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Goal of Logic The goal of logic is to develop formal tests for validity. This is done by finding deductively valid argument forms. In SL, we have two tests to determine valid argument forms: the truth-table test, and derivations. Any argument that has a form that is valid according to the truth-table test is valid, and any argument that has a form that can be proven is valid.
Example For example, here is a valid logical form: Premise: If P, then Q. Premise: Not-Q. Conclusion: Therefore, not-P.
Example We translate it into SL: Premise: (P → Q) Premise: ~Q Conclusion: ~P
(P → Q), ~Q ├ ~P This is how the truth-table test establishes the validity of the argument.
(P → Q), ~Q ├ ~P We can also use a proof to establish validity.
(P → Q), ~Q ├ ~P 1 1. (P → Q) A First, we write down the premises.
(P → Q), ~Q├ ~P 1 1. (P → Q) A 2 2. ~Q A First, we write down the premises.
(P → Q), ~Q ├ ~P 1 1. (P → Q) A 2 2. ~Q A Then we look at what we want to prove.
~I To prove a negation, we often have to use the following strategy: • Assume the opposite of what we’re trying to prove. • Derive a contradiction. • Use ~I to get what we want.
(P → Q), ~Q ├ ~P 1 1. (P → Q) A 2 2. ~Q A 3 3. P A Assume the opposite.
(P → Q), ~Q ├ ~P 1 1. (P → Q) A 2 2. ~Q A 3 3. P A 1,3 4. Q 1,3 →E 1,2,3 5. (Q & ~Q) 4,2 &I Derive a contradiction.
(P → Q), ~Q ├ ~P 1 1. (P → Q) A 2 2. ~Q A 3 3. P A 1,3 4. Q 1,3 →E 1,2,3 5. (Q & ~Q) 4,2 &I 1,2 6. ~P 3,5 ~I Use ~I to get the opposite of what you assumed.
Valid Arguments that Don’t Pass But, there are still valid arguments that don’t pass the truth-table test for validity and aren’t provable in SL. (This is why failing the truth-table test does not show that an argument is invalid). For example: Premise: Michael is human. Conclusion: Therefore, someone is human.
Other Examples Premise: Every chicken is a bird. Premise: Every bird is an animal. Conclusion: Every chicken is an animal. Premise: No one comes to class on Saturday. Conclusion: Michael doesn’t come to class on Saturday.
Other Examples Premise: Every chicken is a bird = C Premise: Every bird is an animal = B Conclusion: Every chicken is an animal = A Premise: No one comes to class on Saturday. Conclusion: Michael doesn’t come to class on Saturday.
Other Examples Premise: C Premise: B Conclusion: A Premise: No one comes to class on Saturday. Conclusion: Michael doesn’t come to class on Saturday.
SL Not Expressive Enough The problem here is that our logic is not expressive enough. All simple English sentences get translated as sentence letters in SL. Therefore, no argument involving only simple sentences has a valid SL form, even though many such arguments are valid.
PL Therefore, we have a more expressive logic, predicate logic PL, which represents the parts of simple sentences. In PL, a sentence like “Michael is human” will have its parts “Michael” and “is human” translated separately. In particular, PL has a specific grammatical category for singular terms.
Singular Terms A singular term is an expression that names or identifies a particular individual, like a person or a city. English singular terms include:
Singular Terms • Proper names: ‘Michael,’ ‘Jenny,’ ‘Hong Kong,’ etc. • ‘the’ + description: ‘the tallest man in the world,’ ‘the country with the second largest economy,’ ‘the third Wednesday of March,’ etc. • ‘that’ ‘this’ or ‘that’ + description, ‘this’ + description: ‘this pencil,’ ‘that table,’ etc.
NOT Singular Terms • ‘Every happy person’ • ‘No one in Hong Kong’ • ‘A bird with red feathers’ • ‘Beautiful dresses’ None of these expressions name a particular individual or thing.
Translating Singular Terms To translate singular terms into PL, we will use lowercase Roman letters: a, b, c, d, e, f, g, etc. So we might translate “Michael” as “m.” If we’re just doing logic (and not translating) usually we choose a, b, and c to be our singular terms.
Variables In addition, PL contains a special grammatical category called variables. Variables are a lot like singular terms, but they do not name or represent anything in particular. Variables: x, y, z (and if need be w, v, and u) We do not use these letters to translate singular terms!
Variables Replace Singular Terms Consider how variables are used in arithmetic: • 5 + 7 = 12 • 5 + x = 12 • y + x = 12 • y + x = z “5,” “7,” and “12” all name particular numbers. “x,” “y,” and “z” do not. But they go in the same places singular terms go.
Variables Replace Singular Terms In logic, things are very similar, except variables can replace more than just singular terms for numbers: • Michael gave that book to Sam • Michael gave x to Sam • y gave x to Sam • y gave x to z
Open Sentences If you take an English sentence, remove one or more singular terms and replace them with variables, the result is an open sentence. • Michael gave x to Sam • x gave y to z • This past winter, x went home to visit x’s grandmother. • John went to the party but z stayed home.
Predicates Predicate logic also contains expressions that translate predicates. In traditional grammar, a sentence like “Michael is human” has “Michael” as its subject and “is human” as its predicate. In logic, we simply identify predicates as open sentences. So “x is human” is a predicate, but “is human” is not.
Translating Predicates Predicates in PL are translated as capital Roman letters: A, B, C, D, E, F, G, etc. We have a preference for the letters F, G, and H when there is no other reason to choose.
Our Fragment Predicate logic is a lot harder than sentential logic, so to make it easier, the system in the reading (PL) only has “monadic” or “one-place” predicates– predicates containing only one place for a singular term.
Only 1-Place Predicates OK: • x is human • Michael gave z to Sam • y gave that book to Sam • Michael gave that book to y NOT OK: • x gave that book to y • x admires z
More Complicated WFFs Just as in SL, we can combine PL WFFs with truth-functional connectives: • (Hm & Bc) • ((Rh ↔ Ij) v (Ad → ~Bc))
Quantifiers Some words in English are not appropriately translated as either singular terms or predicates. Consider the sentence: “Something is red.” Here we know that “x is red” translates as “Rx.” How should we represent “something”?
Not a Singular Term It would be a bad idea to translate “something” as a singular term: First, “something” doesn’t identify a particular individual.
Not a Singular Term It would be a bad idea to translate “something” as a singular term: Second, if we translated “Something” as a singular term “s” and “something is red” as “Rs,” then we would have to translate “something is red and something is not red” as “(Rs & ~Rs).”
Not a Contradiction But it’s not a contradiction if something is red and something (else) is not red. This is why singular terms have to identify particular individuals, and why we can’t translate words like “something” as singular terms.
Quantifiers In PL, there is a special symbol that translates “something”: ∃ (called “backwards E” or “the existential quantifier”). In the grammar of PL, to write “something is red” we write ∃ followed by a variable, followed by an open sentence with that variable in it: ∃xRx
