Symbolic Logic II
Lecture 1
Brandon C. Look: Symbolic Logic II, Lecture 1
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The Nature of Logic
Logical Consequence
Logic is said to deal with the rules or the laws of thought.
But it is more correct to say that it is the science that studies the
relation of consequence; that is, logic concerns the way in which
some set of sentences or propositions entails or has as a
consequence some other sentence or proposition.
Brandon C. Look: Symbolic Logic II, Lecture 1
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On a first pass, we all would say that some statement ψ is a logical
consequence of φ if it is impossible for φ to be true and ψ to be
false.
But the philosophical question is: where does this impossibility
come from? What makes it necessary that ψ be true if φ is? It’s
not obvious.
Brandon C. Look: Symbolic Logic II, Lecture 1
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But note the centrality of the notion of logical consequence.
Logical properties are generally defined in terms of consequence.
Consider the following:
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An argument is valid iff its conclusion is a logical consequence
of its premises.
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A set of claims Γ entails a claim α iff α is a logical
consequence of Γ.
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A set of claims Γ is consistent iff no contradiction is a logical
consequence of it.
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A claim α is independent of a set of claims Γ iff α is not a
logical consequence of Γ.
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A claim α is a logical truth iff it is a logical consequence of
the empty set of claims.
Brandon C. Look: Symbolic Logic II, Lecture 1
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The Nature of Logic
Logic and Meaning
Still, what does it mean to say that a proposition or a set of
propositions implies another proposition?
Consider the following:
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All whales are mammals
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A$rz?&
The first, it would seem, carries with it a number of implications;
the second, not so much.
But this is only if we take into account the meaning of the former
statement. It is, by itself, just as much a string of symbols as the
latter.
Brandon C. Look: Symbolic Logic II, Lecture 1
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More, on the subject of meaning...
Consider the claim:
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All professors are academics
This is, perhaps, an analytic truth, like “All bachelors are
unmarried males.” It is true by virtue of meaning. But so is this:
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If all professors are boring then all professors are boring
What is the difference? The latter is true by virtue of the meaning
of the logical constants. for it simply amounts to φ → φ.
Brandon C. Look: Symbolic Logic II, Lecture 1
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In formal logic, we agree on the meaning of our logical constants
and connectives: ∀, ∃, ¬, ∧, ∨, →, ↔, and, maybe, some others
(e.g. ‘=’).
The idea is that we can avoid confusion and lay bare the logical
connections of statements.
Brandon C. Look: Symbolic Logic II, Lecture 1
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We have to agree that particular strings of symbols are capable of
bearing a meaning. These strings of symbols will be our ‘formulas’.
And if a string of symbols is formed according to the rules that we
choose for our language, then we will call it a ‘well-formed formula’
(or ‘wff’). (More later)
The important thing is that we agree to certain rules in our
language for the creation of formulas. And they could be (almost)
anything.
But note: a string of symbols is a wff iff it is formed according to
certain rules; the meaning that that string of symbols is supposed
or thought to express is irrelevant.
Brandon C. Look: Symbolic Logic II, Lecture 1
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For example, we will say that an atomic sentence φ is a wff; its
negation is a wff, ∼ φ; two atomic sentences can be joined by
certain logical connectives to form a wff (e.g., pφ ∨ ψq); there have
to be as many left-parentheses as right-parentheses, and so on.
This is what will allow us to say that pφ → ψq is a wff and
p)(x∀ ∨ φψq is not.
But we cannot say that a string of symbols constitutes a wff if it
has a particular meaning.
Thus, the rules for wffs concern form and not content or meaning.
Brandon C. Look: Symbolic Logic II, Lecture 1
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The Nature of Logic
A Problem for Consequence?
But formalizing or symbolizing an argument does not necessarily
reveal the relation of logical consequence.
Consider the following argument:
1. Bill and Phil are the same age
2. Bill is 26 years old
3. ∴ Phil is 26 years old
Brandon C. Look: Symbolic Logic II, Lecture 1
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Let language L1 formalize this argument as
1. A(b) = A(p)
2. A(b) = a
3. ∴ A(p) = a
while L2 formalizes it as
1. A(b,p)
2. Y(b)
3. ∴ Y(p)
Both languages represent our original syllogism. But, note: every
argument formalizable by the L1 series is truth-preserving, while
this is not the case with arguments in L2.
Brandon C. Look: Symbolic Logic II, Lecture 1
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The Nature of Logic
Languages and Interpretations
The point is going to be (once we really get into Ch. 2 and
beyond) that we choose or create particular languages which will
have particular interpretations. The meanings of particular claims
will fall out of those languages and interpretations.
After all, suppose that the string of characters ‘Snow is white’
means snow is white in English and pillows are soft in some other
language. Here we have two meanings or interpretations for the
two different languages. In a similar vein, ‘Snow is white’ and
‘Schnee ist weiß’ mean the same thing.
Brandon C. Look: Symbolic Logic II, Lecture 1
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The Nature of Logic
Aside: What are the bearers of truth-values?
When we say that φ is true? What do we mean? What is it that
has the property of being true?
There are basically two options:
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Propositions — abstract objects
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Sentences — concrete objects (things on a page; things said)
What’s the difference? How many truths are here: ‘Snow is white’
and ‘Schnee ist weiß’ ? One? Two?
Brandon C. Look: Symbolic Logic II, Lecture 1
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Logic and Metalogic
In this class, we will be talking about logic. In other words, we will
be reflecting on the nature of logic and so doing ‘metalogic’.
We will have particular languages of first-order logic PL, QL, QML,
and so on. These are our object languages. When we talk about
them, we will use our metalanguage — in this case, English.
Brandon C. Look: Symbolic Logic II, Lecture 1
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Talking about Logic...
More on Use and Mention
When we use words, we will leave them alone. When, however, we
mention words, we will put them in quotation marks. Thus:
Obama is now President.
‘Obama’ sounds like ‘Osama’ (and he still won!).
Michelle calls Barack ‘Barack’.
Chicago is a great town.
‘Chicago’ comes from a native american word for onion or
something else stinky.
Brandon C. Look: Symbolic Logic II, Lecture 1
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When we want to refer to things in our object language as
expressing a general logical relation, we put them in, so-called,
Quine-quotes (or ‘corners’). In Quine’s own terms, this is an
example of ‘quasi-quotation’. (And we leave single, self-standing
Greek letters alone.)
Thus, ‘Jones is away’ stands for the fact that Jones is away; ‘Smith
is at home’ stands for the fact that Smith is at home. If we take
‘Jones is away’ as φ and ‘Smith is at home’ as ψ then pφ ∧ ψq
stands for ‘Jones is away and Smith is at home’.
Huh? Why? The point is that ‘φ ∧ ψ’ stands for this assertion, and
we intend for pφ ∧ ψq to stand for the general case of conjunction
between any φ and any ψ.
Brandon C. Look: Symbolic Logic II, Lecture 1
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Logical Consequence Again
Returning to the original question: What does it mean to say that
one proposition follows from some set of other propositions?
As Sider says, there are a number of different answers — each a
philosophical answer, each unsatisfactory (to some).
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Model-theoretic theory — one defines a notion of model (or
interpretation), defines a notion of truth-in-a-model for
sentences of the language, and then finally represents logical
consequence for the chosen formal language as
truth-preservation in models. (p. 9)
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Proof-theoretic theory — one defines a relation of provability
between sentences of formal languages. (p. 10)
Brandon C. Look: Symbolic Logic II, Lecture 1
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