Intro to Logic: Propositions

by Luke Muehlhauser on June 13, 2009 in Intro to Logic

Welcome to my course Intro to Logic (index). Here, we learn the basic skills of good thinking and their benefits in real life.

Last time, I introduced logic as the science of analyzing an argument to see if it provides good reason to accept its conclusion or not. An argument uses premises to establish a conclusion. For example, here is an argument with two premises and a conclusion:

  1. All men are mortal.
  2. Socrates is a man.
  3. Therefore, Socrates is mortal.

This is a valid argument. If we accept the premises, then we must accept the conclusion.


Arguments are made of propositions, a fancy word for anything that can be asserted or denied.

Here are some example propositions:

  1. God exists.
  2. The largest state in the United States was once an independent republic.
  3. Cars are useful, or they are not useful.
  4. If God did not exist, it would be necessary to invent him.
  5. Irish Spring soap smells manly.

Questions, commands, and exclamations do not assert anything, so they aren’t propositions. None of these are propositions:

  1. Study logic more often, please.
  2. Yay, Cardinals!
  3. What time is it?

But don’t confuse propositions for sentences. For example, each of these sentences express the same proposition:

  • Cars are useful, or they are not useful.
  • Either cars are not useful, or else they are useful.
  • Either cars ain’t useful or they are!

Likewise, the same proposition can look different in different langauges:

  • It is raining.
  • Está lloviendo.
  • 在下雨.

Sometimes, a proposition is compound, meaning it contains several propositions:

The British were at the gates of Hamburg and Bremen and threatening to cut off Germany from Denmark.

This proposition contains three propositions:

  1. The British were at the gates of Hamburg.
  2. The British were at the gates of Bremen.
  3. The British were threatening to cut off Germany from Denmark.

In order for the original proposition above to be true, all three of the propositions it contains must be true. For example, if the British were at the gates of Hamburg and were threatening to cut off Germany from Denmark, but the British were not also at the gates of Bremen, then the original proposition above is false.

Another kind of proposition is the hypothetical proposition. Consider this example:

If God exists, objective moral values exist.

Neither the first part, “God exists,” nor the second part, “objective moral values exist,” is being asserted. Rather, it is only asserted that if God exists, then objective moral values exist. The hypothetical proposition above could be true even if both of its parts are false: it could be the case that if God existed objective moral values would exist, but as it happens God does not exist, and also objective moral values do not exist.

Next time, we’ll learn how to recognize arguments in writing and in speech.

(Also see the post index to this Intro to Logic series.)

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{ 6 comments… read them below or add one }

cartesian June 13, 2009 at 9:21 pm

>>But don’t confuse propositions for sentences.>>
Woh, of course I agree, but I didn’t expect that from you, given what you said during your discussion of Carrier’s book.
So what are propositions, on your view? They’re not sentences, they’re true or false, but what are they? Where do they fit into your austere naturalistic worldview?


lukeprog June 13, 2009 at 10:03 pm


Carrier’s views are not my own.

Also, this is a course on how logic works, not on the metaphysics of propositions. I still don’t know what my metaphysical views are. Some kind of conceptualism seems plausible to me. The problem is that ‘exists’ has so many different meanings, and they are easily confused. I don’t know what it means to say that “7″ exists apart from an idea we use. The same goes for propositions. They certainly don’t have literal physical existences – only particular instantiations of them exist physically (as brain states, as pixels on a screen, etc.).

But were there 7 beetles somewhere before any minds carried the concept of 7? Yes, but it the proposition “There were 7 beetles over there” can only be true today, at a time when the concept “7″ has meaning that is expressed by words and symbols in various human languages. It would not have been true (or possible) to say that “There are 7 beetles over there” before the concept of “7″ existed in minds. There simply was what there was. It is pure human choice to categorize things the way we do so that we can manipulate ideas and navigate the world and divide spacetime up into little chunks such that there are 7 of this and 2 of that and these things that stimulate us with a certain pattern of light we have a (roughly) shared concept of and label “redness” in English.


Dace June 14, 2009 at 1:27 am

“The hypothetical proposition above could be true even if both of its parts are false: it could be the case that objective moral values can’t exist without God, even if it’s true that God exists and also that objective moral values exist.”

I think you meant the reverse here:  ”it could be the case that objective moral values can’t exist without God, even if it’s true that God *does not* exist and that objective moral values *don’t exist* either.”
Nice to have this stuff to hand, by the way.


lukeprog June 14, 2009 at 2:11 am


What I originally wrote was logically correct, but I’ve rewritten it to be clearer and avoid double negatives.


mike June 18, 2009 at 12:40 am

Your Chinese example is a bit weird: it reads as “this is rain” or ”this is raining”. A construction that matches the English and Spanish examples is 在下雨.


lukeprog June 18, 2009 at 7:07 pm

Mike, thanks!


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