Simple Non-Modal Example

Consider these sentences:

To check whether these sentences are consistent. You first need to formalize them. Start by identifying the logical connectives. These often correspond to conjunctions in English ('and', 'or', 'but', 'if-then').

For the first sentence here, there are two statements connected by OR. Pick a lower case letter to represent the first statement. We can use 'a', since 'Athens' is a prominent word in the first statement but not the second. Likewise, we can use 's' for the second statement. We then have "a OR s". To put this in infix notation, change the OR to a 'V'. We then have 'aVs'. This is our representation of the first sentence.

The second sentence looks similar to the second statement of the first sentence, but there is an added 'not'. 'Not' is another logical connective, which modifies one statement instead of two like 'or' did. We have 'NOT Socrates is in Sparta'. Since 'Socrates is in Sparta' was represented earlier as 's', we should use the same exact letter to represent it in this case. So, we have 'NOT s'. In this prefix notation, the 'NOT' becomes an '~'. So, we have '~s'.

Now that we have formalized both sentences, enter them into the input box together, separated by a space. You would type: aVs ~s

When you hit the Check Consistency button, the program will tell you whether the sentences are consistent (i.e. whether they can both be true at the same time). The result in this case is that they are consistent. If the sentences are consistent, the model box shows a situation where all the sentences entered are true. Here, it shows 'a' and '~s'. This model is one where 'a' is true and 's' is false. So, the situation where Socrates is in Athens and Socrates is not in Sparta is a situation where both of the entered statements are true, meaning together they are consistent.

If the entered sentences are inconsistent, there is no situation where the sentences are all true together. So, in that case, there is no model to show.

Modal Example

In modal logic, we want to handle the logical connections between sentences involving phrases like 'possibly', 'could have', 'necessarily', and 'must be'.

There are at least two different ways to interpret a sentence like "John could have gone to the store." One intepretation of the 'could have' is an epistemic one. It tells you what the speaker knows about John's location. A different intepretation of 'could have' is a metaphysical one. The speaker may know that John was at home, not at the store. But if John had decided to go to the store, instead, then there is a sense in which John could have gone to the store.

There are some ways to phrase a sentence to distinguish epistemic from metaphysical readings. Compare:

The logical system used in this website (S5) is more geared toward metaphysical possibility than epistemic possibility. In S5, "John is at home now, but John could have been at the store now" is consistent. In an epistemic logic, knowledge that John is at home will rule out the possibility that he is at the store now.

Now for the example. Consider these sentences:

The first two sentences do not contain any talk of possibility or necessity. So, they can be represented easily, using 'n' for the claim that she was in New York and 'b' for the claim that she was in Boston: '~n' '~b'

The third sentence contains the phrase 'could have'. We can represent the sentence: "POSSIBLY Jill was in New York at noon today". This becomes "POSSIBLY n". 'Possibly' is represented by a capital 'P' in our notation. So, the sentence becomes 'Pn'.

The same process tells us that the fourth sentence is 'Pb'.

The last sentence is more complicated, since it contains multiple connectives ('necessarily', 'not', 'and'). To formalize it, you need to determine the scope of the connectives (i.e. how much of the sentence each connective applies to).

This last sentence can be formalized following these steps:

So, to check these sentences you would enter: ~n ~b Pn Pb N~n&b

The result is that these are consistent. The model showing that these are consistent includes a description of what actually happened and a description of the other possible situations. In this case, the model tells us that Jill was not actually in New York and not actually in Boston, which we already knew. The Other Worlds section gives the other possible situations needed to make all the original sentences true. Here, there are two other situations: one where she is in New York but not Boston, and one where she is in Boston but not New York. In none of the actual or possible situations is she in both New York and Boston at the same time, but she could have been in either place, if she had planned her trip differently.