2.1 Propositions
Critical Thinking Lecture 2: Conditionals and Deduction
Last time we learned that an argument consist of premises and a conclusion, and that premises can be linked or convergent. This week we will look at a very important form of argument in which the premises are linked; namely, deduction. We will begin by investigating conditional claims, which play a crucial role in many deductive arguments. But before we do that, we need some basic information about philosophical terminology and conventions for representing claims and arguments.
Indicative sentences (sentences that say that such and such was/is/will be the case) express propositions. The proposition expressed by an indicative sentence is what that sentence says. Two different sentences can express the same proposition, e.g. “Snow is white” and “Schnee ist weiss”, “Europe is North of Africa” and “Africa is South of Europe”. Propositions can be true or false. (NB There are lots of legitimate uses of language that cannot be true or false, e.g. questions, commands, expressions of attitude, greetings.) Propositions can also be simple/basic or complex. Complex propositions are made out of combinations of simple propositions. In philosophy, it is common to use ‘p’ and ‘q’ to stand for possible propositions that could be slotted into arguments, just as in algebra ‘x’ and ‘y’ stand for possible numbers that could be slotted into equations. When someone makes a claim, they assert an indicative sentence. They claim that p.
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Some propositions ascribe properties to objects or events. e.g. “This lecture is interesting” ascribes the property of being interesting to this lecture. Philosophical convention is to use ‘F’, ‘G’, ‘H’… to stand for properties and ‘a’, ‘b’, ‘c’… to stand for objects or events. Hence, the claim that an object possesses a property can be represented as ‘a is F’ or ‘Fa’. Often, objects that have property F are referred to as Fs.
cf. Those who have the property of being able to surf and doing so regularly are called ‘surfers’.
2.2 Conditional Statements
Many deductive arguments involve what we call “conditional statements” or “conditionals”. These are statements of the form “If p then q”, or any statements which have an equivalent meaning. e.g. “If you are a father then you are a parent”. To assess conditional deductive arguments, we first must understand conditional statements. NB At this stage, we are not asking whether these statements are true or false. We are asking what such statements mean, or, what is said when someone makes such a statement.
Often it is hard to tell the difference between conditional statements and causal statements or causal explanations. Sometimes people express causal explanations by saying things of the form ¡°if p then q¡±.
e.g. Why is Tina drunk? Because if Tina drinks too much champagne,
then Tina gets drunk.
Probably what the speaker means in this situation is that Trev’s drinking to much beer makes Trev drunk, or causes him to be drunk. However, conditional claims do not have the same meaning as causal claims. In the context of this unit of study, conditional statements should not be read as causal explanations. Rather, they are statements about what can be inferred from what. “If p then q” doesn’t mean “p makes q happen”. Instead, it means “If p is true, then q is true”. (Perhaps, more generally, it means “If we can say p then we can say q”.)
This is complicated by the fact that sometimes when someone makes a specific conditional claim that has the form ¡°If p then q¡±, there actually is a causal connection between p and q. Sometimes the fact that p causes q what explains the truth of the conditional claim “If p then q”. In many other cases though, it is true that “If p then q”, but p does not cause q. In order to understand this, we need to consider several examples:
If it rained then the dam levels are higher.
(p = it rained, and q = the dam levels are higher.)
This claim means ¡°If it is true that it rained then it is true that the dam levels are higher.¡± In this specific example, it is true that the event refered to in p happens before the state of affairs referred to in q, and causes that state of affairs. But the claim “If it rained then the dam levels are higher” is not a causal claim. It does not mean “p causes q”. Rather, it is a conditional claim, i.e. If it is the case that it rains, then it is the case that the dam levels are higher. From the fact that it rained we can infer that the dam levels are higher.
If I am a father then I am a parent.
This means that ¡°If it is true that I am a father then it is true that I am a parent.¡± In this case, p does not happen either before or after q, and there is not a causal relationship between p and q. Being a father does not cause you to be a parent. Rather, being a father is one way of being a parent.
If you are bigger than Arnie then you are bigger than me.
In this case, p does not happen either before or after q, and there is not a causal relationship between p and q.
If the dam levels are higher then it rained.
(p = it rained, and q = the dam levels are higher.)
This means ¡°If it is true that the dam levels are higher then it is true that it rained.¡± In this case, the event referred to in p happens after q and is caused by q. According to the statement “If the dam levels are higher then it rained”, from the fact that the dam levels are higher we can infer that it rained. (NB The meaning of this statement is not the same as the meaning of the statement “If it rained then the dam levels are higher.”)
In the context of informal logic, if you are unsure whether a statement of the form ¡°If p then q¡± should be interpreted as either a conditional claim or a causal claim, interpret it as a conditional claim. When we talk about causal claims we will explicitly label them as causal claims.
If you are a police officer then you are allowed to break the law.
The sufficient condition in this claim is “you are a police officer”. According to claim, the fact that you are a police officer is enough to make it true that you are allowed to break the law. Being a police officer is sufficient for being allowed to break the law. NB This conditional claim is FALSE.
If the asteroid hits Earth tomorrow, everyone will die tomorrow.
2.3 Sufficient and Necessary Conditions
A conditional statement contains two parts that slot into the “If … then …” formula. Each of these parts is called a ‘condition’. They are the sufficient condition and the necessary condition. (For people who have studied logic in more depth, in this unit we are treating conditionals as truth-functional material conditionals. This is a simplification, but a useful simplification at this level.)
Sufficient condition. For the standard form of conditional “If p then q”, the sufficient condition is p (i.e. the proposition that comes after the “if” and before the “then”). This conditional statement tells us that the truth of p is sufficient for the truth of q. Hence, the conditional statement can be rewritten as “p is sufficient for q”, i.e. “The truth of p is enough for the truth of q”. (Sometimes the sufficient condition is called the “antecedent condition” or the “antecedent”.)
e.g. If you are a father then you are a parent.
The sufficient condition in this claim is “you are a father”. The fact that you are a father is enough to make it true that you are a parent. Being a father is sufficient for being a parent. This conditional claim is true.
The sufficient condition in this claim is “the asteroid hits Earth tomorrow”. According to this claim, the fact that the asteroid hits the Earth tomorrow is enough to make it true that everyone will die tomorrow. The asteroid hitting the Earth is sufficient for everyone dying tomorrow.
Necessary condition. For the standard form of conditional “If p then q”, the necessary condition is q (i.e. the proposition that comes after the “then”). The conditional statement tells us that the truth of q is necessary for the truth of p. Hence, the conditional statement can be rewritten as “q is necessary for p”, i.e. The truth of q is required by the truth of p”. (Sometimes the necessary condition is called the “consequent”.)
e.g. If you are a father then you are a parent.
The necessary condition in this claim is “you are a parent”. According to this claim, the fact that you are a parent is required by the fact that you are a father. Being a parent is necessary for being a father. This conditional claim is true.
If you are a police officer then you are allowed to break the law.
The necessary condition in this claim is “you are allowed to break the law” (or “you being allowed to break the law”). According to this claim, the fact that you are allowed to break the law is required by your being a police officer. Being allowed to break the law is necessary for being a police officer. This conditional claim is FALSE.
If the asteroid hits Earth tomorrow, everyone will die tomorrow.
The necessary condition in this claim is “everyone will die tomorrow”. According to this claim, the fact that everyone dies tomorrow is required by the fact that the asteroid hits the Earth tomorrow. Everyone dying tomorrow is necessary for the asteroid hitting the Earth tomorrow.
Note that, when it comes to identifying the sufficient and the necessary conditions, there is nothing special about the letters ‘p’ and ‘q’. What matters is the position of the proposition in the conditional claim. Hence, in the conditional claim “If q then p”, the sufficient condition is q and the necessary condition is p.
Note also that variables like ‘p’ and ‘q’ are fixed within each example/question, but can differ from example to example. This is just like ‘x’ and ‘y’ in algebra. The
algebraic equation (e.g. x = 2y +3) sets limits on the possible values for x and y in
that question, and the values of x and y do not carry over into a new question.
2.4 Equivalent Forms of Conditionals
Lots of claims of various forms turn out to be equivalent in meaning to the basic form of the conditional claim “If p then q”. (NB Here we are fixing on the value of p and q in the claim “if p then q”) e.g.
* p is sufficient for q
“If I am a father then I am a parent” means the same as “Being a father is sufficient for being a parent”. NB Not causal.
“If it is more expensive than Paddington then it is more expensive than Marrickville” means the same as “Being more expensive than Paddington is sufficient for being more expensive than Marrickville”.
* q is necessary for p
“If I am a father then I am a parent” means “Being a parent is necessary for being a father”. Think of this as “It must be the case that you are a parent if you are a father”. “If it is more expensive than Paddington then it is more expensive than Marrickville” means the same as “Being more expensive than Marrickville is necessary for being more expensive than Paddington”.
For instance, “If I am a father, then I am a parent” has exactly the same meaning as “I am a parent if I am a father”.
“If it is more expensive than Paddington then it is more expensive than Marrickville” means the same as “It is more expensive than Marrickville if it is more expensive than Paddington”.
* p only if q
“If I am a father then I am a parent” means “I am a father only if I am a parent”. Note that it does not mean “I am a parent only if I am a father”.
“If it is more expensive than Paddington then it is more expensive than Marrickville” means the same as “It is more expensive than Paddington only if it is more expensive than Marrickville”.
* Only if q, then p
“If I am a father then I am a parent” means “Only if I am a parent, then I am a father” “If it is more expensive than Paddington then it is more expensive than Marrickville” means the same as “Only if it is more expensive than Marrickville is it more expensive than Paddington”.
We can see for the past three examples, that the clause after an “if” is the sufficient condition, whereas the clause after an “only if” is the necessary condition.
All we are doing here is trying to spell out systematically the way in which some very basic and very common words and phrases work. Notice how tricky it can be to think clearly about what we ordinarily take for granted.
2.5 “All”, “Every” and “Only” Generalisations
Recall that ‘a’ and ‘b’ stand for objects or events and ‘F’ and ‘G’ stand for properties. Another common form of conditional statement is “If a is F then a is G”. This is equivalent in form to “If p then q”, where p = “a is F” and q = “a is G”. This terminology allows us to translate some conditional claims into equivalent generalisations. This works for conditional claims involving types of thing, e.g. fathers, people, planets, but not for conditional statements involving particular objects rather than types.
* All Fs are Gs (also “Every F is a G”)
“If I am a father then I am a parent” means “All fathers are parents”. Again, it does not mean “All parents are fathers”.
“If it is more expensive than Paddington then it is more expensive than Marrickville” means the same as “All suburbs more expensive than Paddington are more expensive than Marrickville”.
* No Fs are non-Gs
“If I am a father then I am a parent” means “No fathers are non-parents”.
“If it is more expensive than Paddington then it is more expensive than Marrickville” means the same as “No suburbs more expensive than Paddington are not more expensive than Marrickville”.
* Only Gs are Fs
“If I am a father then I am a parent” means “Only parents are fathers”.
“If it is more expensive than Paddington then it is more expensive than Marrickville” means the same as “Only suburbs more expensive than Marrickville are more expensive than Paddington”.
* The only Fs are Gs
“If I am a father then I am a parent” means “The only fathers are parents”.
“If it is more expensive than Paddington then it is more expensive than Marrickville” means the same as “The only suburbs that are more expensive than Paddington are more expensive than Marrickville”.
There are many complications with conditional claims and generalisations that we will briefly mention here, and then ignore. Sometimes when people make claims of the form “All Fs are Gs”, they mean all actual Fs now are Gs, but not that all possible Fs were, are and will be Gs.
e.g. All Members of the Board are bald.
In ordinary language, this would not be taken to imply that all possible Members of the Board are bald, or that all past and future members were and will be bald. Other claims which are superficially of the same form have a much stronger meaning.
cf. All numbers greater than 7 are greater than 2.
All electrons have a negative charge.
All coloured objects are extended (i.e. take up space). All dogs have hair.
All presidents of the USA are born in the USA.
These five claims are true, but there are differing kinds of necessity in each of these cases (stronger to weaker, with the last necessity being merely legal necessity).
When people use the conditional form “If p then q”, and when they speak of necessary and sufficient conditions, they usually mean the stronger form of claim, i.e. not just all actual Fs now are Gs, but all possible Fs are Gs.
1.5 Translations between Conditional Claims
Since all of these statements have the same meaning, we can translate a conditional statement from one form to another. This can be quite tricky, but it is a very useful skill when it comes to assessing conditional deductive arguments.
If someone can smell food cooking then she is near a kitchen.
This is equivalent to:
All people who can smell food cooking are near a kitchen. Only people who are near a kitchen can smell food cooking. Everyone who can smell food cooking is near a kitchen. People can smell food cooking only if they are near a kitchen. No people who can smell food cooking are not near a kitchen. Being near a kitchen is necessary for smelling food cooking. Smelling food cooking is sufficient for being near a kitchen.
If you eat no carbs then you lose weight.
This is equivalent to:
All people who eat no carbs lose weight. Only people who lose weight eat no carbs. Everybody who eats no carbs loses weight. People eat carbs only if they lose weight.
No people who eat no carbs do not lose weight. Losing weight is necessary for eating no carbs. Eating no carbs is sufficient for losing weight.
Some common mistakes that lead to mistranslation of conditional claims:
As we have seen, there is a temptation to read a conditional claim ¡°If p then q¡± as a causal claims, and to think about the actual causal relationship between p and q. This mistake can lead us to mistranslate conditional claims: when q is actually the cause of p, we might automatically assume that q must be the sufficient condition and that p must be the necessary condition.
e.g. I am easy to see only if I am wearing a red jumper.
Many people are tempted to translate this as:
“If I am wearing a red jumper then I am easy to see”. This seems plausible, because wearing a red jumper usually causes one to be easy to see. But it is a mistake to think here about the actual relationship between wearing a red jumper and being easily seen. Instead we must look carefully at what is said in the original claim. The original claim is actually equivalent to the following claims:
If I am easy to see then I am wearing a red jumper. I am wearing a red jumper if I am easy to see. Only if I am wearing a red jumper am I easy to see.
None of these claims mean “If I am wearing a red jumper then I am easy to see”. In the original claim “I am easy to see only if I am wearing a red jumper” the sufficient condition is ¡°I am easy to see¡±, and the necessary condition is ¡°I am wearing a red jumper¡±. According to this claim, me wearing a red jumper is required if I am highly visible, but it is not sufficient for to be highly visible. (This conditional claim is false.)
Remember, conditional claims are not claims about what causes what or what contributes to what. They are claims about what can be inferred from what.
NB “If p then q” does not mean “If q then p”.
When translating conditional claims it is a mistake to think too much about what is true and interesting about the relationship between p and q, rather than what the claimant has actually said about the relationship between p and q. When dealing with conditional claims it is very important to notice carefully what the claimant said, and not confuse this with what you think the claimant ought to have said. Many conditional claims are false, and people have a strong temptation to mistranslate them so that they come out true, or as true and maximally informative.
e.g. Suppose that Trev says “Only kelpies are dogs”. Trev’s claims translates to:
If it is a dog then it is a kelpie. Only if it is a kelpie is it a dog. All dogs are kelpies.
Trev’s claim is false, but there is a strong temptation to mistranslate Trev’s claim so that it comes out true, i.e. “If it is a kelpie then it is a dog”, which is equivalent to “All kelpies are dogs”, “Only dogs are kelpies”. If we are trying to assess what Trev said, we should not mistranslate his claim so that it comes out to be true. Rather, we should say to Trev, “What you actually said was false. Perhaps what you meant to say was that only dogs are kelpies”.
1.6 Counterexamples to Conditional Claims and Generalisations
We can test the truth of a conditional claim by searching for a counterexample to the claim. A counterexample to a conditional statement is an actual or possible object, event or state of affairs which shows that the conditional statement is false. i.e. a thing or event that meets the sufficient condition but not the necessary condition, or a state of affairs in which the sufficient condition is true but the necessary condition is false.
e.g. Only if you are a student are you in this lecture. This is equivalent to
If you are in this lecture, you are a student.
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