Starting out with formal logic

The story so far… So far in this course we have looked at natural language arguments. We have thought about what arguments are, how they are constructed, and how to identify them. We’ve looked at some classic examples of bad reasoning, and we’ve even made some tentative steps towards formalization.

What next? We’ll now be making some less tentative steps towards formalization. From here on out we will be focusing on formal logic – using symbols to construct artificial language that allow us to use specific methods of assessing validity. But throughout these weeks you should remember that this is all one topic. Though things might get a bit abstract, it is all with a view to improving our everyday thinking skills and our capacity to understand ordinary language arguments.

Why formalize? When we analyze arguments we’re really looking at the relationships between the truth-values of the claims that we have. Our gold standard of deductive arguments is validity – and remember, validity is really a question of the relationship between the truth of the premises and the truth of the conclusion. So in a way it doesn’t really matter exactly what the claims say – we are more concerned with how their truth values relate to one another.

Why formalize? One way of summarizing these thoughts is to say that in logic we are really interested in the form of arguments, rather than their content. That’s why we often talk in terms of Ps and Qs (e.g. ‘if P then Q, P, therefore Q’). It doesn’t matter exactly what P and Q stand for, what we want to know is what it means for Q if P is true, and vice versa.

Why formalize? So formalizing allows us to focus on the form, putting the content aside temporarily while we work out the relationships between the claims. Formalizing can also be helpful in allowing us to simplify long arguments.

From standard argument form to formal logic When we looked at standard argument form we worked on simplifying natural language arguments. We can think of formalizing as just going one step further. Let’s return to our old example.

From standard argument form to formal logic It’s just ridiculous to think that real socio-political change is going to come from the ruling elite. After all, they have all the money and power, and history tells us that if people have all the money and power in society, then they won’t give it up freely. As we said in our previous session, this gets rephrased in standard argument form as…

From standard argument form to formal logic (1) The ruling elite have all the money and power in society. (2) If people have all the money and power in society, then they won’t give it up freely. (3) If the ruling elite won’t give up their money and power freely, then real socio-political change is not going to come from them. (4) It’s not the case that real socio-political change is going to come from the ruling elite.

From standard argument form to formal logic We can now start to formalize this argument in formal logic. The first thing to do is to isolate the propositions that we are going to take as atomic. These are the simplest phrases, that don’t involve any terms like ‘and’, ’or’, ‘not’, ‘if… then’ etc. (We will call these connectives – more on this later.)

From standard argument form to formal logic (1) (The ruling elite have all the money and power in society). (2) If (people have all the money and power in society), then (they won’t give it up freely). (3) If (the ruling elite won’t give up their money and power freely), then it is not the case that (real socio-political change is going to come from them). (4) It’s not the case that (real socio-political change is going to come from the ruling elite).

From standard argument form to formal logic As you have probably noticed, some of these propositions are the same (or at least seem to express the same thing in slightly different words). The next step is to take each of the propositions that appear and assign them a letter, that we will use to stand in for them. We’ll call this a transcription guide.

From standard argument form to formal logic P: The ruling elite have all the money and power in society. Q: The ruling elite won’t give up their power and money freely. R: Real socio-political change is going to come from the ruling elite. Some of these are slightly re-phrased – this is all ok, so long as we’re preserving meaning as much as possible.

From standard argument form to formal logic Now all we have to do is substitute in the letters for the phrases in line with the transcription guide. So we get: P If P then Q If Q then it is not the case that R It is not the case that R

From standard argument form to formal logic This is way simpler than the original! And it’s also much easier to see that and why the argument is valid. If it is true that If P then Q, and that if Q then not R – and it is true that P - then it has to be true that not R.

Sentence Letters We call these letters-standing-in-for-propositions sentence letters. It is standard to use capital letters to represent them, and we will follow that convention in this class. They also get called atomic sentences. This is to indicate that they are the smallest things we need to consider when we’re doing sentence logic.

Connectives Using these atomic sentences, we can build up more complex propositions using connectives. Connectives are terms like ‘and’, ‘or’, ‘if… then’, ‘it is not the case that’ that connect independent atomic sentences together into more complex compound sentences. Consider the English sentences: ‘CCNY is in NYC’; ‘NYC is in the USA’; ‘CCNY is in NYC and NYC is in the USA’.

Connectives The first sentence makes a claim (that CCNY is in NYC) and the second sentence makes another claim (that NYC is in the USA). But the third sentence makes two claims (i.e. both of the claims the other two sentences make) and connects them with ‘and’. In logic we replace these connectives with symbols.

Connectives For ‘and’, which often gets called conjunction, we use the symbol ‘&’ or the symbol ‘∧’. For ‘or’, which often gets called disjunction, we use the symbol ‘∨’. For ‘it is not the case that’ or ‘not’, which often gets called negation, we use the symbol ‘~’ or the symbol ‘¬’. For ‘if… then’, which often gets called the conditional, we use the symbol ‘⊃’ or the symbol ‘→’.

Disjunction: inclusive and exclusive A quick side note: the word ‘or’ is ambiguous in English between two different uses. One is called ‘the exclusive or’, and it is used to express ‘A or B, but not both’. The other is called ‘the inclusive or’, and it is used to express ‘A or B, or both’. Compare: ‘lunch special: sandwich with soup or salad’ (exclusive) with ‘if your mother or father has green eyes, you will probably have green eyes too’ (inclusive).

Disjunction: inclusive and exclusive In logic, the symbol ‘∨’ behaves like the inclusive or in English. It is true when either or its disjuncts is true, or when both are true. We will come back to this when we do truth tables in the next session.

Adding connectives We can now go back to our example and formalize further. P becomes P If P then Q P ⊃ Q If Q then it is not the case that R Q ⊃ ~R It is not the case that R ~R

Examples If John is late, George will be angry. And he will be late, so he will be angry.

Examples P: John is late. Q: George will be angry. P ⊃ Q P Q

Examples John is in the park or Jan is in town. And John’s not in the park. So Jan must be in town.

Examples P: John is in the park Q: Jan is in town P ∨ Q ~P Q