1. What is Logic? Flashcards
Logic is about many things, but most centrally it is about _________.
logical consequence
The statement “someone is male” is a _______ of the statement “Grant is male”. If Grant is male, then it logically follows that someone is male.
Put another way: the statement “Grant is male” logically implies the statement “someone is male”.
logical consequence;
The statement __________ is a logical consequence of the statements “It’s not the case that Leisel is male” and “Either Leisel is male or Grant is male” (taken together). The first statement follows from the latter two statements; they logically imply it.
Put another way: the argument whose premises are the latter two statements, and whose conclusion is the former statement, is a logically correct one.
“Grant is male”
________ is truth-preservation by virtue of form
Logical consequence
For φ to be a logical consequence of ψ, it is not enough that we all know that _________.
φ is true if ψ is
We all know that an apple will fall if it is dropped, but the relationship between falling and dropping does not hold by virtue of logic.
Why not?
For one thing, “by virtue of logic” requires the presence of some sort of necessary connection, a connection that is absent in the case of the dropped apple (since it would be possible—in some sense—for a dropped apple not to fall). For another, it requires the relationship to hold by virtue of the forms of the statements involved, whereas the relationship between “the apple was dropped” and “the apple fell” holds by virtue of the contents of these statements and not their form.
The inference from “It’s not the case that Leisel is male” and “Either Leisel is male or Grant is male” to “Grant is male” is said to hold in virtue of form since any argument of the form __________ is logically correct.
“it’s not the case that φ; either φ or ψ; therefore ψ”
Just as logical consequence is truth-preservation by virtue of form, ________ is truth by virtue of form.
logical truth
Logical truth is truth by virtue of form.
Examples might include: “it’s not the case that snow is white and also not white”, “All fish are fish”, and “If Grant is male then someone is male”. As with logical consequence, logical truth is thought to require some sort of ______ and to hold by virtue of ______.
necessity;
form, not content
It is plausible that logical truth and logical consequence are related thus: a logical truth is _______. One can infer a logical truth by _______.
a sentence that is a logical consequence of the empty set of premises;
using logic alone, without the help of any premises
A central goal of logic is to study ______ and ______.
logical truth;
logical consequence
Modern logic is called “mathematical” or “symbolic” logic because its method is ________.
the mathematical study of formal languages
Modern logicians use the tools of mathematics (especially the tools of very abstract mathematics, such as set theory) to ________.
treat sentences and other parts of language as mathematical objects
The formal sentence _____ is a tautology, but since it is uninterpreted, we probably shouldn’t call it a logical truth. Rather, it represents logical truths like “If snow is white then snow is white.” A logical truth ought to at least to be true, after all, and _____ isn’t true, since it doesn’t even have a meaning—what’s the meaning of P?
P→P;
P→P
Why are formal languages called “formal”? (They’re also sometimes called “artificial” languages.)
Because their properties are mathematically stipulated, rather than being pre-existent in flesh-and-blood linguistic populations. We stipulatively define a formal language’s grammar. We must stipulatively define any properties of the symbolic sentences that we want to study, for example, the property of being a tautology. Formal languages often contain abstractions, like the sentence letters P, Q, … of propositional logic.
Metalogic proofs are phrased in _______ and employ _______.
natural language (perhaps augmented with mathematical vocabulary); informal (though rigorous!) reasoning of the sort one would encounter in a mathematics book
Proofs in formal systems, unlike metalogic proofs, are phrased using ________.
sentences of formal languages, and proceed according to prescribed formal rules
Logicians often distinguish the “_____language” from the “_____language”.
object ;
meta
The object language is the language that’s being studied. One example is…
the language of propositional logic. Its sentences look like this:
P∧Q
∼(P∨Q)↔R
The metalanguage is the language we use to talk about the object language.
One example is…
English, as it is used in this book.
Here are some example sentences of the metalanguage:
“P∧Q” is a formal sentence with three symbols
Every sentence of propositional logic has the same number of left parentheses as right parentheses
Every provable formula is a tautology
Thus, we formulate metalogical claims about an object language in the metalanguage, and prove such claims by reasoning in the metalanguage.
i) is it a sentence of the object language or the metalanguage?
ii) is it true?
‘P∨∼P’ is a logical truth.
“‘P∨∼P’ is a logical truth” is a sentence of the metalanguage, and (I would say) is false. ‘P∨∼P’ contains the meaningless letter ‘P’, so it isn’t a logical truth. Rather, it represents logical truths (assuming the law of the excluded middle is correct! See chapter 3.)
i) is it a sentence of the object language or the metalanguage?
ii) is it true?
(P∨Q)→(Q∨P)
‘(P∨Q)→(Q∨P)’ is a sentence of the object language. Since it contains meaningless expressions (‘P’, ‘Q’), it isn’t true. (Not that it’s false!)
i) is it a sentence of the object language or the metalanguage?
ii) is it true?
‘Frank and Joe are brothers’ logically implies ‘Frank and Joe are siblings’.
This is a bit of a trick question. “‘Frank and Joe are brothers’ logically implies ‘Frank and Joe are siblings’” is a sentence of English, which is talking about further sentences of English. So English is functioning here both as the object language and as the metalanguage. As for whether the sentence is true, I would say no, since the implication is not “formal.”
Each of the following sentences confuses use and mention. In each case, fill in quotation marks to x the problem.
* Attorney and lawyer are synonyms.
‘Attorney and lawyer are synonyms’ confuses use and mention; inserting quotation marks thus fixes the problem:
‘Attorney’ and ‘lawyer’ are synonyms.