by Corry Shores
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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part I:
Propositional Logic
1.
Classical Logic and the Material Conditional
1.6
Conditionals
Brief summary:
(1.6.1) We now will examine conditionality in classical propositional logic. (1.6.2) A conditional contains two propositions. One is the consequent, which depends in some sense on the other proposition, called the antecedent. In English they are often formed using “if” or similar constructions. (1.6.3) When writing the antecedent or consequent by themselves, we often need to make changes to the verb tense or mood of the sentence, especially when formulating inferences. (1.6.4) Not all English “if” constructions are conditionals. We can test them by seeing if they can be expressed under the form ‘that A implies B’.
[Preview: Conditionals]
[The Structure of Conditionals]
[Grammar and Isolated Conditional Parts]
[Testing English Constructions for Conditionality]
Summary
[Preview: Conditionals]
[We now will examine conditionality in classical propositional logic.]
[In section 1.3 we learned the semantics for classical propositional logic.] For the rest of chapter 1, Priest will discuss the nature of conditionality that the semantics we have seen so far will give us. We will also evaluate the inadequacies of this notion of the conditional. We begin now with the question, what is a conditional? (11)
[The Structure of Conditionals]
[A conditional contains two propositions. One is the consequent, which depends in some sense on the other proposition, called the antecedent. In English they are often formed using “if” or similar constructions.]
Priest now describes the general structure of a conditional. It has two propositions, with one depending on the other in some sense. The one that doing the depending is the consequent, and the one that is being depended upon is the antecedent. In English, we express conditionality using “if” or equivalent constructions.
Conditionals relate some proposition (the consequent) to some other proposition (the antecedent) on which, in some sense, it depends. They are expressed in English by ‘if’ or cognate constructions:
If the bough breaks (then) the cradle will fall.
The cradle will fall if the bough breaks.
The bough breaks only if the cradle falls. |
If the bough were to break the cradle would fall.
Were the bough to break the cradle would fall.
(11-12)
[Grammar and Isolated Conditional Parts]
[When writing the antecedent or consequent by themselves, we often need to make changes to the verb tense or mood of the sentence, especially when formulating inferences.]
Priest’s next point clarifies something about how certain grammatical structurings are needed when the antecedent or consequent are written by themselves, especially in the context of making inferences.
Note that the grammar of conditionals imposes certain requirements on the tense (past, present, future) and mood (indicative, subjunctive) of the sentences expressing the antecedent and consequent within it. These may be different when the antecedent and consequent stand alone. To see this, just consider the following applications of modus ponens (if A then B; A; hence B):
If he takes a plane he will get there quicker.
He will take a plane.
Hence, he will get there quicker.
If he had come in the window there would have been foot-marks.
He did come in the window.
So, there are foot-marks.
(12)
[Testing English Constructions for Conditionality]
[Not all English “if” constructions are conditionals. We can test them by seeing if they can be expressed under the form ‘that A implies B’.]
Some English constructions use “if,” but they do not form conditional structures. One example Priest gives is: “If I may say so, you have a nice ear-ring” (12). Priest then provides a basic test for “if” constructions to determine whether they are conditionals or not. We see if they can be “rewritten equivalently as ‘that A implies that B’” (12).
Note, also, that not all sentences using ‘if’ are conditionals; consider, for example, ‘If I may say so, you have a nice ear-ring’, ‘(Even) if he was plump, he could still run fast’, or ‘If you want a banana, there is one in the kitchen.’ A rough and ready test for ‘if A, B’ to be a conditional is that it can be rewritten equivalently as ‘that A implies that B’.
(12)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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