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From the book Logic: The laws of truth by Nicholas J.J Smith, page 40, sub-heading: "Syntax of PL".

What does the author mean by the syntax of PL? Can someone explain what the author is trying to convey on page 41 (the paragraph)?

I spent quite a bit time studying this section but could not understand much.

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What Smith means by the syntax of propositional logic are

  1. the basic symbols of the language and
  2. how those symbols can be combined to make sentences of the language.

These sentences and only these sentences are considered well formed formulas (wff).

On the top half of page 41 he defines the syntax.

The basic symbols of propositional logic (PL) are

  1. basic propositions
  2. connectives
  3. punctuation symbols (parentheses)

These basic symbols can be combined to make sentences (wff). He has three rules for how to make wff from the basic symbols:

  1. Basic propositions are by themselves wff. This is the "base clause" of this recursive definition.
  2. Propositions (basic or compound) with connectives used in the ways specified are wff. This is the "recursive clause" of the recursive definition.
  3. Nothing else is a wff.

The above definition is called "recursive" because if you are given a wff with connectives you can break it apart examining each of the wff connected by the connectives. Each of these wff may have connectives which can be broken down. This process continues until you reach the basic propositions without any connective (i.e. is recursive). At this point you can stop.


Smith, N. J. (2012). Logic: The laws of truth. Princeton University Press.

  • what does he mean by recursive definitions? – MinigameZ more Apr 12 at 3:29
  • sorry to keep adding, but one last question: what does he mean by clause? – MinigameZ more Apr 12 at 3:36
  • @MinigameZmore The clause is part of the definition. For example, 2(i) is the "base clause" and 2(ii) is the "recursive cause". They are parts of the definition of a wff. – Frank Hubeny Apr 12 at 3:40
  • "The above definition is called "recursive" because if you are given a wff with connectives you can break it apart examining each of the wff connected by the connectives. Each of these wff may have connectives which can be broken down. This process continues until you reach the basic propositions without any connective (i.e. is recursive). At this point you can stop." - Frank huben , I can't understand why it is called recursive definition why not recursive process?, how is this a definition?, this concept is required for some exercise, so it might be important!. – MinigameZ more Apr 13 at 6:48
  • @MinigameZmore It is a recursive definition for the wff in propositional logic using the symbols defined earlier. Note that the symbols were not defined by a recursive process. The permitted symbols were simply listed, no base clause nor recursive clause were specified for the symbols as it was for the wff. – Frank Hubeny Apr 13 at 11:43

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