For example, how would I turn this sentence into logic symbols: "Provided that the MONEY is deposited, the LENDER will get the payment if the WIRE goes through"?

  • If that question could be answered generally, we would not need propositional logic itself. We can all work your example, but we would all give you different advice, and that divergence of experience would not answer the overall question. My suggestion would be to address it like a language "How do I translate English to French?" Well, uh, carefully? Write some formations and change them until they mean what you want to say.
    – user9166
    Commented Mar 5, 2017 at 1:49

1 Answer 1


Provided that the MONEY is deposited, the LENDER will get the payment if the WIRE goes through

To analyze the sentence into its components and translate it into the constituent conditional symbols, it might help to start with a small portion of the sentence. We can read, "provided that" as "if", and it can be helpful to explicitly articulate the "if's" and "then's". So we have the statement:

  • If the MONEY is deposited, then the LENDER will get the payment.

...and this can be thought of as the proposition:

  • If M, then L.

...such that M represents the general idea of "money deposit" and L represents the general idea of "lender payment." We can then neatly symbolize the conditional statement like so:

  • M --> L

...such that the "arrow" indicates a conditional relationship between the two variables, M and L where the former "implies" (or "sets the condition for" etc.) the latter. (1 - see below)

So what to make of the rest of the sentence and how to fit it into symbolic logical notation, "M → L ...if the WIRE goes through"?

The latter half is also a conditional statement, i.e. "if the WIRE goes through, then the LENDER will get the payment". We can analyze this into "if W, then L" and similarly use the symbols "W → L" - so in short, we've now analyzed the sentence into two propositions that are conditionals resulting in the same consequence:

  • If MONEY deposit, then LENDER paid.


  • If WIRE goes through, then LENDER paid.


  • M → L
  • W → L

But how to combine these two conditionals in a way which reflects their relationship as stated above in the ordinary language formulation?

Since both M and W result in L, how about a conjunction?

  • (M & W) → L

...such that:

  • If M and if W, then L

This conveys the logical relationships of "If the MONEY is deposited AND if the WIRE goes through, then the LENDER gets paid."

...but is this the relationship as stated in the original sentence?

Not quite. While it details the separate conditions - money deposited & wire sent - it doesn't really capture the dependency of the wire being sent upon the money getting deposited. (2 - see below)

So, in order to express the chain of conditions, how about something more like so:

  • "If MONEY gets deposited, then if WIRE gets sent, then LENDER gets paid."


  • "If MONEY gets deposited, then (if W, then L)."

...which can be reduced to:

  • If M, then (if W, then L).

...which can be symbolized as:

  • M → (W → L)

And conveys the logical relationships analyzed in the sentence,

Provided that the MONEY is deposited, the LENDER will get the payment if the WIRE goes through

(1) Note, there are many different acceptable symbols for representing a conditional relationship. In this instance the "-->" arrow between two terms is adequate. If, however, you want to use markup language to display a single arrow, try → and you will see: → ("rarr" is for "R"ight "ARR"ow and the "&" ampersand and ";" semi-colon are just for formatting so the markup can be parsed to display the symbols correctly.)

(2) Of note, the ampersand ( "&" ) is generally no longer used for connective notation and it is common to see "the carrot" ^ or in markup language, ∧ displays "and" as ∧ so likely you'd write the above like so: (M ∧ W) → L

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