6.30
➶ ¬(¬Cube(a) ∧ Cube(b))
¬(¬Cube(b) ∨ Cube(c)) Cube(a)
6.31
➶ Dodec(b) ∨ Cube(b) Small(b) ∨ Medium(b)
¬(Small(b) ∧ Cube(b)) Medium(b) ∧ Dodec(b)
6.32
➶ Dodec(b) ∨ Cube(b) Small(b) ∨ Medium(b)
¬Small(b) ∧ ¬Cube(b)) Medium(b) ∧ Dodec(b)
Section 6.6
Proofs without premises
Not all proofs begin with the assumption of premises. This may seem odd, but in fact it is how we use our deductive system to show that a sentence is a logical truth. A sentence that can be proven without any premises at all is
necessarily true. Here’s a trivial example of such a proof, one that shows that demonstrating logical truth a = a ∧ b = b is a logical truth.
1. a = a = Intro
2. b = b = Intro
3. a = a ∧ b = b ∧ Intro: 1, 2
The first step of this proof is not a premise, but an application of = Intro.
You might think that any proof without premises would have to start with this rule, since it is the only one that doesn’t have to cite any supporting steps earlier in the proof. But in fact, this is not a very representative example of such proofs. A more typical and interesting proof without premises is the following, which shows that ¬(P ∧ ¬P) is a logical truth.
Section 6.6
174 /Formal Proofs and Boolean Logic
1. P ∧ ¬P
2. P ∧ Elim: 1
3. ¬P ∧ Elim: 1
4. ⊥ ⊥ Intro: 2, 3
5. ¬(P ∧ ¬P) ¬ Intro: 1–4
Notice that there are no assumptions above the first horizontal Fitch bar, indicating that the main proof has no premises. The first step of the proof is the subproof ’s assumption. The subproof proceeds to derive a contradiction, based on this assumption, thus allowing us to conclude that the negation of the subproof’s assumption follows without the need of premises. In other words, it is a logical truth.
When we want you to prove that a sentence is a logical truth, we will use Fitch notation to indicate that you must prove this without assuming any premises. For example the above proof shows that the following “argument”
is valid:
¬(P ∧ ¬P)
We close this section with the following reminder:
Remember
A proof without any premises shows that its conclusion is a logical truth.
Exercises
6.33
➶
(Excluded Middle) Open the file Exercise 6.33. This contains an incomplete proof of the law of excluded middle, P ∨ ¬P. As it stands, the proof does not check out because it’s missing some sentences, some support citations, and some rules. Fill in the missing pieces and submit the completed proof as Proof 6.33. The proof shows that we can derive excluded middle in F without any premises.
Proofs without premises/ 175
In the following exercises, assess whether the indicated sentence is a logical truth in the blocks language.
If so, use Fitch to construct a formal proof of the sentence from no premises (using Ana Con if necessary, but only applied to literals). If not, use Tarski’s World to construct a counterexample. (A counterexample here will simply be a world that makes the purported conclusion false.)
6.34
➶
¬(a = b ∧ Dodec(a) ∧ ¬Dodec(b))
6.35
➶
¬(a = b ∧ Dodec(a) ∧ Cube(b))
6.36
➶
¬(a = b ∧ b = c ∧ a 6= c)
6.37
➶
¬(a 6= b ∧ b 6= c ∧ a = c)
6.38
➶
¬(SameRow(a, b) ∧ SameRow(b, c) ∧ FrontOf(c, a))
6.39
➶
¬(SameCol(a, b) ∧ SameCol(b, c) ∧ FrontOf(c, a))
The following sentences are all tautologies, and so should be provable in F . Although the informal proofs are relatively simple, F makes fairly heavy going of them, since it forces us to prove even very obvious steps. Use Fitch to construct formal proofs. You may want to build on the proof of Excluded Middle given in Exercise 6.33. Alternatively, with the permission of your instructor, you may use Taut Con, but only to justify an instance of Excluded Middle. The Grade Grinder will indicate whether you used Taut Con or not.
6.40
➶?
A ∨ ¬(A ∧ B)
6.41
➶?
(A ∧ B) ∨ ¬A ∨ ¬B
6.42
➶?
¬A ∨ ¬(¬B ∧ (¬A ∨ B))
Section 6.6
Chapter 7
Conditionals
There are many logically important constructions in English besides the Boolean connectives. Even if we restrict ourselves to words and phrases that connect two simple indicative sentences, we still find many that go beyond the Boolean operators. For example, besides saying:
Max is home and Claire is at the library, and
Max is home or Claire is at the library,
we can combine these same atomic sentences in the following ways, among others:
Max is home if Claire is at the library, Max is home only if Claire is at the library,
Max is home if and only if Claire is at the library, Max is not home nor is Claire at the library, Max is home unless Claire is at the library, Max is home even though Claire is at the library,
Max is home in spite of the fact that Claire is at the library, Max is home just in case Claire is at the library,
Max is home whenever Claire is at the library, Max is home because Claire is at the library.
And these are just the tip of the iceberg. There are also constructions that combine three atomic sentences to form new sentences:
If Max is home then Claire is at the library, otherwise Claire is concerned,
and constructions that combine four:
If Max is home then Claire is at the library, otherwise Claire is concerned unless Carl is with him,
and so forth.
Some of these constructions are truth functional, or have important truth-functional uses, while others do not. Recall that a connective is truth func-tional if the truth or falsity of compound statements made with it is completely
Material conditional symbol: →/ 177
determined by the truth values of its constituents. Its meaning, in other words, can be captured by a truth table.
Fol does not include connectives that are not truth functional. This is non-truth-functional connectives not to say that such connectives aren’t important, but their meanings tend to
be vague and subject to conflicting interpretations. The decision to exclude them is analogous to our assumption that all the predicates of fol have precise interpretations.
Whether or not a connective in English can be, or always is, used truth functionally is a tricky matter, about which we’ll have more to say later in the chapter. Of the connectives listed above, though, there is one that is very clearly not truth functional: the connective because. This is not hard to prove.
Proof: To show that the English connective because is not truth functional, it suffices to find two possible circumstances in which the sentence Max is home because Claire is at the library would have different truth values, but in which its constituents, Max is home and Claire is at the library, have the same truth values.
Why? Well, suppose that the meaning of because were captured by a truth table. These two circumstances would correspond to the same row of the truth table, since the atomic sentences have the same values, but in one circumstance the sentence is true and in the other it is false. So the purported truth table must be wrong, contrary to our assumption.
For the first circumstance, imagine that Max learned that Claire would be at the library, hence unable to feed Carl, and so rushed home to feed him. For the second circumstance, imagine that Max is at home, expecting Claire to be there too, but she unexpectedly had to go the library to get a reference book for a report. In both circumstances the sentences Max is home and Claire is at the library are true. But the compound sentence Max is home because Claire is at the library is true in the first, false in the second.
The reason because is not truth functional is that it typically asserts some sort of causal connection between the facts described by the constituent sen-tences. This is why our compound sentence was false in the second situation:
the causal connection was missing.
In this chapter, we will introduce two new truth-functional connectives, known as the material conditional and the material biconditional, both stan-dard features of fol. It turns out that, as we’ll show at the end of the chapter, these new symbols do not actually increase the expressive power of fol. They
Section 7.1
178 /Conditionals
do, however, make it much easier to say and prove certain things, and so are valuable additions to the language.
Section 7.1