(1) Explain what a reductio ad absurdum argument is. Give your own example to il
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Question
(1) Explain what a reductio ad absurdum argument is. Give your own example to illustrate your definition; when you do so, please make use of one of the valid forms of argument we've talked about so far. What is the usefulness of a reductio argument?
(2) In one paragraph, explain the significance of truth functional meaning with reference to Gottlob Frege’s principle of compositionality (PC). In your answer, make sure to address what phenomena Frege proposes PC to explain; why he thinks that PC could, if true, help to explain these phenomena; and how truth-functional meaning would help support PC.
Explanation / Answer
(1)
Description: A mode of argumentation or a form of argument in which a proposition is disproven by following its implications logically to an absurd conclusion. Arguments that use universals such as, “always”, “never”, “everyone”, “nobody”, etc., are prone to being reduced to absurd conclusions. The fallacy is in the argument that could be reduced to absurdity -- so in essence, reductio ad absurdum is a technique to expose the fallacy.
Logical Form:
Assume P is true.
From this assumption, deduce that Q is true.
Also, deduce that Q is false.
Thus, P implies both Q and not Q (a contradiction, which is necessarily false).
Therefore, P itself must be false.
Example #1:
I am going into surgery tomorrow so please pray for me. If enough people pray for me, God will protect me from harm and see to it that I have a successful surgery and speedy recovery.
Explanation: We first assume the premise is true: if “enough” people prayed to God for the patient's successful surgery and speedy recovery, then God would make it so. From this, we can deduce that God responds to popular opinion. However, if God simply granted prayers based on popularity contests, that would be both unjust and absurd. Since God cannot be unjust, then he cannot both respond to popularity and not respond to popularity, the claim is absurd, and thus false.
Example #2:
If everyone lived his or her life exactly like Jesus lived his life, the world would be a beautiful place!
Explanation: We first assume the premise is true: if everyone lived his or her life exactly like Jesus lived his, the world would be a beautiful place. If this were true, we would have 7 billion people on this earth roaming from town to town, living off the charity of others, preaching about God (with nobody listening). Without anyone creating wealth, there would be nobody to get charity from -- there would just be 7 billion people all trying to tell each other about God. After a few weeks, everyone would eventually starve and die. This world might be a beautiful place for the vultures and maggots feeding on all the Jesus wannabes, but far from a beautiful world from a human perspective. Since the world cannot be both a beautiful place and a horrible place, the proposition is false.
Exception: Be sure to see the appeal to extremes fallacy.
(2)
The overwhelming majority of theories of semantics presuppose the Principle of
Compositionality in (1), standardly attributed to Gottlob Frege.
(1) Principle of Compositionality
The meaning of a complex expression is a ‘function of ’
(i) the meanings of its component expressions and
(ii) their mode of syntactic composition.
• Within Montague Grammar, this principle is implemented as a homomorphism
between the syntax and the semantics (Montague 1970). The denition in (2) is from
Westerståhl (1998: 635f.).
(2) Compositionality as a Homomorphism
Given the syntactic algebra A = (A, (F?)???) and a meaning function m from
A to a set of meanings M. Let F be a k-ary operation of A. m is F-compositional
if there is a k-ary partial function G on M such that whenever F(a1 . . . , ak ) is
defined,
m(F(a1 .. . . , ak )) = G(m(a1), . . . , m(ak )).
With G as above we say that m is F-compositional with G, and we say that m is
compositional if it is F-compositional for all operations F of A.
• The Problem with Idioms:
Idioms seem to be in direct contrast with the requirement in (2). The meaning of,
e.g., kick the bucket seems to be unrelated to its parts and their combination.
• Central Question:
Do idioms force us to abandon (2) (and perhaps the basic idea that natural languages
are compositional)?
• A Terminological Point:
Given the denition in (2), it does not make sense to ask whether certain expressions.
re compositional. Only entire interpretations can be. What we can ask is whether we
can capture idioms in a compositional semantics.
• Outlook:
Westerståhl (2002) is a detailed study of idioms and compositionality. He argues that
idioms are unproblematic for compositionality.
2 The Compositionality of Idioms: Westerståhl (2002)
• Westerståhl (2002) develops three ways in which idioms can be implemented in a
compositional system. His rst develops a system and then shows three ways this
system has to be amended to include idioms. We will go through them and then step
back to draw some general conclusions about compositionality.
2.1 The Basic System
(3) GrammarA grammar
E = (E, A, ?)
consists of a set E of expressions, a set A ? E of atom ic expressions and
for each function symbol ? ? ? a corresponding syntact ic rule: a partial
map ? from E nt toE, for some n.
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