PHIL12A Section Answers, 14 February 2011

Home , Atomic sentence, Tautology (logic)

PHIL12A Section answers, 14 February 2011

Julian Jonker

1 How much do you know?

1. You should understand why a truth table is constructed the way it is: why are the truth values listed in the order they are? In principle, it doesn’t matter how they are listed, but there is a natural way to list them systematically, which we use.

(a) My favourite slot machine gives me money if I line up four cherries in a row. When I pull the lever, each of the four displayed items can come up as either a cherry or a banana. How many different combinations can the machine display? List them. This is exactly like listing all possible truth values for four atomic sentences, except with say, cherry for True and banana for False. There are 2 × 2 × 2 × 2 = 24 = 16 possible combinations. Think of it in the following way: Suppose you were betting on just one item. It could come up as a cherry or banana, so there are two possible ‘combinations’. Now suppose you were betting on two items. The ﬁrst item can be a cherry, but there are two possible ways for it to be a cherry if we include the possible values of the second item: cherry cherry cherry banana But this is true also if it comes up as a banana. banana cherry banana banana So there are four possible combinations. More particularly, we have 2 × 2 = 22 = 4 possibilities. If we add a third item, then the ﬁrst item can come up in four different ways: if it comes up as a cherry, then there are two different ways the second item can come up (cherry or banana), and for each way the second item comes up there are two different ways the third item can come up. So all in all there are 2 × 2 × 2 = 23 = 8 possibilities. The same holds for a truth table. If you have n atomic sentences, you will have 2n rows in your truth table. In other words, the number of rows grows exponentially, which is why you will never be asked to do a truth table for more than a handful of atomic sentences!

1 (b) Suppose the machine displays three different items at a time – my goal is to line up three cherries. But each item could be either a cherry, a banana, a palm tree or a robot. How many combinations are possible? It would take a long time to list them all, but convince yourself that there is a way to list them systematically so that you are sure you don’t leave any out. What would the 17th item on your list be?

There are 4 × 4 × 4 = 43 = 64 possible combinations. Think of the list as a sort of tree. For the ﬁrst item, draw four branches for the different possibilities: cherry, banana, palm tree, robot. At the end of the cherry branch, draw another four branches for the possible values of the second item (cherry, banana, palm tree, robot), and then do this three more times at the end of each of the other branches. You should now have sixteen end points. At the end of each end point, draw four branches for the possible values of the third item. All in all you should have 64 end points now, which you could number if you wish. If you drew the branches in the order I mentioned them here (you didn’t have to, but you should have stuck with some kind of order), your 17th possibility would have been: banana cherry cherry

2. Draw up truth tables for the following sentences, and say whether they are tautologies or not.

(a) (A ∨ ¬B) ∨ (¬A ∨ B)

A B (A ∨ ¬ B) ∨ (¬ A ∨ B) T T TTFT T FTTT T F TTTF T FTTF F T FFFT T TFTT F F FTTF T TFTF

(b) (Ex 4.4) ¬(B ∧ ¬C ∧ ¬B) B C ¬( B ∧ ¬ C ∧ ¬ B) T T T TFFTFFT T F T TTTFFFT F T T FFFTFTF F F T FFTFFTF

2 (c) (¬A ∨ (¬B ∨ C)) ∨ (¬(A ∧ B) ∨ C) A B C (¬ A ∨ (¬ B ∨ C)) ∨ (¬ (A ∧ B) ∨ C) T T T FTTFTTT T FTTTTT T T F FTFFTFF F FTTTFF T F T FTTTFTT T TTFFTT T F F FTTTFTF T TTFFTF F T T TFTFTTT T TFFTTT F T F TFTFTFF T TFFTTF F F T TFTTFTT T TFFFTT F F F TFTTFTF T TFFFTF

(d) (Ex 4.6) ¬[¬A ∨ ¬(B ∧ C) ∨ (A ∧ B)] A B C ¬[ ¬ A ∨ ¬ (B ∧ C) ∨ (A ∧ B)] T T T F FTFFTTTTTTT T T F F FTTTTFFTTTT T F T F FTTTFFTTTFF T F F F FTTTFFFTTFF F T T F TFTFTTTTFFT F T F F TFTTTFFTFFT F F T F TFTTFFTTFFF F F F F TFTTFFFTFFF

2 Something slightly harder, if there’s time.

A sentence S is tautological if and only if every row of its truth table assigns True to S. S is logically necessary if and only if it is true in every logically possible circumstance. S is TW-necessary if and only it is true in every world that could possibly be constructed in Tarski’s World.

1. Is the following sentence a tautology? ‘It is not the case that both all men are mortal and Socrates is a man, otherwise Socrates is mortal.’ Is it logically necessary?

The sentence is logically necessary, since there is no possible world in which it is false. There may be some possible worlds in which Socrates is mortal, and in that case the sentence is true. In the other worlds, at least one of the following must be true: not all men are mortal (since Socrates isn’t), or Socrates isn’t a man and isn’t mortal like them. So in these cases the sentence is true.

Now consider whether the sentence is a tautology or not. The atomic sentences are: ‘all men are mortal’, ‘Socrates is a man’, and ‘Socrates is mortal’, and they are combined by the connectives ¬, ∧ and ∨. So we

3 could abbreviate the sentence in the following way: ¬(P ∧ Q) ∨ R, where P stands for ‘all men are mortal’, Q stands for ‘Socrates is a man’, and R stands for ‘Socrates is mortal’. This sentence is not a tautology, as drawing up a truth table will show.

The moral of the story is that our truth table method is not ﬁne-grained enough to capture the meanings of predicates and names. There is logical structure embedded within the atomic sentences that cannot be captured using truth tables. (In fact, many logic textbooks teach logic in two stages: propositional logic, dealing only with complex sentences made up of atomic sentences connected by truth-functional connectives; and predicate logic, which allows you to get into the structure of atomic sentences by working with predicates, names and quantiﬁers. Truth tables are taught in propositional logic, but drop out of the picture with predicate logic.)

2. Name at least two sentences that are logically necessary but not tautological.

For example:

(a) All bachelors are unmarried men.

(b) There are inﬁnitely many prime numbers.

3. Name at least two sentences that are TW-necessary but not logically necessary.

For example:

(a) Cube(a) ∨ Tet(a) ∨ Dodec(a)

(b) (Large(a) ∧ Large(b)) ∧ ¬Adjoins(a,b)

4. For each of the following, determine whether the sentence is a tautology, a logical necessity, or a TW necessity.

(a) a=b ∨ b=c ∨ c=c This is not a necessity of any sort, though it is logically possible.

(b) BackOf(a,b) ∨ ¬BackOf(a,b) This is a tautology, and therefore also a logical necessity and a TW necessity.

(c) ¬(Cube(b) ∧ Cube(e)) ∨ Cube(b) A truth table shows that this is a tautology, and therefore also a logical necessity and a TW necessity.

(d) ¬(Cube(b) ∧ Cube(e)) ∨ SameShape(b,e) This is a TW necessity, since in Tarski’s World there are one of two possibilities: either b and e are the same kind of block, for example they are both tetrahedrons or they are both cubes, in which case the sentence is true; or they are different shapes. But if they are different shapes, then they are certainly not both cubes, which means that the sentence is true. The sentence is not a tautology, as a truth table will show; however it is a logical necessity, as far as I can tell.

4 (e) SameRow(a,b) ∨ ¬(FrontOf(a,b) ∨ BackOf(a,b)) This is a TW necessity. Either a and b are in the same row; or it is not the case that either a is in front of b or b is in front of a. I believe this is a logical necessity; it is not a tautology.

(f) SameCol(a,b) ∧ SameRow(a,b) ∧ a=b This is not a necessity of any kind, since it is false if a and b are not in the same place. The following would be a TW necessity: SameCol(a,b) ∧ SameRow(a,b) ∨ ¬(a=b), but it would not be a logical necessity. (Why?)

5. (Based on Ex 4.10) A sentence S is a logically possibility if it is true in some logically possible circumstances. S is a TW-possibility if there is at least one world that can be constructed in Tarski’s World and in which S is true. S is a TT-possibility if at least one row of its truth table assigns True to S. Draw a set of nested circle indicating the relationship between logical necessities, logical possibilities, TW-possibilities, TT-possibilities, and sentences which do not belong to any of these kinds. Give an example of each kind of sentence.

You should have drawn three nested circles, with the innermost being the TW possibilities, the middle circle comprising the logical possibilities, and the outer circle comprising the TT possibilities. Beyond this are those sentences which are not TT possible, and thus not logically possible and not TW possible (sentences such as a ∧ ¬a).

Here’s a TW possibility (and therefore also a logical possibility and a TT possibility): SameRow(a,b) ∧ SameCol(a,b) ∧ a=b.

Here’s a logical possibility (and therefore also a TT possibility) that is not a TW possibility: SameRow(a,b)∧ SameCol(a,b) ∧ ¬(a=b).

Here’s a TT possibility that is not a logical possibility: ¬SameRow(a,b) ∧ ¬SameCol(a,b) ∧ a=b.

You might want to think about these relationships along the following lines: truth tables don’t look at the logical structure embedded within atomic sentences, and so they simply consider what is possible in terms of the structure of the logical connectives. The meanings of names and predicates add additional constraints: think of these as the constraints placed upon logical truths by our language. The constraints narrow the sphere of TT possibilities to the logical possibilities.The structure of the world itself then adds extra constraints on the true things we can say about the world. These constraints narrow the sphere of the logical possibilities to the TW possibilities.

3 Challenge questions

1. When determining whether a sentence is a tautology, you typically draw up a truth table and assign all possible combinations of truth values to its atomic sentences. This is pretty laborious once you have more

5 than three atomic sentences. Can you come up with a quicker way to determine whether a sentence is a tautology? You should try your method with the following sentence: (¬(¬(¬(¬A ∨ B) ∨ C) ∨ D) ∨ E) ∨ (A ∨ ¬E).

A tautology is true in every row of its truth table. If we try to make a tautology false, we will end up with a contradiction. On the other hand, if the sentence is not a tautology, we will end up with a valid assignment of truth values to its atomic sentences. So let’s try this, starting with the connective that has widest scope: (¬ (¬ (¬ (¬ A ∨ B) ∨ C) ∨ D) ∨ E) ∨ (A ∨ ¬ E) F We know that for a disjunction to be false both of its disjuncts must be false, so we now ﬁll these in: (¬ (¬ (¬ (¬ A ∨ B) ∨ C) ∨ D) ∨ E) ∨ (A ∨ ¬ E) FFF The disjunction on the left with the widest scope is false, so we need two false disjuncts again. On the right hand side, we have another false disjunction, allowing us to ﬁnd values for A and E: (¬ (¬ (¬ (¬ A ∨ B) ∨ C) ∨ D) ∨ E) ∨ (A ∨ ¬ E) FFFFFFFT You should see the problem immediately. We need to make E on the right hand side true, but on the left hand side E comes out false. Since an atomic sentence must have the same truth value wherever it appears in a complex sentence, we have discovered a contradiction. All this means not only that we have failed to make the sentence false, but that we cannot do so. So it is a tautology.

2. The textbook says that the notion of logical necessity is ‘annoyingly vague’. Why? Can you think of a sentence whose status as a logical necessity is controversial?

A concept is vague if there are borderline cases which the concept does not help us to settle. Such cases may arise for various reasons. One problem with our deﬁnition of logical necessity is its circularity. We said that a sentence is logically necessary just in case there is no logical possible circumstance in which it is false. But now we have based our deﬁnition of logical necessity on the notion of a logically possible circumstance, and in particular, on the notion of logical possibility. Which circumstances are logically possible? Those which are compatible with what is logically necessary.

Another problem is that there are a number of ways in which a set of circumstances can be possible or impos- sible. It is not possible that 2+2=5, but this kind of impossibility is not the kind of impossibility that prohibits faster than light speed travel. Potentially, then, our notion of necessity is ambiguous. There are further kinds of impossibility: sentences that can’t be true because of the logical relations between their words (‘There is an unmarried person who is married.’), and sentences which can’t be true because of the meanings of their words (‘There is a bachelor who is married.’) Quine famously thought that it was too difﬁcult to come up with a precise identiﬁcation of the latter class of sentences, and that while our sentences are made true both by our

6 language and the way the world is, it is not possible to sort out the contributions of language and world sentence by sentence.

Consider a sentence such as: ‘A square has four equal sides.’ This, you might venture, is not true of Tahrir Square. How can I convince you that I did not mean to include Tahrir Square amongst the things I referred to as squares? Only by saying that what I meant by ‘square’ was ‘’shape with four equal sides’. And that would be begging the question.

There is another problem with logical necessity, separate from the Quine’s doubts that we can get clear on what it is for a sentence to be true in virtue of its meaning. For one might forego such sentences as logically necessary, and point simply to those sentences that are true in virtue of their logical structure. We know that a ∨ ¬a is tautological, and thus the clearest case of a logical necessity. But we can doubt that every sentence that follows this schema is logically necessary. ‘John has a beard or John does not have a beard’ follows the schema, but how could we deny that there are interesting intermediate cases in which John neither has a proper beard nor lacks one entirely?

Or consider the schema ¬(a ∧ ¬a) - again, as clear cut a case of a logical necessity as one could possibly ﬁnd. But ‘it is not the case that the particle is spin up and the particle is spin down’ is an instantiation of the schema, and quantum physics seems to tell us that such sentences are often false.

There are deep problems here about what counts as a possibility, and about what counts as a possible circumstance. Formal languages, such as First Order Logic, help us make these ideas more precise, but they do not necessarily solve the problems for natural language.