentences are tautologically equivalent, we construct a joint truth table for them. The two sentences are tautologically equivalent if they are assigned the same truth value on every row.

Note that sentences may be logically equivalent without being tautologically equivalent. A good example is given on pp. 107-8:

a = b Cube(a) a = b Cube(b)

These sentences are logically equivalentthere is no possible circumstance in which they could differ in truth value. But they are not tautologically equivalent, as the truth table on p. 108 shows.


Note that this truth table contains two rows (rows 2 and 3) that do not represent real logical possibilities, although they do represent truth table possibilities. Row 2, for example, assigns T to a = b, T to Cube(a), and F to Cube(b). This assignment does not represent a real logical possibility, since it is not possible for a to be a cube while b is not a cube if a and b are the same block.

How, then, can this be a truth table possibility? The answer is that, as we saw above, Boole cant see inside atomic sentences and doesnt understand the predicates they contain. As far as Boole is concerned, a = b, Cube(a), and Cube(b) are just three different atomic sentences, to which it can assign any values it likes.

4.3 Logical and tautological consequence

Consequence is the core notion

Q is a logical consequence of P if it is impossible for P to be true and Q false. That is, there is no possible circumstance in which P is true and Q is false.

Both logical truth and logical equivalence are special cases of logical consequence:

A sentence is a logical truth if it is a logical consequence of the empty set of sentences.

Two sentences are logically equivalent if they are logical consequences of one another.

Tautological consequence and truth tables

Q is a tautological consequence of P if in the joint truth table for the two sentences there is no row on which P is true and Q is false.

The relation between logical and tautological consequence

As with tautological truth (and equivalence) vs. logical truth (and equivalence), tautological consequence is a special case of logical consequence. That is, every tautological consequence is also a logical consequence, but the converse does not holdin some cases, Q might be a logical consequence of P but not a tautological consequence.

Examples

Cube(b) is a logical consequence of a = b Cube(a), but not a tautological consequence of it. Thats because theres no possible circumstance in which a = b Cube(a) is true and Cube(b) is false. But the truth table for these sentences does not show this, since (as we saw above) it is allowed to assign T to a = b, T to Cube(a), and F to Cube(b). This is a truth table possibility that is not a real possibility.

The same distinction obtains between logical possibility and TT-possibility. The sentence Cube(a) Tet(a) is TT-possible, since it takes a T in row 1. But that TT-possibility is not a real possibility, for that row represents the (impossible) case in which a is both a cube and a tetrahedron.

We will look at both of these examples again in a moment.


4.4 Tautological consequence in Fitch

Using Fitch to check for consequence

Truth tables are mechanicalthere is an automatic procedure (an algorithm) that always give you an answer to the question whether a sentence is a tautology or whether a sentence is a tautological consequence of another sentence or set of sentences.

Instead of using Boole to construct a truth table, you can use the program Fitch to check whether one sentence is a tautological consequence of a given set of sentences.

Do the You try it on p. 114 to see how to do this.

Taut Con, FO Con, and Ana Con

These are three methods, of increasing strength, that Fitch uses to check for consequence.

Taut Con checks to see whether a sentence is a tautological consequence of some others. It pays attention only to the truth functional connectives. It is the weakest procedure of the three because it only catches tautological consequence, and misses the broader notions of consequence.

FO Con checks to see whether a sentence is a first-order consequence of some others. It pays attention not only to the truth functional connectives but also to the identity predicate and to the quantifiers.

Ana Con checks to see whether a sentence is an analytic consequence of some others. It pays attention not only to the truth functional connectives, the identity predicate, and the quantifiers, but also to the meanings of most (but not all!) of the predicates in the blocks language. This notion comes the closest of the three to that of (unrestricted) logical truth.

If a sentence is a tautological consequence of some others it is clearly also a first-order consequence and an analytic consequence of those sentences. But the converse does not holdsome first-order consequences are not tautological consequences, and some analytic consequences are not first-order consequences.

Examples

Cube(a) Cube(b) is a tautological consequence of Cube(a). This is obviousthere is no assignment of truth-values to these sentences that makes Cube(a) true and Cube(a) Cube(b) false.

Cube(b) is a first-order consequence, but not a tautological consequence, of a = b Cube(a). We can check this out, first in Boole (see file Ch4Ex1.tt), then in Fitch (see file Ch4Ex1.prf).

SameSize(a, b) is an analytic consequence, but not a first-order consequence (and hence not a tautological consequence), of Larger(a, b) Larger(b, a). We can check this out in Fitch (see file Ch4Ex3.prf).

Cube(a) Tet(a) is FO-possible (and hence TT-possible), but not logically possible. We can use Boole (see file Ch4Ex2.tt) to show that it is TT-possible. Notice how, with a little trickery, we can also use Fitch (see file Ch4Ex2.prf) to show both that it is TT-possible and FO-possible, but not logically possible.


A warning about Ana Con

The Ana Con mechanism does not distinguish between logical necessity and TW-necessity. That is, it counts at least some Tarski World consequences as analytic consequences along with logical consequences more narrowly conceived. An example will make this clear.

According to Ana Con, Cube(b) is an analytic consequence of Tet(b) Dodec(b). (Obviously, this is not a first-order consequence, and hence not a tautological consequence either.)

This happens because Ana Con pays attention not only to the meanings of some of the predicates, but also to some of the special features of Tarskis World. Since in Tarskis World there are only three shapes of blocks, it follows that there cannot be a Tarski World in which an object is neither a tetrahedron nor a cube nor a dodecahedron.

But while that may be true for every Tarski World, it does not hold for every possible world. In general, it does not follow logically, from the fact that b is neither a tetrahedron nor a dodecahedron, that b is a cubeb might be a sphere. So this example does not seem to be a logical necessity, but only something weakera TW-necessity.

Ana Con also has some other limitations. It misses certain TW-necessities, namely, those involving the predicates Adjoins and Between, which it does not understand. For example, Large(a) is a TW-consequence of Adjoins(a, b), since it is impossible in a Tarski world for a large block to adjoin another block. But Ana Con will not recognize this consequence.

Similarly, Ana Con does not understand any predicates that are not in the blocks language. Hence, it will not know that Older(b, a) is a logical consequence of Younger(a, b), since these predicates are not in the blocks language. So you must use Ana Con with caution!


Chapter 5: Methods of Proof for Boolean Logic

5.1 Valid inference steps

Conjunction elimination

Sometimes called simplification. From a conjunction, infer any of the conjuncts.

From P Q, infer P (or infer Q). Conjunction introduction

Sometimes called conjunction. From a pair of sentences, infer their conjunction.

From P and Q, infer P Q. 5.2 Proof by cases

This is another valid inference step (it will form the rule of disjunction elimination in our formal deductive system and in Fitch), but it is also a powerful proof strategy.

In a proof by cases, one begins with a disjunction (as a premise, or as an intermediate conclusion already proved). One then shows that a certain consequence may be deduced from each of the disjuncts taken separately. One concludes that that same sentence is a consequence of the entire disjunction.

From P Q, and from the fact that S follows from P and S also follows from Q, infer S.

The general proof strategy looks like this: if you have a disjunction, then you know that at least one of the disjuncts is trueyou just dont know which one. So you consider the individual cases (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from it. You repeat this process for each disjunct. So it doesnt matter which disjunct is trueyou get the same conclusion in any case. Hence you may infer that it follows from the entire disjunction.

In practice, this method of proof requires the use of subproofswe will take these up in the next chapter when we look at formal proofs.

5.3 Indirect proof: proof by contradiction

Also called indirect proof or reductio ad absurdum, this is a powerful method of proof commonly used in mathematics.

In a proof by contradiction, one assumes that ones conclusion is false, and then tries to show that this assumption (together with the arguments premises) leads to a contradiction. This shows that the conclusion cannot be false if all the premises are truei.e., that the conclusion must be true if the premises are true. That is to say, that the conclusion is a logical consequence of the premises.

We will develop this idea as a way of establishing a negative conclusion. Suppose you wish to establish that a conclusion of the form S is a logical consequence of a set of premises P1, P2, Pn. You assume S (equivalent to the negation of the arguments conclusion) and treat it as a premise along with P1, P2, Pn. You then try to deduce from these assumptions a contradictiona pair of sentences that contradict one another, e.g., Q and Q. You may then (no longer assuming S) conclude that S.


For an example of indirect proof, see the proof on p. 136. In this example, we get these contradictions: Cube(b) contradicts Tet(b), and Cube(b) contradicts Dodec(b). These are not TT-contradictions (like Cube(b) Cube(b)), but they are still logically contradictory, in that it is impossible for them both to be true.

The contradiction symbol

In constructing proofs we will use the symbol (an upside down tee) to indicate that a contradiction has been reached. Rather than struggle for a way to pronounce this symbol, we will read simply as contradiction.

TT-contradictions vs. other types

Not all contradictions are TT-contradictions. Consider these examples:

Cube(b) Cube(b) TT-contradiction a a FO-contradiction (but not a TT- contradiction) Cube(b) Tet(b) Logical contradiction (but not a FO-

contradiction)

Large(b) Adjoins(b, c) TW-contradiction (but not a logical contradiction)

The reasons for this classification are as follows:

Cube(b) Cube(b) A truth table shows that this sentence cannot be true. Hence, it is a TT-contradiction.

a a No truth table can show that this sentence cannot be true, for a truth table can assign F to a = a. So it is not a TT-contradiction. But the meaning FOL assigns to = and its rules for using names like a make the sentence a = a true in every world. That is, an identity sentence is true in any world in which the names it contains name the same object, and no name can name two different objects in the same world. So a a is an FO-contradiction.

Cube(b) Tet(b) FOL does not assign any particular meaning to the predicates Cube and Tet. For all FOL knows, this sentence can be true. So it is not an FO-contradiction. But given the meanings of these predicates, this sentence cannot be trueit is logically impossible for something to be both a cube and a tetrahedron. Hence, this sentence is a logical contradiction.

Large(b) Adjoins(b, c) Given the meanings of the predicates Large and Adjoins, it should be perfectly possible for this sentence to be true. That is, we can describe a situation in which a large object adjoins another object. So this sentence is not a logical contradiction. However, there is no Tarski World in which this sentence is true. Hence, it is a TW-contradiction.


The basic rule (called introduction) that we will use in our formal system (and in Fitch) to show that a contradiction has been reached will require that our contradictory sentences be TT-contradictory. This will require some extra footwork in cases in which we have other kinds of contradictions.

5.4 Arguments with inconsistent premises

If a set of premises is inconsistent, any argument having those premises is valid. (If the premises are inconsistent, there is no possible circumstance in which they are all true. So no matter what the conclusion is, there is no possible circumstance in which the premises are all true and the conclusion is false.

But no such argument is sound, since a sound argument is not only valid but has true premises.

Why be interested in arguments with inconsistent premises? Well, we know that if you can derive a contradiction from a set of premises, the set is inconsistent. (If it were possible for the premises all to be true, then since we have derived from them, it would have to be possible for to be true, and this clearly is not possible.)

We may not know, at the start, that our premises are inconsistent, but if we derive from them, we have established that they are inconsistent. If a set of premises, or assumptions, is inconsistent, it is important to know this. And being able to deduce a contradiction from them is an excellent way of showing this. We may not be able to show, using logic alone, which premise is false, but we can establish that at least one of them is false.

Inconsistent premises vs. impossible sentences

If a set of premises (or any set of sentences, actually) is inconsistent, then at least one of the sentences in the set must be false. But which one is false depends on the worldthere need not be a single sentence which is always the culprit, independent of what the facts happen to be.

To see this, open Ch5Ex1.sen and Ch5Ex1.wld on the Supplementary Exercises web page. You will see that it is impossible for all of the sentences to come out trueno matter how you change the world, at least one sentence comes out false. You can make any three of them true, but you cant make all four true.

Contrast this case with the case of Ch5Ex2.sen and Ch5Ex2.wld. Here we have an inconsistent set of sentences where there is a culpritthe last sentence cannot be true (it is, in fact, TT-impossible). It is truly a bad apple: it will make any set of sentences it belongs to inconsistent.

To see the inconsistency of these sets of sentences, open Ch5Ex1.prf and Ch5Ex2.prf. These two Fitch proofs contain the sets of sentences above as their premise-sets.

Notice that in both cases, the arguments are valid. That is, in both cases, is a tautological consequence of the premises. (Check this out using Taut Con.) Notice, too, that in Ex1, the argument checks out only if all four premises are cited. But in Ex2, the argument checks out if (and only if) the culprit premise is cited.

Here we see the difference between two kinds of inconsistent sets: one (Ex2) contains an impossible sentence, the other (Ex1) does not. Each sentence in Ex1 is possible; what is impossible is the conjunction of all four.


A connection between validity and inconsistency

When an argument is valid, its conclusion is a logical consequence of its premises. Another way to put this is to say that it would be inconsistent to assert the premises and deny the conclusion.

This means that for an argument to be valid is for the set of sentences consisting of all of the premises together with the negation of the conclusion to be inconsistent.

Examples

This set of sentences is inconsistent:

{Cube(a) Cube(b), Cube(a), Cube(b)}

And so this argument is valid:

Cube(a) Cube(b)

Cube(a)

Cube(b)

This set of sentences is consistent:

{Tet(a) Tet(b), Tet(a), Tet(b)}

And so this argument is invalid:

Tet(a) Tet(b)

Tet(a)

Tet(b)

Remember: for an argument to be valid is for its premises to be inconsistent with the negation of its conclusion.


Chapter 6: Formal Proofs and Boolean Logic

The Fitch program, like the system F, uses introduction and elimination rules. The ones weve seen so far deal with the logical symbol =. The next group of rules deals with the Boolean connectives , , and .

6.1 Conjunction rules

Conjunction Elimination ( Elim)

P1 Pi Pn

Pi

This rule tells you that if you have a conjunction in a proof, you may enter, on a new line, any of its conjuncts. (Pi here represents any of the conjuncts, including the first or the last.)

Notice this important point: the conjunction to which you apply Elim must appear by itself on a line in the proof. You cannot apply this rule to a conjunction that is embedded as part of a larger sentence. For example, this is not a valid use of Elim:

1. (Cube(a) Large(a))

2. Cube(a) x Elim: 1 The reason this is not valid use of the rule is that Elim can only be applied to conjunctions, and the line that this proof purports to apply it to is a negation. And its a good thing that this move is not allowed, for the inference above is not validfrom the premise that a is not a large cube it does not follow that a is not a cube. a might well be a small cube (and hence not a large cube, but still a cube).

This same restrictionthe rule applies to the sentence on the entire line, and not to an embedded sentenceholds for all of the rules of F, by the way. And so Fitch will not let you apply Elim or any of the rules of inference to sentences that are embedded within larger sentences.

Conjunction Introduction ( Intro)

P1

Pn

P1 Pn

This rule tells you that if you have a number of sentences in a proof, you may enter, on a new line, their conjunction. Each conjunct must appear individually on its own line, although they may occur in any order. Thus, if you have A on line 1 and B on line 3, you may enter B A on a subsequent line. (Note that the lines need not be consecutive.) You may, of course, also enter A B.


Default and generous uses of rules

Unlike system F, Fitch has both default and generous uses of its rules. A default use of a rule is what will happen if you cite a rule and a previous line (or lines) as justification, but do not enter any new sentence. If you ask Fitch to check out the step, it will enter a sentence for you. A generous use of a rule is one that is not is not strictly in accordance with the rule as stated in F (i.e., F would not allow you to derive it in a single step), but is still a valid inference. Fitch will often let you do this in one step.

Default and generous uses of the rules

Default use: if you cite a conjunction and the rule Elim, and ask Fitch to check out the step, Fitch will enter the leftmost conjunct on the new line.

Generous use: if you cite a conjunction and the rule Elim, you may manually enter any of its conjuncts, or you may enter any conjunction whose conjuncts are among those in the cited line. Fitch will check out the step as a valid use of the rule.

Note just how generous Fitch is about Elimfrom the premise

A B C D

Fitch will allow you to obtain any of the following (among others!) by a generous use of the rule:

A B C D A B A C A D B C B D C D A B C B A D D A C B A C D

6.2 Disjunction rules

Disjunction Introduction ( Intro)

Pi

P1 Pi Pn

This rule tells you that if you have a sentence on a line in a proof, you may enter, on a new line, any disjunction of which it is a disjunct. (Pi here represents any of the disjuncts, including the first or the last.)


Disjunction Elimination ( Elim)

This is the formal rule that corresponds to the method of proof by cases. It incorporates the formal device of a subproof.

A subproof involves the temporary use of an additional assumption, which functions in a subproof the way the premises do in the main proof under which it is subsumed.

We place a subproof within a main proof by introducing a new vertical line, inside the vertical line for the main proof. We begin the subproof with an assumption (any sentence of our choice), and place a new Fitch bar under the assumption:

Premise

Assumption for subproof

The subproof may be ended at any time. When the subproof ends, the vertical line stops, and the next line either jumps out to the original vertical proof line, or a new subproof may be begun. As well see, Elim involves the use of two (or more) subproofs, typically (although not necessarily) entered one immediately after the other.

The rule:

P1 Pn

P1

S

Pn

S

S

What the rule says is this: if have a disjunction in a proof, and you have shown, through a sequence of subproofs, that each of the disjuncts (together with any other premises in the main proof) leads to the same conclusion, then you may derive that conclusion from the disjunction (together with any main premises cited within the subproofs).

This is clearly a formal version of the method of proof by cases. Each of the Pi represents one of the cases. Each subproof represents a demonstration that, in each case, we may conclude S. Our conclusion is that S is a consequence of the disjunction together with any of the main premises cited within the subproofs.

When you do the You try it on p. 151, notice, as you proceed through the proof, that after step 4 you must end the subproof first, before you begin the next subproof.

To do these things, you can click on the options in the Proof menu. But it is easier and quicker to use the keyboard shortcuts: to end a subproof, press Control-E; to begin a new subproof, press Control-P. Another handy shortcut is Control-A for adding a new line after the current line, as part of the same proof or subproof. (Any time you add a new line, Fitch will wait for you to write in a sentence and cite a justification for it.)


Note also that the use of Reit is strictly optional. For example, in the proof on p. 151, step 5 is not required. The proof might look like the one in Page 151.prf (on Supplementary Exercises page) and it will check out.


Default uses

o Elim: if you cite a disjunction and some subproofs, with each subproof beginning with a different disjunct of the disjunction, and all subproofs ending in the same sentence, S, cite the rule Elim, and ask Fitch to check it out, Fitch will enter S.

o Intro: if you cite a sentence and the rule Intro, and ask Fitch to check it out, Fitch will enter the cited sentence followed by a , and wait for you to enter whatever disjunct you wish.

Generous use: if your cited disjunction contains more than two disjuncts, you dont need a separate subproof for each disjunct. A subproof may begin with a disjunction of just some of the disjuncts of the cited disjunction. When you ask Fitch to check the step, Fitch will check it out as a valid use of the rule, so long as every disjunct of the cited disjunction is either a subproof assumption or a disjunct of such an assumption.

6.3 Negation rules

Negation Elimination ( Elim)

This simple rule allows us to eliminate double negations.

P

P

Negation Introduction ( Intro)

This is our formal version of the method of indirect proof, or proof by contradiction. It requires the use of a subproof. The idea is this: if an assumption made in a subproof leads to , you may close the subproof and derive as a conclusion the negation of the sentence that was the assumption.

P

P

To use this rule, we will need a way of getting the contradiction symbol, , into a proof. We will have a special rule for that, one which allows us to enter a if we have, on separate lines in our proof (or subproof) both a sentence and its negation.

Introduction ( Intro)

P

P


Note that the cited lines must be explicit contradictories, i.e., sentences of the form P and P. This means that the two sentences must be symbol-for-symbol identical, except for the negation sign at the beginning of one of them. It is not enough that the two sentences be TT-inconsistent with one another, such as A B and A B. Although these two are contradictories (semantically speaking) since they must always have opposite truth-values, they are not explicit contradictories (syntactically speaking) since they are not written in the form P and P.

To try out these two rules, do the You try it on p. 156.

Other kinds of contradictions

The rule of Intro lets us derive whenever we have a pair of sentences that are explicit contradictories. But there are other kinds of contradictory pairs: non-explicit TT-contradictories, FO-contradictories that are TT-consistent, logical contradictories that are FO-consistent, and TW-contradictories that are logically consistent. Here are some examples of these other types of contradictory pairs:

1. Tet(a) Tet(b) and Tet(a) Tet(b)

2. Cube(b) a = b and Cube(a)

3. Cube(b) and Tet(b)

4. Tet(a) Cube(a) and Dodec(a)

In example (1) we have TT-contradictory sentences but not an explicit contradiction, as defined above. In (2) we have a pair of sentences that are FO-inconsistent (they cannot both be true in any possible circumstance), but not TT-inconsistent (a truth-table would not detect their inconsistency). In (3) we have a pair that are logically inconsistent but not FO-inconsistent (or TT-inconsistent). Finally, in (4) we have a pair that are TW-contradictories (there is no Tarski world in which both of these sentences are simultaneously true), although they are logically consistentit is possible for an object to be neither a tetrahedron nor a cube nor a dodecahedron (it may be a sphere).

The rule of Intro does not apply directly in any of these examples. In each case it takes a bit of maneuvering first before we come up with an explicitly contradictory pair of sentences, as required by the rule.

Example 1

1. Tet(a) Tet(b) 2. Tet(a) Tet(b)

3. Tet(a) Elim: 2 4. Tet(b) Elim: 2

5. Tet(a) 6. Intro: 3, 5

7. Tet(b) 8. Intro: 4, 7 9. Elim: 1, 5-6, 7-8


Here we used Elim twice, to get the two conjuncts of (2) separately, and then constructed a proof by cases to show that whichever disjunct of line (1) we choose, we get to an explicit contradiction.

Example 2

1. Cube(b) a = b 2. Cube(a)

3. Cube(b) Elim: 1 4. a = b Elim: 1 5. Cube(b) = Elim: 2, 4 6. Intro: 3, 5

Here we used Elim to get Cube(b) and a = b to stand alone, and then = Elim (substituting b for a in line 2) to get the explicit contradictory of Cube(b).

Example 3

1. Cube(b) 2. Tet(b)

3. Tet(b) Ana Con: 1 4. Intro: 2, 3

Here we had to use Ana Con. Of course, as long as we were going to use Ana Con at all, we could have used it instead of Intro to get our contradiction, as follows:

1. Cube(b) 2. Tet(b)

3. Ana Con: 1, 2

Example 4

1. Tet(a) Cube(a) 2. Dodec(a)

3. Ana Con: 1, 2

To see these different forms of contradictions in action, do the You try it on p. 159. Its an excellent illustration of these differences. Youll find that you often need to use the Con mechanisms to introduce a into a proof, since Intro requires that there be an explicit contradiction in the form of a pair of sentences P and P.


Elimination ( Elim)

P

The rule of elimination is added to our system strictly as a conveniencewe do not really need it. It allows us, once we have a in a proof, to enter any sentence we like. (Weve already seen that every sentence follows from a contradiction.) As p. 161 shows, we can easily do without this rule with a four step workaround.


Note the default and generous uses of these rules in Fitch (p. 161). With Elim, you dont need two steps to get from P to P (passing through the intermediate step P). You can do it in one step. In fact, this is also the default use of the rule (if you cite the rule and ask Fitch to fill in the derived line).

In the case of Intro, where the subproof assumption is a negation, P, and the subproof ends with a :

Default use: if you end the subproof, cite the subproof and rule Intro, and ask Fitch to check the step, Fitch will enter the line P.

Generous use: if you end the subproof, enter the line P manually, cite the subproof and rule Intro, and ask Fitch to check the step, Fitch will check it out as a valid use of the rule.

6.4 The proper use of subproofs

Once a subproof has ended, none of the lines in that subproof may be cited in any subsequent part of the proof. Look at the proof on p. 163 to see what can happen if this restriction is violated.

How Fitch keeps you out of trouble

When you are working in system F, you can enter erroneous lines like line 8 on p. 163 and never be aware of it. But Fitch wont let you do this! To see what happens, look at Page163.prf.

Notice that when we try to justify line 8 by means of Intro, Fitch will not let us cite the line that occurs inside the subproof that has already been closed.

When a subproof ends, we say that its assumption has been discharged. After an assumption is discharged, one may not cite any line that depended on that assumption.

Note that it is permissible, while within a subproof, to cite lines that occur outside that subproof. So, for example, one may, while within a subproof, refer back to the original premises, or conclusions derived from them. One must just take care not to cite lines that occur in subproofs whose assumptions have been discharged.

Subproofs may be nestedone subproof may begin before another is ended. In such cases, the last subproof begun must be ended first. The example on p. 165 illustrates such a nested subproof.

6.5 Strategy and tactics

Keep in mind what the sentences in your proof mean

Dont just look at the sentences in your proof as meaningless collections of symbols. Remember what the sentences mean as you try to discover whether the argument is valid.


If youre not told whether the argument is valid, you can use Fitchs Taut Con mechanism to check it out. If you discover that the argument is not valid, you should not waste time trying to find a proof.

Try to sketch out an informal proof

This will often give you a good formal proof strategy. An informal indirect proof can be turned into a use of Intro in F. An informal proof by cases can be turned into a use of Elim in F.

Try working backwards

This is a very basic strategy. It involves figuring out what intermediate conclusion you might reach that would enable you to obtain your ultimate conclusion, and then taking that intermediate conclusion as your new goal. You can then work backwards to achieve this new goal: figure out what other intermediate conclusion you might reach that would enable you to obtain your first intermediate conclusion, and so on. Working backward in this way, you may discover that it is obvious to you how to obtain one of those intermediate conclusions. You then have all the pieces you need to assemble the proof.

Fitch is very helpful to you in using this strategy, for you can work from the bottom up as well as from the top down. To see this, do the You try it on p. 168 (open the file Strategy 1). You will note that you can cite a line, or a subproof, as part of a justification even before you have justified the line itself. This shows up with the two innermost subproofs (3-5 and 6-8) which can be used in the justification of line 9 even before lines 5 and 8 themselves have been justified.

This gives you a good method for checking out your strategy.

An example

(A B) (C D) C B

D

Open Ch6Ex2a youll find this problem on the Supplementary Exercises page of the web site. We can start by working backwards. We can get D from D by assuming D and using Intro. So our goal will be to get D.

Our first premise is a disjunction, so that suggests a proof by cases. We will have a separate subproof for each case, deriving D at the end of each subproof. Open Ch6Ex2b Notice that our strategy checks out when we apply Elim, and that our strategy for obtaining D also checks out.

Case 1: A B

The second conjunct, B, contradicts the second conjunct of premise 2. So we can derive by Intro and then derive D by Elim.

Case 2: C

The first conjunct, C, contradicts the first conjunct of premise 2. So we can derive by Intro and then derive D by Elim.


Case 3: D We already have D, so we can use Reit to enter it as a conclusion in the subproof. In fact, we can even skip the Reit step, as well see.

Well now go after case 1. Open Ch6Ex2c, and follow the strategy above. Notice that the steps check out. Finally, well complete cases 2 and 3. Open Ch6Ex2d, and follow the strategy for cases 2 and 3.

Notice that in case 3, we did not need to use Reit. In this case, our subproof contains only the assumption line. In such a case, we count the assumption line itself as the last line in the subproof, and hence we take that line to have been established, given the assumption. This is obviously acceptable, since every sentence is a consequence of itself.

6.6 Proofs without premises

Both system F and the program Fitch are set up so that a proof may begin with some line other than a premise. For example, it might begin with a use of = Intro. Or, it may begin with a subproof assumption.

This means that we may have a proof that has no premises at all! What does such a proof establish? Since a proof establishes that a conclusion is a logical consequence of its premises (i.e., that it must be true if they are), a proof without premises establishes that its conclusion is a logical consequence of the empty set of premises. That is, it establishes that its conclusion must be true, period.

In other words, such a proof establishes that its conclusion is a logical truth. See pages 173-4 for examples of such proofs. (The conclusion of a proof without premises is often called a theorem, although Barwise and Etchemendy do not use that terminology.)

For a try at proving a logical truth in Fitch, try exercise 6.33. Can you think of a simpler proof of the same logical truth? You can find one at Proof 6.33 simpler.

A Difficult example: 6.41

Well try to prove this starred tautology: (A B) A B (And we will do so the hard way, without using Taut Con to justify an instance of Excluded Middle.)

Our tautology says that either A and B are both true, or at least one of them is false. To prove a tautology, it is often easiest to use indirect proof: assume the negation of what were trying to prove, and show that it leads to a contradiction. That is the method we will use.

To see the general strategy for our proof, open Proof 6.41a. We assume the negation of our desired conclusion, and aim to derive . We can then apply rule Intro (generous version) to get our theorem. Note that this last step checks out.

Now what we have assumed is the negation of a disjunction. So if what weve assumed is true, each of the disjuncts is false. In particular, both A and B are false. So, given our assumption, we should be able to prove both A and B. That is what we will do next. We will do so by indirect proof: open Proof 6.41b. Note that our indirect method for proving both A and B checks out.


Our strategy is to assume A, reach a , and deduce A. Then we do the same for B. So we have three questions to answer. (1) How do we show that A leads to a contradiction? (2) How do we show that B leads to a contradiction? (3) Having established both A and B, how do we show that that in turn leads to a contradiction? The answer is the same in every case: by using judiciously chosen applications of Intro.

Our two (inner) assumptions (A and B) are, in fact, disjuncts of the theorem were trying to prove. Hence, we can get from each of those assumptions to the theorem in one application of Intro. That wont prove the theorem (we still have an open assumption), but it will give us a sentence that contradicts our assumption, which is exactly what we want.

To see the complete proof, open Proof 6.41c.

An alternative strategy for 6.41: proof by cases

Notice that a different strategy might yield an equally correct, but much more complicated proof. To see the alternative strategy, open Proof 6.41d.

The idea here is to do a proof by cases:

Case 1

Assume (A B) and derive the theorem.

Case 2

Assume (A B) and derive the theorem.

We can use Taut Con to obtain the disjunction (A B) (A B) that we need. Then we can apply rule Elim and complete the proof by cases. Note that both of these rule applications check out in Fitch.

Case 1 is easy: it takes only one step of Intro. But case 2 is complicated. To develop the alternative strategy further, open Proof 6.41e. The idea is to try to obtain the right-most disjunction, A B, by indirect proof. So we assume its negation, viz., (A B).

Note what this sentence says: neither not A nor not B, which is equivalent to A and B. But (A B) contradicts our assumption line, so once we have it in our proof, we can obtain . (Note that our use of Intro will check out in Fitch.) Our next task will be to obtain the conjunction (A B) from our indirect proof assumption. We know that it follows, because its an instance of one of DeMorgans laws. But those laws are not part of system F, so we will need a different strategy. We will obtain each of A and B separately, and then use Intro to get A B.

To obtain A we use an indirect proof; then we do the same for B. To see how the strategy now looks, open Proof 6.41f. The remaining steps are simple. We assume A for indirect proof. The line we need to contradict is (A B). But A is one of the disjuncts of our negated disjunction. So we use Intro to get the disjunction A B, and we have our contra-diction. This lets us obtain A by means of a generous use of Intro. We repeat this for B.

To see the complete proof, open Proof 6.41g.


Chapter 7: Conditionals

We next turn to the logic of conditional, or if then, sentences. We will be treating if then as a truth-functional connective in the sense defined in chapter 3: the truth value of a compound sentence formed with such a connective is a function of (i.e., is completely determined by) the truth value of its components.

Not all sentence-forming connectives are truth-functional. Consider because. It is obvious that we could not fill out a truth table for the sentence P because Q. How would we fill out the value of P because Q in the row where P and Q are both true? There is no way to do this.

Consider a sentence like Tom left the party because Lucy sneezed. Suppose that both component sentences are true. What is the truth value of the entire compound? You cant tellit could be either. If Tom and Lucy had prearranged that Lucy would sneeze as a signal to Tom that it was time to leave, the sentence would be true. But if Lucy just happened to sneeze and Tom left, but for some other reason, it would be false. So because is not a truth-functional connective.

This should be a tip-off that you should not read any kind of causal connection into the if then that we will be introducing into FOL.

7.1 Material conditional symbol:

Truth table definition of

Here is the truth table that appears on p. 178:

P Q P Q

T T F F

T F T F

T F T T

Here P is the antecedent and Q is the consequent. (The antecedent is on the left, with the arrow pointing from it; the consequent is on the right, with the arrow pointing to it.)

As the truth table shows, a conditional sentence comes out true in every case except the one where the antecedent is true and the consequent false. That is, P Q is equivalent to both of these Boolean forms:

P Q (P Q)

Hence, adds no new expressive power to FOL (anything we can say using we can also say without it, just using and or and ). But the new symbol makes it easier to produce FOL sentences that correspond naturally to sentences of English.

English forms of the material conditional

It is convenient to read sentences in English using if then. That is, we read P Q ( P arrow Q) as if P, then Q. But there are many other ways in English of saying the same thing, and hence many other ways of reading sentences in English:

Q if P P only if Q Q provided that P Q in case P

Provided P, Q In the event that P, Q


Note the variation in word order: in English (unlike FOL) the antecedent (in this case P) doesnt always come first.

If you are looking for a way of reading P Q in English that begins with the sentence that replaces P, the only formulation that works is P only if Q.

People sometimes read P Q as p implies q. This is handy, in that it gives you a way to read the FOL sentence from left to right, symbol-for-symbol, maintaining the word order. But there is something misleading about it, for it suggests a confusion between the truth of an ifthen sentence and a logical implication. That is because p implies q is even more often used as a s

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